eISSN: 2378-315X BBIJ

Biometrics & Biostatistics International Journal
Research Article
Volume 7 Issue 3

Linear inference under alpha–stable errors

Donald R Jensen
Virginia Polytechnic Institute and State University, USA
Received: May 02, 2018 | Published: May 23, 2018

Correspondence: Donald R Jensen, Professor Emeritus, Virginia Polytechnic Institute and State University, Blacksburg VA, 24061, USA, Tel 540 639 0865, Email

Citation: Jensen DR. Linear inference under alpha–stable errors. Biom Biostat Int J. 2018;7(3):205‒210. DOI: 10.15406/bbij.2018.07.00210

Abstract

Linear inference remains pivotal in statistical practice, despite errors often having excessive tails and thus deficient of moments required in conventional usage. Such errors are modeled here via spherical α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabeg 7aHbaa@3A9E@ –stable measures on n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaatuuDJXwAK1 uy0HMmaeHbfv3ySLgzG0uy0HgiuD3BaGqbaKqzGeGae8xhHiLcdaah aaWcbeqcbauaaKqzadGaamOBaaaaaaa@4659@ with stability index α(0,2], MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabeg 7aHjaaygW7cqGHiiIZcaaMb8UaaGikaiaaicdacaaISaGaaGjcVlaa ikdacaaIDbGaaGilaaaa@4542@ arising in turn through multivariate central limit theory devoid of the second moments required for Gaussian limits. This study revisits linear inference under α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabeg 7aHbaa@3A9E@ –stable errors, focusing on aspects to be salvaged from the classical theory even without moments. Critical entities include Ordinary Least Squares (OLS) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaiI cacaWGpbGaaGzaVlaadYeacaaMb8Uaam4uaiaaiMcaaaa@3FF5@ solutions, residuals, and conventional F MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadA eaaaa@39CA@ ratios in inference. Closure properties are seen in that OLS solutions and residual vectors under α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabeg 7aHbaa@3A9E@ –stable errors also have α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabeg 7aHbaa@3A9E@ –stable distributions, whereas F ratios remain exact in level and power as for Gaussian errors. Although correlations are undefined for want of second moments, corresponding scale parameters are seen to gauge degrees of association under α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabeg 7aHbaa@3A9E@ –stable symmetry.

AMS subject classification: 62E15, 62H15, 62J20

Keywords:excessive errors, central limit theory, stable laws, linear inference

Introduction

Models here are {Y=Xβ+ε} MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaiU hacaWHzbGaaGzaVlaai2dacaaMb8UaaCiwaGGabiab=j7aIjab=Tca Riab=v7aLjaai2haaaa@44CD@ with error vector ε n . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaiiaajugibi ab=v7aLjaaygW7cqGHiiIZcaaMb8+efv3ySLgznfgDOjdaryqr1ngB PrginfgDObcv39gaiuaacqGFDeIukmaaCaaaleqajeaqbaqcLbmaca WGUbaaaKqzGeGaaGOlaaaa@4DE3@ Classical linear inference rests heavily on means, variances, correlations, skewness and kurtosis parameters, these requiring moments to fourth order. To the contrary, distributions having excessive tails, and devoid of moments even of first or second order, arise in a variety of circumstances. These encompass acoustics, image processing, radar tracking, biometrics, portfolio analysis and risk management in finance, and other venues in contemporary practice. Supporting references include1–6 monographs of note are,7–9 together with the recent work of Nolan.10 In these settings the classical foundations necessarily must be reworked.

To place this study in perspective, alternatives to Gaussian laws long have been sought in theory and practice, culminating in the class { S n (0,Σ)} MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaiU hacaqGtbGcdaWgaaqcbauaaKqzadGaamOBaaqcbauabaqcLbsacaaI OaGaaCimaiaaiYcaiiaacqWFJoWucaaIPaGaaGyFaaaa@43AC@ consisting of elliptically contoured distributions in n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaatuuDJXwAK1 uy0HMmaeHbfv3ySLgzG0uy0HgiuD3BaGqbaKqzGeGae8xhHiLcdaah aaWcbeqcbauaaKqzadGaamOBaaaaaaa@4659@ centered at 0 with scale parameters Σ. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaiiaajugibi ab=n6atjab=5caUaaa@3B68@ These typically are taken to be rich in moments, and to provide alternatives to the use of large-sample approximate Gaussian distributions under conditions for central limit theory. Comprehensive treatises on the theory and applications of these models are.11–13

In contrast, errors having excessive tails are modeled on occasion via spherically symmetric α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabeg 7aHbaa@3A9E@ stable (SαS) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaiI cacaqGtbGaeqySdeMaae4uaiaaiMcaaaa@3DAF@ distributions in n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaatuuDJXwAK1 uy0HMmaeHbfv3ySLgzG0uy0HgiuD3BaGqbaKqzGeGae8xhHivcfa4a aWbaaKqaafqabaqcLbmacaWGUbaaaaaa@46D2@ with index α(0,2]. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabeg 7aHjaaygW7cqGHiiIZcaaMb8UaaGikaiaaicdacaaISaGaaGjcVlaa ikdacaaMi8UaaGyxaiaai6caaaa@46D5@ These comprise the limit distributions of standardized vector sums, specifically, Gaussian limits at  Cauchy limits at α=2, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabeg 7aHjaaygW7caaI9aGaaGzaVlaaikdacaaISaaaaa@3FEB@ and corresponding stable limits otherwise. These distributions are contained in the class { S n (0, I n )}, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaiU hacaqGtbGcdaWgaaqcbauaaKqzadGaamOBaaWcbeaajugibiaaiIca caWHWaGaaGilaiaahMeajuaGdaWgaaqcbauaaKqzadGaamOBaaqcba uabaqcLbsacaaIPaGaaGyFaiaaiYcaaaa@4760@ thus sharing its essential geometric features, but instead are deficient in moments usually ascribed to { S n (0, I n )}. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaiU hacaqGtbGcdaWgaaqcbauaaKqzadGaamOBaaWcbeaajugibiaaiIca caWHWaGaaGilaiaahMeakmaaBaaajeaqbaqcLbmacaWGUbaaleqaaK qzGeGaaGykaiaai2hacaaIUaaaaa@469F@ Despite the venues cited, α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabeg 7aHbaa@3A9E@ stable errors have seen limited usage for want of closed expressions for stable density functions, known only in selected cases but topics of continuing research. Nonetheless, findings reported here rest on well defined characteristic functions ( chfs MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaabo gacaqGObGaaeOzaiaadohaaaa@3CB1@ ), on critical representations for these, and on the inversion of the latter in order to represent the α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabeg 7aHbaa@3A9E@ -stable densities themselves. Even here a divide emerges between independent, identically distributed α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabeg 7aHbaa@3A9E@ -stable sequences, and dependent SαS MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaabo facqaHXoqycaqGtbaaaa@3C4A@ variables, as reported in Jensen14 and as summarized here for completeness in an Appendix. In addition, many findings of the present study are genuinely nonparametric, in applying for all or portions of distributions in the range α(0,2], MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabeg 7aHjaaygW7cqGHiiIZcaaMb8UaaGikaiaaicdacaaISaGaaGjcVlaa ikdacaaIDbGaaGilaaaa@4542@ and thus remaining distribution-free within that class. An outline follows.

Notation and technical foundations are provided in the next major section, Preliminaries, to include Notation and accounts of Special Distributions, Central Limit Theory and Essentials of SαS MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaabo facqaHXoqycaqGtbaaaa@3C4A@ Distributions as subsections. The principal sections following these address Linear Models under SαS MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaabo facqaHXoqycaqGtbaaaa@3C4A@ Errors, with a separate subsection on Models Having Cauchy Errors, and Conclusions. Collateral topics are contained for completeness in Appendix A.

Preliminaries

Notation

Spaces of note include n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaatuuDJXwAK1 uy0HMmaeHbfv3ySLgzG0uy0HgiuD3BaGqbaKqzGeGae8xhHiLcdaah aaWcbeqcbauaaKqzadGaamOBaaaaaaa@4659@ as Euclidean n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaad6 gaaaa@39F2@ space, with S n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaatuuDJXwAK1 uy0HMmaeHbfv3ySLgzG0uy0HgiuD3BaGqbaKqzGeGae8NKWpLcdaWg aaqcbauaaKqzadGaamOBaaqcbauabaaaaa@47DC@ as the real symmetric (n×n) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaiI cacaWGUbGaaGzaVlabgEna0kaaygW7caWGUbGaaGykaaaa@4175@ matrices and S n + MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaatuuDJXwAK1 uy0HMmaeHbfv3ySLgzG0uy0HgiuD3BaGqbaKqzGeGae8NKWpvcfa4a a0baaKqaafaajugWaiaad6gaaKqaafaajugWaiabgUcaRaaaaaa@4A71@ as their positive definite varieties. Vectors and matrices are set in bold type; the transpose, inverse, trace, and determinant of are A', MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaahg eacaaINaGaaGilaaaa@3B30@ A 1 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaahg eajuaGdaahaaqcbauabeaajugWaiabgkHiTiaaigdaaaqcLbsacaaI Saaaaa@3EDE@ tr(A), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaads hacaWGYbGaaGikaiaahgeacaaIPaGaaGilaaaa@3DD4@ and |A|; MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaiY hacaWHbbGaaGiFaiaaiUdaaaa@3C9A@ the unit vector in n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaatuuDJXwAK1 uy0HMmaeHbfv3ySLgzG0uy0HgiuD3BaGqbaKqzGeGae8xhHiLcdaah aaWcbeqcbauaaKqzadGaamOBaaaaaaa@4659@ is ; and I n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaahM eakmaaBaaajeaqbaqcLbmacaWGUbaaleqaaaaa@3C72@ is the (n×n) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaiI cacaWGUbGaaGzaVlabgEna0kaaygW7caWGUbGaaGykaaaa@4175@ identity.

Moreover, Diag( A 1 ,, A k ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaads eacaWGPbGaamyyaiaadEgacaaIOaGaaCyqaOWaaSbaaKqaafaajugW aiaaigdaaSqabaqcLbsacaaISaGaeSOjGSKaaGilaiaahgeajuaGda WgaaqcbauaaKqzadGaam4AaaqcbauabaqcLbsacaaIPaaaaa@48F7@ is a block-diagonal array, and Σ 1 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaiiaajugibi ab=n6atTWaaWbaaKqaafqabaWcdaWcaaqcbauaaKqzGcGaaGymaaqc bauaaKqzGcGaaGOmaaaaaaaaaa@3F40@ is the spectral square root of

Special distributions

Given Y=[ Y 1 ,, Y n ] n , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaahM facaaMb8UaaGypaiaaygW7caaIBbGaaGjcVlaadMfajuaGdaWgaaqc bauaaKqzadGaaGymaaqcbauabaqcLbsacaaISaGaeSOjGSKaaGilai aadMfakmaaBaaajeaqbaqcLbmacaWGUbaaleqaaKqzGeGaaGjcVlqa i2fagaqbaiabgIGiolaaygW7tuuDJXwAK1uy0HMmaeHbfv3ySLgzG0 uy0HgiuD3BaGqbaiab=1risPWaaWbaaSqabKqaafaajugWaiaad6ga aaqcLbsacaaISaaaaa@5F98@ its distribution, expected value, and dispersion matrix are designated as L(Y), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaatuuDJXwAK1 uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqbaKqzGeGae8NeHWKaaGik aiaahMfacaaIPaGaaGilaaaa@468A@ E(Y)=μ, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadw eacaaIOaGaaCywaiaaiMcacaaMb8UaaGypaiaaygW7cqaH8oqBcaaI Saaaaa@4257@ and V(Y)=, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadA facaaIOaGaaCywaiaaiMcacaaMb8UaaGypaiaaygW7cqGHris5caaI Saaaaa@4256@ with variance Var(Y)= σ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadA facaWGHbGaamOCaiaaiIcacaWGzbGaaGykaiaaygW7caaI9aGaaGza Vlabeo8aZPWaaWbaaSqabKqaafaajugWaiaaikdaaaaaaa@4603@ on 1 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaatuuDJXwAK1 uy0HMmaeHbfv3ySLgzG0uy0HgiuD3BaGqbaKqzGeGae8xhHivcfa4a aWbaaKqaafqabaqcLbmacaaIXaaaaKqzGeGaaGOlaaaa@47E1@ Specifically, L(Y)= N n (m,) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaatuuDJXwAK1 uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqbaKqzGeGae8NeHWKaaGik aiaahMfacaaIPaGaaGzaVlaai2dacaaMb8UaaeOtaOWaaSbaaKqaaf aajugWaiaad6gaaSqabaqcLbsacaaIOaGaaOyBaiaaiYcacaaMi8Ua eyyeIuUaaGykaaaa@53F9@ is Gaussian on n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaatuuDJXwAK1 uy0HMmaeHbfv3ySLgzG0uy0HgiuD3BaGqbaKqzGeGae8xhHivcfa4a aWbaaKqaafqabaqcLbmacaWGUbaaaaaa@46D2@ with parameters (μ,). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaiI cacqaH8oqBcaaISaGaeyyeIuUaaGykaiaai6caaaa@3F2C@ Distributions on 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaatuuDJXwAK1 uy0HMmaeHbfv3ySLgzG0uy0HgiuD3BaGqbaKqzGeGae8xhHivcfa4a aWbaaKqaafqabaqcLbmacaaIXaaaaaaa@469A@ of note include the χ 2 (u;ν,λ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabeE 8aJLqbaoaaCaaajyaibeqaaKqzadGaaGOmaaaajugibiaaiIcacaWG 1bGaaG4oaiabe27aUjaaiYcacqaH7oaBcaaIPaaaaa@4554@ and related χ(u;ν,λ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabeE 8aJjaaiIcacaWG1bGaaG4oaiabe27aUjaaiYcacqaH7oaBcaaIPaaa aa@41FC@ distributions, together with the Snedecor -Fisher F(u; ν 1 , ν 2 ,λ), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadA eacaaIOaGaamyDaiaaiUdacqaH9oGBkmaaBaaajeaqbaqcLbkacaaI XaaaleqaaKqzGeGaaGilaiabe27aUPWaaSbaaKqaafaajugOaiaaik daaSqabaqcLbsacaaISaGaeq4UdWMaaGykaiaaiYcaaaa@49E5@ these having (ν, ν 1 , ν 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaiI cacqaH9oGBcaaISaGaeqyVd4wcfa4aaSbaaKqaafaajugWaiaaigda aSqabaqcLbsacaaISaGaeqyVd4wcfa4aaSbaaKqaafaajugWaiaaik daaSqabaqcLbsacaaIPaaaaa@47F1@ as degrees of freedom and λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabeU 7aSbaa@3AB3@ a noncentrality parameter. The characteristic function (chf) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaiI cacaqGJbGaaeiAaiaabAgacaaIPaaaaa@3D1E@ for Y n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaahM facaaMb8UaeyicI4SaaGzaVprr1ngBPrwtHrhAYaqeguuDJXwAKbst HrhAGq1DVbacfaGae8xhHiLcdaahaaWcbeqcbauaaKqzadGaamOBaa aaaaa@4BD3@ is the expectation ϕ Y (t)=E[ e ιt'Y ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabew 9aMLqbaoaaBaaajeaqbaqcLbmacaWHzbaajeaqbeaajugibiaaygW7 caaIOaGaaCiDaiaaiMcacaaMb8UaaGypaiaaygW7caqGfbGaaG4wai aadwgakmaaCaaaleqajeaqbaqcLbmacaaMi8UaeqyUdKMaaCiDaiaa iEcacaWHzbaaaKqzGeGaaGyxaaaa@51FC@ with argument t'=[ t 1 ,, t n ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaahs hacaaINaGaaGzaVlaai2dacaaMb8UaaG4waiaayIW7caWG0bqcfa4a aSbaaKqaafaajugWaiaaigdaaKqaafqaaKqzGeGaaGilaiablAcilj aaiYcacaWG0bGcdaWgaaqcbauaaKqzadGaamOBaaWcbeaajugibiaa yIW7caaIDbaaaa@4EE1@ and ι= 1 ; MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabeM 7aPjaaygW7caaI9aGaaGzaVRWaaOaaaeaajugibiabgkHiTiaaigda aSqabaqcLbsacaaI7aaaaa@4239@ a standard source is Lukacs & Laha.15 Attention is drawn subsequently to probability density (pdf) and cumulative distribution (cdf) functions. Moreover, the class {L(Z) S n (0,)} MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaiU hatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqbaiab=jri mjaaiIcacaWHAbGaaGykaiaaygW7cqGHiiIZcaaMb8Uaae4uaOWaaS baaKqaafaajugWaiaad6gaaSqabaqcLbsacaaIOaGaaCimaiaaiYca cqGHris5caaIPaGaaGyFaaaa@54F7@ consists of elliptically contoured distributions in n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaatuuDJXwAK1 uy0HMmaeHbfv3ySLgzG0uy0HgiuD3BaGqbaKqzGeGae8xhHiLcdaah aaWcbeqcbauaaKqzadGaamOBaaaaaaa@4659@ centered at 0 and having chf’s of type ϕ Z (t)=ψ(t'St). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabew 9aMPWaaSbaaKqaafaajugWaiaahQfaaSqabaqcLbsacaaMb8UaaGik aiaahshacaaIPaGaaGzaVlaai2dacaaMb8UaeqiYdKNaaGikaiaahs hacaaINaGaaK4uaiaahshacaaIPaGaaGOlaaaa@4D22@ We adopt the following.

Definition 1 A distribution P on n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaatuuDJXwAK1 uy0HMmaeHbfv3ySLgzG0uy0HgiuD3BaGqbaKqzGeGae8xhHiLcdaah aaWcbeqcbauaaKqzadGaamOBaaaaaaa@4659@ is said to be monotone unimodal about 0 n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaahc dacaaMb8UaeyicI4SaaGzaVprr1ngBPrwtHrhAYaqeguuDJXwAKbst HrhAGq1DVbacfaGae8xhHiLcdaahaaWcbeqcbauaaKqzadGaamOBaa aaaaa@4BAA@ if for every y n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaahM hacaaMb8UaeyicI4SaaGzaVprr1ngBPrwtHrhAYaqeguuDJXwAKbst HrhAGq1DVbacfaGae8xhHiLcdaahaaWcbeqcbauaaKqzadGaamOBaa aaaaa@4BF3@ and every convex set C symmetric about 0 n , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaahc dacaaMb8UaeyicI4SaaGzaVprr1ngBPrwtHrhAYaqeguuDJXwAKbst HrhAGq1DVbacfaGae8xhHiLcdaahaaWcbeqcbauaaKqzadGaamOBaa aajugibiaaiYcaaaa@4CEF@ P[C+ky] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadc facaaIBbGaam4qaiaaygW7cqGHRaWkcaaMb8Uaam4AaiaahMhacaaI Dbaaaa@4250@ is no increasing in k[0,). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadU gacaaMb8UaeyicI4SaaGzaVlaaiUfacaaMi8UaaGimaiaaiYcacaaM i8UaeyOhIuQaaGykaiaai6caaaa@46DA@ See reference.16

Central limit theory

For iid MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaabM gacaaMb8UaaeyAaiaaygW7caqGKbaaaa@3ED2@ vectors { Z 1 , Z 2 , Z 3 ,} MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaiU hacaWHAbGcdaWgaaqcbauaaKqzadGaaGymaaWcbeaajugibiaaiYca caWHAbGcdaWgaaqcbauaaKqzadGaaGOmaaWcbeaajugibiaaiYcaca WHAbGcdaWgaaqcbauaaKqzadGaaG4maaWcbeaajugibiaaiYcacqWI MaYscaaI9baaaa@49E3@ in n , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaatuuDJXwAK1 uy0HMmaeHbfv3ySLgzG0uy0HgiuD3BaGqbaKqzGeGae8xhHiLcdaah aaWcbeqcbauaaKqzadGaamOBaaaajugibiaaiYcaaaa@479E@ let Z ¯ N = N 1 [ Z 1 ++ Z N ], MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaqdaaqaaK qzGeGaaCOwaaaakmaaBaaaleaajugibiaaygW7caWGobaaleqaaKqz GeGaaGzaVlaai2dacaaMb8UaamOtaOWaaWbaaSqabKqaafaajugWai abgkHiTiaaigdaaaqcLbsacaaIBbGaaCOwaOWaaSbaaKqaafaajugW aiaaigdaaSqabaqcLbsacqGHRaWkcqWIMaYscqGHRaWkcaWHAbGcda WgaaqcbauaaKqzadGaamOtaaWcbeaajugibiaai2facaaISaaaaa@5379@ and consider limit distributions of type { L (c Z ¯ N )=liminfL(c Z ¯ N )} MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaiU hatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqbaiab=jri mPWaaSbaaKqaafaajugWaiabg6HiLcWcbeaajugibiaaiIcacaWGJb GcdaqdaaqaaKqzGeGaaCOwaaaakmaaBaaaleaajugibiaaygW7caWG obaaleqaaKqzGeGaaGykaiaaygW7caaI9aGaaGzaVlGacYgacaGGPb GaaiyBaiaacMgacaGGUbGaaiOzaiab=jrimjaaiIcacaWGJbGcdaqd aaqaaKqzGeGaaCOwaaaakmaaBaaaleaajugibiaaygW7caWGobaale qaaKqzGeGaaGykaiaai2haaaa@62EB@ for suitably chosen C. On specializing from the elliptical class S n (d,) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaabo fakmaaBaaajeaqbaqcLbmacaWGUbaaleqaaKqzGeGaaGikaiaaksga caaISaGaeyyeIuUaaGykaaaa@41B4@ having location-scale parameters (d,S), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaiI cacaGIKbGaaGilaiaajofacaaIPaGaaGilaaaa@3D9E@ we consider α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabeg 7aHbaa@3A9E@ stable limit distributions as follow on identifying L (c Z ¯ N ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaatuuDJXwAK1 uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqbaKqzGeGae8NeHWKcdaWg aaqcbauaaKqzadGaeyOhIukaleqaaKqzGeGaaGikaiaadogakmaana aabaqcLbsacaWHAbaaaOWaaSbaaSqaaKqzGeGaaGzaVlaad6eaaSqa baqcLbsacaaIPaaaaa@4ED1@ with

Definition 2 Let L(Z) S n α (d,) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaatuuDJXwAK1 uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqbaKqzGeGae8NeHWKaaGik aiaahQfacaaIPaGaaGzaVlabgIGiolaaygW7caqGtbqcfa4aa0baaK qaafaajugWaiaad6gaaKqaafaajugWaiabeg7aHbaajugibiaaiIca caGIKbGaaGilaiabggHiLlaaiMcaaaa@56B3@ designate an elliptical α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabeg 7aHbaa@3A9E@ -Stable law on n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaatuuDJXwAK1 uy0HMmaeHbfv3ySLgzG0uy0HgiuD3BaGqbaKqzGeGae8xhHiLcdaah aaWcbeqcbauaaKqzadGaamOBaaaaaaa@4659@ centered at d n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaks gacaaMb8UaeyicI4SaaGzaVprr1ngBPrwtHrhAYaqeguuDJXwAKbst HrhAGq1DVbacfaGae8xhHiLcdaahaaWcbeqcbauaaKqzGdGaamOBaa aaaaa@4C01@ with scale parameters MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabgg HiLdaa@3AA3@ and stable index α(0,2], MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabeg 7aHjaaygW7cqGHiiIZcaaMb8UaaGikaiaaicdacaaISaGaaGjcVlaa ikdacaaMi8UaaGyxaiaaiYcaaaa@46D3@ having the chf MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaabo gacaqGObGaaeOzaaaa@3BB9@ ϕ Z (t)=exp{ιt'd 1 2 (t'St) α 2 }. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabew 9aMPWaaSbaaKqaafaajugWaiaahQfaaSqabaqcLbsacaaMb8UaaGik aiaahshacaaIPaGaaGzaVlaai2dacaaMb8UaciyzaiaacIhacaGGWb GaaG4EaiabeM7aPjaayIW7caWH0bGaaG4jaiaaksgacaaMb8UaeyOe I0IaaGzaVNqbaoaalaaajaaibaqcLbmacaaIXaaajaaibaqcLbmaca aIYaaaaKqzGeGaaGikaiaahshacaaINaGaaK4uaiaahshacaaIPaGc daahaaWcbeqcbauaaKqbaoaalaaajeaqbaqcLbmacqaHXoqyaKqaaf aajugWaiaaikdaaaaaaKqzGeGaaGyFaiaai6caaaa@6665@ Each marginal distribution of S n α (δ 1 n , I n ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaabo fajuaGdaqhaaqcbauaaKqzadGaamOBaaqcbauaaKqzadGaeqySdega aKqzGeGaaGikaiabes7aKjaahgdakmaaBaaajeaqbaqcLbmacaaMb8 UaamOBaaWcbeaajugibiaaiYcacaWHjbGcdaWgaaqcbauaaKqzadGa amOBaaWcbeaajugibiaaiMcaaaa@4DCC@ on 1 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaatuuDJXwAK1 uy0HMmaeHbfv3ySLgzG0uy0HgiuD3BaGqbaKqzGeGae8xhHivcfa4a aWbaaKqaafqabaqcLbmacaaIXaaaaKqzGeGaaGilaaaa@47DF@ namely S 1 α (δ,1), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaabo fajuaGdaqhaaqcbauaaKqzadGaaGymaaqcbauaaKqzadGaeqySdega aKqzGeGaaGikaiabes7aKjaaiYcacaaIXaGaaGykaiaaiYcaaaa@458F@ has the chf ϕ Z i (t)=exp{ιtδ 1 2 |t | α }. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabew 9aMPWaaSbaaKqaafaajugWaiaadQfajuaGdaWgaaqcbauaaKqzadGa amyAaaqcbauabaaaleqaaKqzGeGaaGzaVlaaiIcacaWG0bGaaGykai aaygW7caaI9aGaaGzaVlGacwgacaGG4bGaaiiCaiaaiUhacqaH5oqA caaMi8UaamiDaiaayIW7cqaH0oazcqGHsisljuaGdaWcaaqcaasaaK qzadGaaGymaaqcaasaaKqzadGaaGOmaaaajugibiaaiYhacaaMi8Ua amiDaiaaiYhakmaaCaaaleqajeaqbaqcLbmacqaHXoqyaaqcLbsaca aI9bGaaGOlaaaa@64C5@ Let SαS={ S n α (d,S);(d,)( n n )} MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaabo facqaHXoqycaqGtbGaaGzaVlaai2dacaaMb8UaaG4EaiaabofajuaG daqhaaqcbauaaKqzadGaamOBaaqcbauaaKqzadGaeqySdegaaKqzGe GaaGikaiaaksgacaaISaGaaK4uaiaaiMcacaaI7aGaaGjcVlaaiIca caGIKbGaaGilaiabggHiLlaaiMcacaaMb8UaeyicI4SaaGzaVlaaiI catuuDJXwAK1uy0HMmaeHbfv3ySLgzG0uy0HgiuD3BaGqbaiab=1ri sPWaaWbaaSqabKqaGfaajugWaiaad6gaaaqcLbsacaaMb8Uaey4LIq SaaGzaVlabggHiLRWaaSbaaKqaafaajugWaiaad6gaaSqabaqcLbsa caaIPaGaaGyFaaaa@72AD@ designate the class of all such distributions.

Remark 1 L(Z) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaatuuDJXwAK1 uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqbaKqzGeGae8NeHWKaaGik aiaahQfacaaIPaaaaa@45D5@ is of full rank and has a density in n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaatuuDJXwAK1 uy0HMmaeHbfv3ySLgzG0uy0HgiuD3BaGqbaKqzGeGae8xhHiLcdaah aaWcbeqcbauaaKqzadGaamOBaaaaaaa@4659@ if and only if MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabgg HiLdaa@3AA3@ is of full rank in S n + ; MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaatuuDJXwAK1 uy0HMmaeHbfv3ySLgzG0uy0HgiuD3BaGqbaKqzGeGae8NKWpvcfa4a a0baaKqaafaajugWaiaad6gaaKqaafaajugWaiabgUcaRaaajugibi aaiUdaaaa@4BC5@ otherwise L(Z) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaatuuDJXwAK1 uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqbaKqzGeGae8NeHWKaaGik aiaahQfacaaIPaaaaa@45D5@ is concentrated in a subspace of n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaatuuDJXwAK1 uy0HMmaeHbfv3ySLgzG0uy0HgiuD3BaGqbaKqzGeGae8xhHivcfa4a aWbaaKqaafqabaqcLbmacaWGUbaaaaaa@46D2@ of dimension equal to the rank of

To continue, designate by D α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaatuuDJXwAKz KCHTgD1jharyqr1ngBPrgigjxyRrxDYbacfaqcLbsacqWFdaprkmaa BaaajeaqbaqcLbmacaaMb8UaeqySdegaleqaaaaa@4A53@ the domain of attraction of each element Z i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaahQ fakmaaBaaajeaqbaqcLbmacaWGPbaaleqaaaaa@3C7E@ in { Z 1 , Z 2 , Z 3 ,} MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaiU hacaWHAbGcdaWgaaqcbauaaKqzadGaaGymaaWcbeaajugibiaaiYca caWHAbGcdaWgaaqcbauaaKqzadGaaGOmaaWcbeaajugibiaaiYcaca WHAbGcdaWgaaqcbauaaKqzadGaaG4maaWcbeaajugibiaaiYcacqWI MaYscaaI9baaaa@49E3@ in n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaatuuDJXwAK1 uy0HMmaeHbfv3ySLgzG0uy0HgiuD3BaGqbaKqzGeGae8xhHiLcdaah aaWcbeqcbauaaKqzGcGaamOBaaaaaaa@4639@ having liminfL(c Z ¯ N ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiGacY gacaGGPbGaaiyBaiaacMgacaGGUbGaaiOzamrr1ngBPrwtHrhAXaqe guuDJXwAKbstHrhAG8KBLbacfaGae8NeHWKaaGikaiaadogakmaana aabaqcLbsacaWHAbaaaOWaaSbaaSqaaKqzGeGaaGzaVlaad6eaaSqa baqcLbsacaaIPaaaaa@50BC@ in SαS. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaabo facqaHXoqycaqGtbGaaGOlaaaa@3D02@ That is, their chfs satisfy {liminf ϕ c Z ¯ N (t)=exp[ιt'd 1 2 (t'St) α 2 ]} MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaiU haciGGSbGaaiyAaiaac2gacaGGPbGaaiOBaiaacAgacqaHvpGzkmaa BaaajeaqbaqcLbkacaWGJbWcdaqdaaqcbauaaKqzGcGaaCOwaaaalm aaBaaajeaqbaqcLbkacaaMb8UaamOtaaqcbauabaaaleqaaKqzGeGa aGzaVlaaiIcacaWH0bGaaGykaiaaygW7caaI9aGaaGzaVlGacwgaca GG4bGaaiiCaiaaiUfacqaH5oqAcaaMi8UaaCiDaiaaiEcacaGIKbGa aGzaVlabgkHiTiaaygW7lmaalaaajqgaa+FaaKqzGcGaaGymaaqcKb aG=haajugOaiaaikdaaaqcLbsacaaIOaGaaCiDaiaaiEcacaqItbGa aCiDaiaaiMcakmaaCaaaleqajeaqbaWcdaWcaaqcbauaaKqzGcGaeq ySdegajeaqbaqcLbkacaaIYaaaaaaajugibiaai2facaaI9baaaa@7579@ when scaled suitably. Specifically, the distributions D 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaatuuDJXwAKz KCHTgD1jharyqr1ngBPrgigjxyRrxDYbacfaqcLbsacqWFdaprkmaa BaaajeaqbaqcLbmacaaIYaaaleqaaaaa@47E6@ attracted to Gaussian limits comprise all distributions L( Z i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaatuuDJXwAK1 uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqbaKqzGeGae8NeHWKaaGik aiaahQfakmaaBaaajeaqbaqcLbmacaWGPbaaleqaaKqzGeGaaGykaa aa@4900@ in n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaatuuDJXwAK1 uy0HMmaeHbfv3ySLgzG0uy0HgiuD3BaGqbaKqzGeGae8xhHiLcdaah aaWcbeqcbauaaKqzadGaamOBaaaaaaa@4659@ having second moments. More generally, domains of attraction to distributions in SαS MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaabo facqaHXoqycaqGtbaaaa@3C4A@ have been studied in references,17–20 to include Lindeberg conditions in Barbosa & Dorea,21 together with rates of convergence to stable limits in Paulauskas.22

Remark 2 That Φ Z (t)=exp[ιt'd 1 2 (t'St) α 2 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9rOYxHaVhbbf9v8qqaqFr0xc9pk 0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr 0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGGaaKqzGe Gae8NPdyucga4aaSbaaKqaafaajugWaiaadQfaaSqabaqcLbsacaGG OaGaaiiDaiaacMcacqGH9aqpcaGGLbGaaiiEaiaacchacaGGBbGaeq yUdKMaaGjcVlaahshacaaINaGaaOizaiaaygW7cqGHsisljuaGdaWc aaqcgasaaKqzadGaaGymaaqcgasaaKqzadGaaGOmaaaajugibiaaiI cacaWH0bGaaG4jaiaajofacaWH0bGaaGykaKGbaoaaCaaabeqcgasa aKqbaoaalaaajyaibaqcLbmacqaHXoqyaKGbGeaajugWaiaaikdaaa aaaKqzGeGaaiyxaaaa@6047@ has elliptical contours derives from the spherical chf ϕ U (t)=exp[ιt'q 1 2 (t't) α 2 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabew 9aMPWaaSbaaKqaafaajugWaiaahwfaaSqabaqcLbsacaaMb8UaaGik aiaahshacaaIPaGaaGzaVlaai2dacaaMb8UaciyzaiaacIhacaGGWb GaaG4waiabeM7aPjaayIW7caWH0bGaaG4jaiaakghacqGHsisljuaG daWcaaqcaasaaKqzadGaaGymaaqcaasaaKqzadGaaGOmaaaajugibi aaiIcacaWH0bGaaG4jaiaahshacaaIPaqcfa4aaWbaaKqaafqabaqc fa4aaSaaaKqaafaajugWaiabeg7aHbqcbauaaKqzadGaaGOmaaaaaa qcLbsacaaIDbaaaa@61FC@ through the transformation a

Essentials for SαS MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaho facaaMb8UaeqySdeMaaGzaVlaahofaaaa@3F6A@ distributions

As noted, closed expressions for SαS MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaabo facqaHXoqycaqGtbaaaa@3C4A@ densities are known in selected cases only, to be complemented by results to follow. Here g n (u;d,) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadE gajuaGdaWgaaqcbauaaKqzadGaamOBaaqcbauabaqcLbsacaaIOaGa aCyDaiaaiUdacaGIKbGaaGilaiabggHiLlaaiMcaaaa@4450@ is the Gaussian density on n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaatuuDJXwAK1 uy0HMmaeHbfv3ySLgzG0uy0HgiuD3BaGqbaKqzGeGae8xhHivcfa4a aWbaaKqaafqabaqcLbmacaWGUbaaaaaa@46D2@ having parameters (δ,), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaiI cacqaH0oazcaaISaGaeyyeIuUaaGykaiaaiYcaaaa@3F19@ and f n α (μ;δ,) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajaaOcaWGMb qcfa4aa0baaKqaafaajugWaiaad6gaaKqaafaajugWaiabeg7aHbaa jaaOcaaIOaGaeqiVd0MaaG4oaiabes7aKjaaiYcacqGHris5caaIPa aaaa@48BE@ is the provisional SαS MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaabo facqaHXoqycaqGtbaaaa@3C4A@ density corresponding to ϕ Z (t)=exp[ιt'd 1 2 (t'St) α 2 ]. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabew 9aMPWaaSbaaKqaafaajugWaiaahQfaaKqaGgqaaKqzGeGaaGzaVlaa iIcacaWH0bGaaGykaiaaygW7caaI9aGaaGzaVlGacwgacaGG4bGaai iCaiaaiUfacqaH5oqAcaaMi8UaaCiDaiaaiEcacaGIKbGaeyOeI0sc fa4aaSaaaKaaafaajugWaiaaigdaaKaaafaajugWaiaaikdaaaqcLb sacaaIOaGaaCiDaiaaiEcacaqItbGaaCiDaiaaiMcakmaaCaaajeaO beqcbauaaKqbaoaalaaajeaqbaqcLbmacqaHXoqyaKqaafaajugWai aaikdaaaaaaKqzGeGaaGyxaiaai6caaaa@648F@ The following properties are essential.

Theorem 1 Let L(Z) S n α (d,S) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaatuuDJXwAK1 uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqbaKqzGeGae8NeHWKaaGik aiaahQfacaaIPaGaaGzaVlabgIGiolaaygW7caqGtbqcfa4aa0baaK qaafaajugWaiaad6gaaKqaafaajugWaiabeg7aHbaajugibiaaiIca caGIKbGaaGilaiaajofacaaIPaaaaa@55ED@ have the chf ϕ Z (t)=exp[ιt'd 1 2 (t'St) α 2 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabew 9aMPWaaSbaaKqaafaajugWaiaahQfaaSqabaqcLbsacaaMb8UaaGik aiaahshacaaIPaGaaGzaVlaai2dacaaMb8UaciyzaiaacIhacaGGWb GaaG4waiabeM7aPjaayIW7caWH0bGaaG4jaiaaksgacaaMb8UaeyOe I0IaaGzaVNqbaoaalaaajaaibaqcLbmacaaIXaaajaaibaqcLbmaca aIYaaaaKqzGeGaaGikaiaahshacaaINaGaaK4uaiaahshacaaIPaGc daahaaWcbeqcbauaaKqbaoaalaaajeaqbaqcLbmacqaHXoqyaKqaaf aajugWaiaaikdaaaaaaKqzGeGaaGyxaaaa@656D@ and density function f n α (z;δ,) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadA gajuaGdaqhaaqcbauaaKqzadGaamOBaaqcbauaaKqzadGaeqySdega aKqzGeGaaGikaiaahQhacaaI7aGaeqiTdqMaaGilaiabggHiLlaaiM caaaa@47D7@ if defined. Then the following properties hold.

  1. For nonsingular, L(Z) S n α (δ,) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaatuuDJXwAK1 uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqbaKqzGeGae8NeHWKaaGik aiaahQfacaaIPaGaaGzaVlabgIGiolaaygW7caqGtbqcfa4aa0baaK qaafaajugWaiaad6gaaKqaafaajugWaiabeg7aHbaajugibiaaiIca cqaH0oazcaaISaGaeyyeIuUaaGykaaaa@5768@ is absolutely continuous in n , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaatuuDJXwAK1 uy0HMmaeHbfv3ySLgzG0uy0HgiuD3BaGqbaKqzGeGae8xhHiLcdaah aaWcbeqcbauaaKqzadGaamOBaaaajugibiaaiYcaaaa@479E@ having a density function f n α (z;δ,); MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadA gajuaGdaqhaaqcbauaaKqzadGaamOBaaqcbauaaKqzadGaeqySdega aKqzGeGaaGikaiaahQhacaaI7aGaeqiTdqMaaGilaiabggHiLlaaiM cacaaI7aaaaa@489C@
  2. The Gaussian mixture ϕ Z (t)= 0 e ιt'dt'St/2s dΨ(s;α) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabew 9aMPWaaSbaaKqaafaajugWaiaahQfaaSqabaqcLbsacaaMb8UaaGik aiaahshacaaIPaGaaGzaVlaai2dacaaMb8UcdaWdXaqabKqaafaaju gWaiaaicdaaKqaafaajugWaiabg6HiLcqcLbsacqGHRiI8aiaadwga juaGdaahaaqcbauabeaajugWaiaayIW7cqaH5oqAcaWH0bGaaG4jai aaksgacqGHsislcaaMi8UaaCiDaiaaiEcacaqItbGaaCiDaiaai+ca caaIYaGaam4CaaaajugibiaayIW7caWGKbGaeuiQdKLaaGikaiaado hacaaI7aGaeqySdeMaaGykaaaa@6831@ holds with Ψ(s;α) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabfI 6azjaaiIcacaWGZbGaaG4oaiabeg7aHjaaiMcaaaa@3F4F@ as a mixing on 1 ; MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaatuuDJXwAK1 uy0HMmaeHbfv3ySLgzG0uy0HgiuD3BaGqbaKqzGeGae8xhHivcfa4a aWbaaSqabKqaafaajugWaiaaigdaaaqcLbsacaaI7aaaaa@47F9@
  3. The Gaussian mixture f n α (z;δ,)= 0 g n (z;δ, s 1 )dΨ(s;α) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadA gajuaGdaqhaaqcbauaaKqzadGaamOBaaqcbauaaKqzadGaeqySdega aKqzGeGaaGikaiaahQhacaaI7aGaeqiTdqMaaGilaiabggHiLlaaiM cacaaMb8UaaGypaiaaygW7kmaapedabeqcbauaaKqzadGaaGimaaqc bauaaKqzadGaeyOhIukajugibiabgUIiYdGaam4zaOWaaSbaaKqaaf aajugWaiaad6gaaSqabaqcLbsacaaIOaGaaCOEaiaaiUdacqaH0oaz caaISaGaam4CaOWaaWbaaSqabKqaafaajugWaiabgkHiTiaaigdaaa qcLbsacqGHris5caaIPaGaaGjcVlaadsgacqqHOoqwcaaIOaGaam4C aiaaiUdacqaHXoqycaaIPaaaaa@6C68@ holds with Ψ(s;α) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabfI 6azjaaiIcacaWGZbGaaG4oaiabeg7aHjaaiMcaaaa@3F4F@ as a mixing cdf as before;
  4. L(Z) S n α (δ,) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaatuuDJXwAK1 uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqbaKqzGeGae8NeHWKaaGik aiaahQfacaaIPaGaaGzaVlabgIGiolaaygW7caqGtbqcfa4aa0baaK qaafaajugWaiaad6gaaKqaafaajugWaiabeg7aHbaajugibiaaiIca cqaH0oazcaaISaGaeyyeIuUaaGykaaaa@5768@ is monotone unimodal with mode at d, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaks gacaaISaaaaa@3AA5@ for each α(0,2); MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabeg 7aHjaaygW7cqGHiiIZcaaMb8UaaGikaiaaicdacaaISaGaaGjcVlaa ikdacaGGPaGaaG4oaaaa@4517@
  5. Let T(Z)=U k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaads facaaIOaGaaCOwaiaaiMcacaaMb8UaaGypaiaaygW7caWHvbGaaGza VlabgIGiolaaygW7tuuDJXwAK1uy0HMmaeHbfv3ySLgzG0uy0HgiuD 3BaGqbaiab=1risPWaaWbaaSqabKqaafaajugWaiaadUgaaaaaaa@52C8@ be scale-invariant; then for L(Z) S n α (δ,), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaatuuDJXwAK1 uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqbaKqzGeGae8NeHWKaaGik aiaahQfacaaIPaGaaGzaVlabgIGiolaaygW7caqGtbqcfa4aa0baaK qaafaajugWaiaad6gaaKqaafaajugWaiabeg7aHbaajugibiaaiIca cqaH0oazcaaISaGaeyyeIuUaaGykaiaaiYcaaaa@581E@ the distribution L(U) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaatuuDJXwAK1 uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqbaKqzGeGae8NeHWKaaGik aiaahwfacaaIPaaaaa@45D0@ is identical to its normal-theory form under L(Z)= N n (z;δ,). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaatuuDJXwAK1 uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqbaKqzGeGae8NeHWKaaGik aiaahQfacaaIPaGaaGzaVlaai2dacaaMb8UaaeOtaOWaaSbaaKqaaf aajugWaiaad6gaaSqabaqcLbsacaaIOaGaaCOEaiaaiUdacqaH0oaz caaISaGaeyyeIuUaaGykaiaai6caaaa@5595@

Proof: Conclusion (i) is Theorem 6.5.4 of Press.23 Conclusion (ii) invokes a result of Hartman et al.24 namely, the process { Z t ;t=1,2,} MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaiU hacaWGAbGcdaWgaaWcbaqcLbsacaWG0baaleqaaKqzGeGaaG4oaiaa dshacaaMb8UaaGypaiaaygW7caaIXaGaaGilaiaaikdacaaISaGaeS OjGSKaaGyFaaaa@47E0@ is spherically invariant if and only if, for each n and Z=[ Z 1 ,, Z n ], MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaahQ facaaMb8UaaGypaiaaygW7caaIBbGaaGjcVlaadQfakmaaBaaajeaq baqcLbmacaaIXaaaleqaaKqzGeGaaGilaiablAciljaaiYcacaWGAb GcdaWgaaqcbauaaKqzadGaamOBaaWcbeaajugibiaayIW7caaIDbGa aGilaaaa@4DD5@ the chf ϕ Z (t) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabew 9aMLqbaoaaBaaajeaqbaqcLbmacaWHAbaajeaqbeaajugibiaaygW7 caaIOaGaaCiDaiaaiMcaaaa@4296@ is a scale mixture of spherical Gaussian s on n , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaatuuDJXwAK1 uy0HMmaeHbfv3ySLgzG0uy0HgiuD3BaGqbaKqzGeGae8xhHiLcdaah aaWcbeqcbauaaKqzadGaamOBaaaajugibiaaiYcaaaa@479E@ to give conclusion (ii) on transforming from spherical to elliptical symmetry. To continue, f Z (z)= (2π) n n e it'z ϕ Z (t)Λ(dt) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadA gakmaaBaaajeaqbaqcLbmacaWHAbaaleqaaKqzGeGaaGikaiaahQha caaIPaGaaGzaVlaai2dacaaMb8UaaGikaiaaikdacqaHapaCcaaIPa GcdaahaaWcbeqcbauaaKqzadGaeyOeI0IaamOBaaaakmaapebabeWc baWefv3ySLgznfgDOjdaryqr1ngBPrginfgDObcv39gaiuaajugibi ab=1risPWaaWbaaSqabeaajugibiaad6gaaaaaleqajugibiabgUIi YdGaamyzaOWaaWbaaSqabKqaafaajugWaiabgkHiTiaadMgacaaMi8 UaaCiDaiaaiEcacaWH6baaaKqzGeGaeqy1dyMcdaWgaaqcbauaaKqz adGaaCOwaaWcbeaajugibiaaygW7caaIOaGaaCiDaiaaiMcacqqHBo atcaaIOaGaamizaiaahshacaaIPaaaaa@7135@ is the standard inversion formula from s to densities in n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeGabaaubmrr1n gBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbacfaqcLbsacqWFDeIu kmaaCaaaleqajeaqbaqcLbmacaWGUbaaaaaa@46AD@ with Λ() MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabfU 5amjaaiIcacqGHflY1caaIPaaaaa@3E23@ as Lebesgue measure, so that from conclusion (ii) we recover

f n α (μ;δ, I n )= 1 (2π) n k e it'x 0 e it'δt't/2s dΨ(s;α)Λ(dt). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadA gajuaGdaqhaaqcbauaaKqzadGaamOBaaqcbauaaKqzadGaeqySdega aKqzGeGaaGikaiabeY7aTjaaiUdacqaH0oazcaaISaGaaCysaOWaaS baaKqaafaajugWaiaad6gaaSqabaqcLbsacaaIPaGaaGzaVlaai2da caaMb8UcdaWcaaqaaKqzGeGaaGymaaGcbaqcLbsacaaIOaGaaGOmai abec8aWjaaiMcakmaaCaaaleqajeaqbaqcLbmacaWGUbaaaaaakmaa pebabeWcbaWefv3ySLgznfgDOjdaryqr1ngBPrginfgDObcv39gaiu aajugibiab=1risPWaaWbaaSqabeaajugibiaadUgaaaaaleqajugi biabgUIiYdGaamyzaOWaaWbaaSqabKqaafaajugWaiabgkHiTiaadM gacaaMi8UaaCiDaiaaiEcacaWH4baaaOWaa8qmaeqajeaqbaqcLbma caaIWaaajeaqbaqcLbmacqGHEisPaKqzGeGaey4kIipacaWGLbGcda ahaaWcbeqcbauaaKqzadGaamyAaiaahshacaaINaGaeqiTdqMaeyOe I0IaaGjcVlaahshacaaINaGaaCiDaiaai+cacaaIYaGaam4Caaaaju gibiaayIW7caWGKbGaeuiQdKLaaGikaiaadohacaaI7aGaeqySdeMa aGykaiabfU5amjaaiIcacaWGKbGaaCiDaiaaiMcacaaIUaaaaa@949A@ (1)

Reversing the order of integration inverts the Gaussian chf to give conclusion (iii). Conclusion (iv) follows as in Wolfe25 conjunction with conclusion (iii). Finally observe from conclusion (iii), with 0 g n (Z;δ,/s)dΨ(s;α), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWdXaqabK qaafaajugWaiaaicdaaKqaafaajugWaiabg6HiLcqcLbsacqGHRiI8 aiaadEgakmaaBaaajeaqbaqcLbmacaWGUbaaleqaaKqzGeGaaGikai aadQfacaaI7aGaeqiTdqMaaGilaiabggHiLlaai+cacaWGZbGaaGyk aiaayIW7caWGKbGaeuiQdKLaaGikaiaadohacaaI7aGaeqySdeMaaG ykaiaaiYcaaaa@5681@ that the change of variables ZU=T(Z) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaahQ facaaMb8UaeyOKH4QaaGzaVlaahwfacaaMb8UaaGypaiaaygW7caWG ubGaaGikaiaahQfacaaIPaaaaa@46BD@ behind the integral is independent of Ψ(s;α) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabfI 6azjaaiIcacaWGZbGaaG4oaiabeg7aHjaaiMcaaaa@3F4F@ since T(Z) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaads facaaIOaGaaCOwaiaaiMcaaaa@3C20@ is scale-invariant independently of s to give conclusion (v).

It remains to reconsider degrees of association in SαS MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaabo facqaHXoqycaqGtbaaaa@3C4A@ distributions, as distinct from the classical second-moment correlation parameters { ρ ij = σ ij /( σ ii σ jj ) 1 2 }. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaiU hacqaHbpGCkmaaBaaajeaqbaqcLbmacaWGPbGaamOAaaWcbeaajugi biaaygW7caaI9aGaeq4WdmNcdaWgaaqcbauaaKqzadGaamyAaiaadQ gaaSqabaqcLbsacaaIVaGaaGikaiabeo8aZPWaaSbaaKqaGfaajug4 aiaadMgacaWGPbaaleqaaKqzGeGaeq4WdmNcdaWgaaqcbauaaKqzad GaamOAaiaadQgaaSqabaqcLbsacaaIPaGcdaahaaWcbeqcbauaaKqb aoaalaaajeaqbaqcLbmacaaIXaaajeaqbaqcLbmacaaIYaaaaaaaju gibiaai2hacaaIUaaaaa@5DF8@ For L(Z) S k α (δ,) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaatuuDJXwAK1 uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqbaKqzGeGae8NeHWKaaGik aiaahQfacaaIPaGaaGzaVlabgIGiolaaygW7caqGtbqcfa4aa0baaK qaafaajugWaiaadUgaaKqaafaajugWaiabeg7aHbaajugibiaaiIca cqaH0oazcaaISaGaeyyeIuUaaGykaaaa@5765@ with α<2, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabeg 7aHjaaygW7caaI8aGaaGzaVlaaikdacaaISaaaaa@3FEA@ the elements of s serve instead as scale parameters, since U= 1 2 Z MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaahw facaaMb8UaaGypaiaaygW7cqGHris5juaGdaahaaqcbauabeaajugW aiabgkHiTKqbaoaalaaajeaqbaqcLbmacaaIXaaajeaqbaqcLbmaca aIYaaaaaaajugibiaahQfaaaa@48E8@ and U'U=Z' 1 Z MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaahw facaaINaGaaCyvaiaaygW7caaI9aGaaGzaVlaahQfacaaINaGaeyye IuUcdaahaaWcbeqcbauaaKqzadGaeyOeI0IaaGymaaaajugibiaahQ faaaa@4748@ are dimensionless. As to whether { ρ ij } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaiU hacqaHbpGCkmaaBaaajeaqbaqcLbmacaWGPbGaamOAaaWcbeaajugi biaai2haaaa@40E5@ again might quantify associations for α<2, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabeg 7aHjaaygW7caaI8aGaaGzaVlaaikdacaaISaaaaa@3FEA@ a definitive answer is supplied in the following.

Lemma 1 Let L(Z) S n α (δ,). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaatuuDJXwAK1 uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqbaKqzGeGae8NeHWKaaGik aiaahQfacaaIPaGaaGzaVlabgIGiolaaygW7caqGtbqcfa4aa0baaK qaafaajugWaiaad6gaaKqaafaajugWaiabeg7aHbaajugibiaaiIca cqaH0oazcaaISaGaeyyeIuUaaGykaiaai6caaaa@5820@ For , the parameters { ρ ij = σ ij /( σ ii σ jj ) 1 2 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaiU hacqaHbpGCkmaaBaaajeaqbaqcLbmacaWGPbGaamOAaaWcbeaajugi biaaygW7caaI9aGaaGzaVlabeo8aZPWaaSbaaKqaafaajugWaiaadM gacaWGQbaaleqaaKqzGeGaaG4laiaaiIcacqaHdpWCkmaaBaaajeaq baqcLbmacaWGPbGaamyAaaWcbeaajugibiabeo8aZPWaaSbaaKqaaf aajugWaiaadQgacaWGQbaaleqaaKqzGeGaaGykaOWaaWbaaSqabKqa afaajuaGdaWcaaqcbauaaKqzadGaaGymaaqcbauaaKqzadGaaGOmaa aaaaqcLbsacaaI9baaaa@5E8A@ serve to quantify degrees of association between ( Z i , Z j ), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaiI cacaWGAbGcdaWgaaqcbauaaKqzadGaamyAaaWcbeaajugibiaaiYca caWGAbGcdaWgaaqcbauaaKqzadGaamOAaaWcbeaajugibiaaiMcaca aISaaaaa@43E5@ the extent of their association increasing with

Proof: It suffices to consider ( Z 1 , Z 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaiI cacaWGAbGcdaWgaaqcbauaaKqzadGaaGymaaWcbeaajugibiaaiYca caWGAbGcdaWgaaqcbauaaKqzadGaaGOmaaWcbeaajugibiaaiMcaaa a@42C9@ centered at (0,0) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaiI cacaaIWaGaaGilaiaayIW7caaIWaGaaGykaaaa@3E1F@ with S=[ 1 ρ ρ 1 ]. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcaqItb GaaGzaVlaai2dacaaMb8+aamWaaeaafaqabeGacaaabaGaaGymaaqa aiabeg8aYbqaaiabeg8aYbqaaiaaigdaaaaacaGLBbGaayzxaaGaaG Olaaaa@4567@ On taking U=( Z 1 Z 2 ), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadw facaaMb8UaaGypaiaaygW7caaIOaGaamOwaOWaaSbaaKqaafaajugW aiaaigdaaSqabaqcLbsacaaMb8UaeyOeI0IaaGzaVlaadQfakmaaBa aajeaqbaqcLbmacaaIYaaaleqaaKqzGeGaaGykaiaaiYcaaaa@4B7F@ L(U) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaatuuDJXwAK1 uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqbaKqzGeGae8NeHWKaaGik aiaadwfacaaIPaaaaa@45CC@ clearly is symmetric about 0 with scale parameter σ U =2(1ρ). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcqaHdp WCdaWgaaqaaiaadwfaaeqaaiaaygW7caaI9aGaaGzaVlaaikdacaaI OaGaaGymaiaaygW7cqGHsislcaaMb8UaeqyWdiNaaGykaiaai6caaa a@48EC@ A result of Fefferman et al.26 shows for each c>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaado gacaaMb8UaaGOpaiaaygW7caaIWaaaaa@3E7D@ that P(U(c,c)) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadc facaaIOaGaamyvaiaaygW7cqGHiiIZcaaMb8UaaGikaiabgkHiTiaa dogacaaISaGaaGjcVlaadogacaaIPaGaaGykaaaa@4714@ is decreasing in σ U MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcqaHdp WCdaWgaaqaaiaadwfaaeqaaaaa@3BBC@ thus increasing in ρ. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcqaHbp GCcaaIUaaaaa@3B76@ Equivalently, P(| Z 1 Z 2 |c)1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadc facaaIOaGaaGiFaiaadQfakmaaBaaajeaqbaqcLbmacaaIXaaaleqa aKqzGeGaaGzaVlabgkHiTiaaygW7caWGAbGcdaWgaaqcbauaaKqzad GaaGOmaaWcbeaajugibiaaiYhacaaMb8UaeyizImQaam4yaiaaiMca caaMb8UaeyyKH0QaaGzaVlaaigdaaaa@52D6@ as ρ1, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabeg 8aYjaaygW7cqGHrgsRcaaMb8UaaGymaiaaiYcaaaa@412F@ identifying the sense in which ( Z 1 , Z 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaiI cacaWGAbGcdaWgaaqcbauaaKqzadGaaGymaaWcbeaajugibiaaiYca caWGAbGcdaWgaaqcbauaaKqzadGaaGOmaaWcbeaajugibiaaiMcaaa a@42C9@ become increasingly indistinguishable, thus associated, with increasing values of

Definition 3 For L(Z) S n α (δ,S) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaatuuDJXwAK1 uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqbaKqzGeGae8NeHWKaaGik aiaahQfacaaIPaGaaGzaVlabgIGiolaaygW7caqGtbGcdaqhaaqcba uaaKqzadGaamOBaaqcbauaaKqzadGaeqySdegaaKqzGeGaaGikaGGa aiab+r7aKjaaiYcacaqItbGaaGykaaaa@5622@ with α<2, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabeg 7aHjaaygW7caaI8aGaaGzaVlaaikdacaaISaaaaa@3FEA@ the entities { ρ ij = σ ij /( σ ii σ jj ) 1 2 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaiU hacqaHbpGCkmaaBaaajeaqbaqcLbmacaWGPbGaamOAaaWcbeaajugi biaaygW7caaI9aGaaGzaVlabeo8aZPWaaSbaaKqaafaajugWaiaadM gacaWGQbaaleqaaKqzGeGaaG4laiaaiIcacqaHdpWCkmaaBaaajeaq baqcLbmacaWGPbGaamyAaaWcbeaajugibiabeo8aZPWaaSbaaKqaaf aajugWaiaadQgacaWGQbaaleqaaKqzGeGaaGykaOWaaWbaaSqabKqa afaajuaGdaWcaaqcbauaaKqzadGaaGymaaqcbauaaKqzadGaaGOmaa aaaaqcLbsacaaI9baaaa@5E8A@ are called pseudo–correlation, specifically, α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabeg 7aHbaa@3A9E@ -association parameters

Linear models under errors

The principal findings

Take L(Y) S n α (Xβ, σ 2 I n ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaatuuDJXwAK1 uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqbaKqzGeGae8NeHWKaaGik aiaahMfacaaIPaGaaGzaVlabgIGiolaaygW7caqGtbGcdaqhaaqcba uaaKqzadGaamOBaaqcbauaaKqzadGaeqySdegaaKqzGeGaaGikaiaa hIfacqaHYoGycaaISaGaaGjcVlabeo8aZPWaaWbaaSqabKqaafaaju gWaiaaikdaaaqcLbsacaWHjbGcdaWgaaqcbauaaKqzadGaamOBaaWc beaajugibiaaiMcaaaa@606C@ with (Xβ, σ 2 I n ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaiI cacaWHybGaeqOSdiMaaGilaiaayIW7cqaHdpWCkmaaCaaaleqajeaq baqcLbmacaaIYaaaaKqzGeGaaCysaOWaaSbaaKqaafaajugWaiaad6 gaaSqabaqcLbsacaaIPaaaaa@47EC@ as centering and scale parameters, where {Y n ,X F n×k ,β k }. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaiU hacaWHzbGaaGzaVlabgIGiolaaygW7tuuDJXwAK1uy0HMmaeHbfv3y SLgzG0uy0HgiuD3BaGqbaiab=1risPWaaWbaaSqabKqaafaajugWai aad6gaaaqcLbsacaaISaGaaCiwaiaaygW7cqGHiiIZcaaMb8Uae8xH WBKcdaWgaaqcbauaaKqzadGaaGzaVlaad6gacaaMb8Uaey41aqRaaG zaVlaadUgaaSqabaqcLbsacaaISaGaeqOSdiMaaGzaVlabgIGiolaa ygW7cqWFDeIukmaaCaaaleqajeaqbaqcLbmacaWGRbaaaKqzGeGaaG yFaiaai6caaaa@6D9C@ OLS MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaad+ eacaaMb8UaamitaiaaygW7caWGtbaaaa@3E90@ solutions β = (X'X) 1 X'Y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiqbek 7aIzaataGaaGzaVlaai2dacaaMb8UaaGikaiaahIfacaaINaGaaGza VlaahIfacaaIPaGcdaahaaWcbeqcbauaaKqzadGaaGzaVlabgkHiTi aaigdaaaqcLbsacaWHybGaaG4jaiaahMfaaaa@4BDB@ , as minimally dispersed unbiased linear estimates, are available here only for α=2, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabeg 7aHjaaygW7caaI9aGaaGzaVlaaikdacaaISaaaaa@3FEB@ whereas alternative moment criteria necessarily are subject to moment constraints. Specifically, for scalars ( θ ,θ) 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaiI cacuaH4oqCgaWeaiaaiYcacqaH4oqCcaaIPaGaaGzaVlabgIGiolaa ygW7tuuDJXwAK1uy0HMmaeHbfv3ySLgzG0uy0HgiuD3BaGqbaiab=1 risPWaaWbaaSqabKqaafaajugWaiaaigdaaaaaaa@505A@ under loss L( θ ,θ)=| θ θ|, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadY eacaaIOaGafqiUdeNbambacaaISaGaeqiUdeNaaGykaiaaygW7caaI 9aGaaGzaVlaaiYhacuaH4oqCgaWeaiabgkHiTiaaygW7cqaH4oqCca aI8bGaaGilaaaa@4C0B@ the risk R( θ )=E[L( θ ,θ)] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadk facaaIOaGafqiUdeNbambacaaIPaGaaGzaVlaai2dacaaMb8Uaaeyr aiaaiUfacaWGmbGaaGikaiqbeI7aXzaataGaaGilaiabeI7aXjaaiM cacaaIDbaaaa@49EC@ is undefined for α<1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabeg 7aHjaaygW7caaI8aGaaGzaVlaaigdaaaa@3F33@ as for Cauchy errors at α=1. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabeg 7aHjaaygW7caaI9aGaaGzaVlaaigdacaaIUaaaaa@3FEC@ Moreover, risk functions {R( θ )=E(| θ θ | κ )} MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaiU hacaWGsbGaaGikaiqbeI7aXzaataGaaGykaiaaygW7caaI9aGaaGza VlaabweacaaIOaGaaGiFaiqbeI7aXzaataGaeyOeI0IaaGzaVlabeI 7aXjaaiYhakmaaCaaaleqajeaqbaqcLbmacqaH6oWAaaqcLbsacaaI PaGaaGyFaaaa@5118@ are defined but concave for {κ<α<1}, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaiU hacqaH6oWAcaaMb8UaaGipaiaaygW7cqaHXoqycaaMb8UaaGipaiaa ygW7caaIXaGaaGyFaiaaiYcaaaa@4781@ and for {1<κ<α2} MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaiU hacaaIXaGaaGzaVlaaiYdacaaMb8UaeqOUdSMaaGzaVlaaiYdacaaM b8UaeqySdeMaaGzaVlabgsMiJkaaygW7caaIYaGaaGyFaaaa@4C50@ are convex, at issue in attaining global optima. Versions of these apply also for vector parameters; however, minimal risk estimation would require not only knowledge regarding α, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabeg 7aHjaaiYcaaaa@3B54@ but also optimizing algorithms. Instead we seek what might be salvaged from classical linear models under the constraints of SαS MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaabo facqaHXoqycaqGtbaaaa@3C4A@ errors. In addition, portions of our findings extend beyond Gauss–Markov theory and OLS MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaad+ eacaaMb8UaamitaiaaygW7caWGtbaaaa@3E90@ to include the much larger class of equivariant estimators.

Definition 4 An estimator δ(Y) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabes 7aKjaaiIcacaWHzbGaaGykaaaa@3CEB@ for β k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabek 7aIjabgIGiolaaygW7tuuDJXwAK1uy0HMmaeHbfv3ySLgzG0uy0Hgi uD3BaGqbaiab=1risPWaaWbaaSqabKqaafaajugWaiaadUgaaaaaaa@4B05@ is translation –equivariant if for {YY+Xb}, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaiU hacaWHzbGaaGzaVlabgkziUkaaygW7caWHzbGaaGzaVlabgUcaRiaa ygW7caWHybGaaCOyaiaai2hacaaISaaaaa@4848@ then {δ(Y+Xb)=δ(Y)+b} MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaiU hacqaH0oazcaaIOaGaaCywaiaaygW7cqGHRaWkcaaMb8UaaCiwaiaa hkgacaaIPaGaaGzaVlaai2dacaaMb8UaeqiTdqMaaGikaiaahMfaca aIPaGaaGzaVlabgUcaRiaaygW7caWHIbGaaGyFaaaa@5161@ for every b k . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaahk gacaaMb8UaeyicI4SaaGzaVprr1ngBPrwtHrhAYaqeguuDJXwAKbst HrhAGq1DVbacfaGae8xhHiLcdaahaaWcbeqcbauaaKqzadGaam4Aaa aajugibiaai6caaaa@4D20@

On taking P=[ I n X (X'X) 1 X'], MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaahc facaaMb8UaaGypaiaaygW7caaIBbGaaCysaOWaaSbaaKqaafaajugW aiaad6gaaSqabaqcLbsacaaMb8UaeyOeI0IaaGzaVlaahIfacaaIOa GaaCiwaiaaiEcacaaMb8UaaCiwaiaaiMcakmaaCaaaleqajeaqbaqc LbmacaaMb8UaeyOeI0IaaGymaaaajugibiaahIfacaaINaGaaGyxai aaiYcaaaa@557D@ the elements of e=PY MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaahw gacaaMb8UaaGypaiaaygW7caWHqbGaaCywaaaa@3F83@ comprise the observed residuals and S 2 =e'e/(nk) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaado fakmaaCaaaleqajeaqbaqcLbmacaaIYaaaaKqzGeGaaGzaVlaai2da caaMb8UaaCyzaiaaiEcacaWHLbGaaG4laiaaiIcacaWGUbGaaGzaVl abgkHiTiaaygW7caWGRbGaaGykaaaa@4B3B@ the residual mean square. Normal–theory tests for H 0 :β= β 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadI eakmaaBaaajeaqbaqcLbmacaaIWaaaleqaaKqzGeGaaGzaVlaaiQda caaMb8UaeqOSdiMaaGzaVlaai2dacaaMb8UaeqOSdiMcdaWgaaqcba uaaKqzadGaaGimaaWcbeaaaaa@4A20@ against H 1 :β β 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadI eakmaaBaaajeaqbaqcLbmacaaIXaaaleqaaKqzGeGaaGzaVlaaiQda caaMb8UaeqOSdiMaaGzaVlabgcMi5kaaygW7cqaHYoGykmaaBaaaje aqbaqcLbmacaaIWaaaleqaaaaa@4B21@ utilize F=( β β 0 ) X'X( β β 0 )/ S 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadA eacaaMb8UaaGypaiaaygW7caaIOaGccuaHYoGygaWeaKqzGeGaaGza VlabgkHiTiaaygW7cqaHYoGykmaaBaaajeaqbaqcLbmacaaIWaaale qaaKqzGeGabGykayaafaGaaCiwaiaaiEcacaaMb8UaaCiwaiaaiIca kiqbek7aIzaataqcLbsacaaMb8UaeyOeI0IaaGzaVlabek7aIPWaaS baaKqaafaajugWaiaaicdaaSqabaqcLbsacaaIPaGaaG4laiaadofa kmaaCaaaleqajeaqbaqcLbmacaaIYaaaaaaa@5E4E@ having the distribution F(u;k,nk,λ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadA eacaaIOaGaamyDaiaaiUdacaWGRbGaaGilaiaad6gacaaMb8UaeyOe I0IaaGzaVlaadUgacaaISaGaeq4UdWMaaGykaaaa@46E2@ with λ=(β β 0 ) X'X(β β 0 )/ σ 2 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabeU 7aSjaaygW7caaI9aGaaGzaVlaaiIcacqaHYoGycqGHsislcaaMb8Ua eqOSdiMcdaWgaaWcbaqcLbsacaaIWaaaleqaaKqzGeGabGykayaafa GaaCiwaiaaiEcacaaMb8UaaCiwaiaaiIcacqaHYoGycaaMb8UaeyOe I0IaaGzaVlabek7aIPWaaSbaaSqaaKqzGeGaaGimaaWcbeaajugibi aaiMcacaaIVaGaeq4WdmNcdaahaaWcbeqaaKqzGeGaaGOmaaaacaaI Uaaaaa@5B45@ We proceed to examine essential properties of S n α (Xβ, σ 2 I n ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaabo fakmaaDaaajeaqbaqcLbmacaWGUbaajeaqbaqcLbmacqaHXoqyaaqc LbsacaaIOaGaaCiwaiabek7aIjaaiYcacaaMi8Uaeq4WdmNcdaahaa WcbeqcbauaaKqzadGaaGOmaaaajugibiaahMeakmaaBaaajeaqbaqc LbmacaWGUbaaleqaaKqzGeGaaGykaaaa@4EFF@ as α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabeg 7aHbaa@3A9E@ ranges over (0,2), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaiI cacaaMi8UaaGimaiaaiYcacaaMi8UaaGOmaiaayIW7caGGPaGaaGil aaaa@41F3@ where some expressions simplify on taking σ 2 =1, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabeo 8aZPWaaWbaaSqabKqaafaajugWaiaaikdaaaqcLbsacaaMb8UaaGyp aiaaygW7caaIXaGaaGilaaaa@4308@ then reinstating σ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabeo 8aZPWaaWbaaSqabKqaafaajugWaiaaikdaaaaaaa@3D2D@ as needed. The following properties are fundamental.

Theorem 2 Given L(Y)= S n α (Xβ, σ 2 I n ), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaatuuDJXwAK1 uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqbaKqzGeGae8NeHWKaaGik aiaahMfacaaIPaGaaGzaVlaai2dacaaMb8Uaae4uaOWaa0baaKqaaf aajugWaiaad6gaaKqaafaajugWaiabeg7aHbaajugibiaaiIcacaWH ybGaeqOSdiMaaGilaiabeo8aZPWaaWbaaSqabKqaafaajugWaiaaik daaaqcLbsacaWHjbGcdaWgaaqcbauaaKqzadGaamOBaaWcbeaajugi biaaiMcacaaISaaaaa@5ED4@ consider [ β ,e] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaiU fakiqbek7aIzaataqcLbsacaaISaGaaGjcVlaahwgacaaIDbaaaa@4054@ with e=PY MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaahw gacaaMb8UaaGypaiaaygW7caWHqbGaaCywaaaa@3F83@ as the residual vector, and U=(nk) S 2 / σ 2 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadw facaaMb8UaaGypaiaaygW7caaIOaGaamOBaiaaygW7cqGHsislcaaM b8Uaam4AaiaaiMcacaWGtbGcdaahaaWcbeqcbauaaKqzadGaaGOmaa aajugibiaai+cacqaHdpWCkmaaCaaaleqajeaqbaqcLbmacaaIYaaa aKqzGeGaaGOlaaaa@4EFD@ Then

(i) L( β ,e)= S n+k α ([β,0],S), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaatuuDJXwAK1 uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqbaKqzGeGae8NeHWKaaGik aOGafqOSdiMbambajugibiaaiYcacaWHLbGaaGykaiaaygW7caaI9a GaaGzaVlaabofakmaaDaaajeaqbaqcLbmacaWGUbGaaGzaVlabgUca RiaaygW7caWGRbaajeaqbaqcLbmacqaHXoqyaaqcLbsacaaIOaGaaG 4waiabek7aIjaaiYcacaWHWaGaaGyxaiaaiYcacaaMi8UaaK4uaiaa iMcacaaISaaaaa@62DA@ with = σ 2 Diag((X'X ) 1 ,P), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabgg HiLlaaygW7caaI9aGaaGzaVlabeo8aZPWaaWbaaSqabKqaafaajugW aiaaikdaaaqcLbsacaWGebGaamyAaiaadggacaWGNbGaaGikaiaaiI cacaWHybGaaG4jaiaaygW7caWHybGaaGykaOWaaWbaaSqabKqaafaa jugWaiaaygW7cqGHsislcaaIXaaaaKqzGeGaaGilaiaahcfacaaIPa GaaGilaaaa@5540@ a distribution on n+k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaatuuDJXwAK1 uy0HMmaeHbfv3ySLgzG0uy0HgiuD3BaGqbaKqzGeGae8xhHiLcdaah aaWcbeqcbauaaKqzadGaamOBaiaaygW7cqGHRaWkcaaMb8Uaam4Aaa aaaaa@4B3F@ of rank s

(ii) The marginal’s are L( β )= S k α (β, σ 2 (X'X) 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaatuuDJXwAK1 uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqbaKqzGeGae8NeHWKaaGik aOGafqOSdiMbambajugibiaaiMcacaaMb8UaaGypaiaaygW7caqGtb GcdaqhaaqcbauaaKqzadGaam4AaaqcbauaaKqzadGaeqySdegaaKqz GeGaaGikaiabek7aIjaaiYcacaaMi8Uaeq4WdmNcdaahaaWcbeqcba uaaKqzadGaaGOmaaaajugibiaaiIcacaWHybGaaG4jaiaaygW7caWH ybGaaGykaOWaaWbaaSqabKqaafaajugWaiaaygW7cqGHsislcaaIXa aaaKqzGeGaaGykaaaa@670D@ centered at β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabek 7aIbaa@3AA0@ with scale parameters σ 2 (X'X) 1 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabeo 8aZPWaaWbaaSqabKqaafaajugWaiaaikdaaaqcLbsacaaIOaGaaCiw aiaaiEcacaaMb8UaaCiwaiaaiMcakmaaCaaaleqajeaqbaqcLbmaca aMb8UaeyOeI0IaaGymaaaajugibiaaiYcaaaa@4944@ and

(iii) L(e)= S n α (0, σ 2 P) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaatuuDJXwAK1 uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqbaKqzGeGae8NeHWKaaGik aiaahwgacaaIPaGaaGzaVlaai2dacaaMb8Uaae4uaOWaa0baaKqaaf aajugWaiaad6gaaKqaafaajugWaiabeg7aHbaajugibiaaiIcacaWH WaGaaGilaiaayIW7cqaHdpWCkmaaCaaaleqajeaqbaqcLbmacaaIYa aaaKqzGeGaaCiuaiaaiMcaaaa@5AC9@ on n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaatuuDJXwAK1 uy0HMmaeHbfv3ySLgzG0uy0HgiuD3BaGqbaKqzGeGae8xhHiLcdaah aaWcbeqcbauaaKqzadGaamOBaaaaaaa@4659@ of rank nk MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaad6 gacaaMb8UaeyOeI0IaaGzaVlaadUgaaaa@3EE3@ centered at 0with scale parameters σ 2 P; MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabeo 8aZPWaaWbaaSqabKqaafaajugWaiaaikdaaaqcLbsacaWHqbGaaG4o aaaa@3F5A@

(iv) U=(nk) S 2 / σ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadw facaaMb8UaaGypaiaaygW7caaIOaGaamOBaiaaygW7cqGHsislcaaM b8Uaam4AaiaaiMcacaWGtbGcdaahaaWcbeqcbauaaKqzadGaaGOmaa aajugibiaai+cacqaHdpWCkmaaCaaaleqajeaqbaqcLbmacaaIYaaa aaaa@4DB6@ has density f(u;ν,α)= 0 h(u;ν,s)dΨ(s;α) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadA gacaaIOaGaamyDaiaaiUdacqaH9oGBcaaISaGaeqySdeMaaGykaiaa ygW7caaI9aGaaGzaVRWaa8qmaeqajeaqbaqcLbmacaaIWaaajeaqba qcLbmacqGHEisPaKqzGeGaey4kIipacaWGObGaaGikaiaadwhacaaI 7aGaeqyVd4MaaGilaiaadohacaaIPaGaaGjcVlaadsgacqqHOoqwca aIOaGaam4CaiaaiUdacqaHXoqycaaIPaaaaa@5CFD@ with h(u;ν,s) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadI gacaaIOaGaamyDaiaaiUdacqaH9oGBcaaISaGaam4CaiaaiMcaaaa@4076@ as the central chi–squared density on ν=(nk) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabe2 7aUjaaygW7caaI9aGaaGzaVlaaiIcacaWGUbGaaGzaVlabgkHiTiaa ygW7caWGRbGaaGykaaaa@45DB@ degrees of freedom, scaled by S, and with Ψ(s;α) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabfI 6azjaaiIcacaWGZbGaaG4oaiabeg7aHjaaiMcaaaa@3F4F@ as a mixing distribution.

Proof. Let L'= (X'X) 1 X' MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaahY eacaaINaGaaGzaVlaai2dacaaMb8UaaGikaiaahIfacaaINaGaaGza VlaahIfacaaIPaGcdaahaaWcbeqcbauaaKqzadGaaGzaVlabgkHiTi aaigdaaaqcLbsacaaMb8UaaCiwaiaaiEcaaaa@4C4E@ and P=[ I n X (X'X) 1 X' ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaahc facaaMb8UaaGypaiaaygW7kmaadmaabaqcLbsacaWHjbGcdaWgaaqc bauaaKqzadGaamOBaaWcbeaajugibiaaygW7cqGHsislcaaMb8UaaC iwaiaaiIcacaWHybGaaG4jaiaaygW7caWHybGaaGykaOWaaWbaaSqa bKqaafaajugWaiaaygW7cqGHsislcaaIXaaaaKqzGeGaaGzaVlaahI facaaINaaakiaawUfacaGLDbaaaaa@571A@ to project onto the error space, so that G=[L,P] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaahE eacaaMb8UaaGypaiaaygW7caaIBbGaaCitaiaaiYcacaWHqbGaaGyx aaaa@41DA@ operates on y to give

Z=G'Y=[ β e ]=[ L' P' ]Y n+k andG'G=[ (X'X) 1 0 0 P ], MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaahQ facaaI9aGaaC4raiaaiEcacaWHzbGaaGypaOWaamWaaeaajugibuaa beqaceaaaKqaGgaajugWaiqbek7aIzaataaaleaajugibiaahwgaaa aakiaawUfacaGLDbaajugibiaai2dakmaadmaabaqcLbsafaqabeGa baaaleaajugibiaahYeacaaINaaaleaajugibiaahcfacaaINaaaaa GccaGLBbGaayzxaaqcLbsacaWHzbGaeyicI48efv3ySLgznfgDOjda ryqr1ngBPrginfgDObcv39gaiuaacqWFDeIukmaaCaaaleqajeaqba qcLbmacaWGUbGaaGzaVlabgUcaRiaaygW7caWGRbaaaKqzGeGaaGjb VlaadggacaWGUbGaamizaiaaysW7caWHhbGaaG4jaiaahEeacaaI9a GcdaWadaqaaKqzGeqbaeqabiGaaaWcbaqcLbsacaaIOaGaaCiwaiaa iEcacaaMb8UaaCiwaiaaiMcakmaaCaaaleqajeaqbaqcLbmacaaMb8 UaeyOeI0IaaGymaaaaaOqaaKqzGeGaaCimaaGcbaqcLbsacaWHWaaa keaajugibiaahcfaaaaakiaawUfacaGLDbaajugibiaaiYcaaaa@7F61@ (2)

 the latter of order [(n+k)×(n+k)] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaiU facaaIOaGaamOBaiaaygW7cqGHRaWkcaaMb8Uaam4AaiaaiMcacaaM b8Uaey41aqRaaGzaVlaaiIcacaWGUbGaaGzaVlabgUcaRiaaygW7ca WGRbGaaGykaiaai2faaaa@4E72@ and rank n. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaad6 gacaaIUaaaaa@3AAA@ The chf with argument s'=[ s 1 ,, s n+k ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaho hajugWaiaaiEcajugibiaaygW7caaI9aGaaGzaVlaaiUfacaaMi8Ua am4CaOWaaSbaaKqaafaajugWaiaaigdaaSqabaqcLbsacaaISaGaeS OjGSKaaGilaiaadohakmaaBaaajeaqbaqcLbmacaWGUbGaaGzaVlab gUcaRiaaygW7caWGRbaaleqaaKqzGeGaaGyxaaaa@532D@ is E[exp(ιs'Z)]=E[exp(ιs'G'Y)]=E[exp(ιv'Y)] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaabw eacaaIBbGaciyzaiaacIhacaGGWbGaaGikaiabeM7aPjaayIW7caWH ZbGaaG4jaiaahQfacaaIPaGaaGyxaiaaygW7caaI9aGaaGzaVlaabw eacaaIBbGaciyzaiaacIhacaGGWbGaaGikaiabeM7aPjaayIW7caWH ZbGaaG4jaiaahEeacaaINaGaaCywaiaaiMcacaaIDbGaaGzaVlaai2 dacaaMb8UaaeyraiaaiUfaciGGLbGaaiiEaiaacchacaaIOaGaeqyU dKMaaGjcVlaahAhacaaINaGaaCywaiaaiMcacaaIDbaaaa@6823@ = ϕ Y (v) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabew 9aMPWaaSbaaKqaafaajugWaiaahMfaaSqabaqcLbsacaaMb8UaaGik aiaahAhacaaIPaaaaa@41D4@ with argument v=Gs MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaahA hacaaMb8UaaGypaiaaygW7caWHhbGaaC4Caaaa@3FA5@ replacing t, to give conclusion (i). Next partition s'=[ s 1 ', s 2 '] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaho hacaaINaGaaGzaVlaai2dacaaMb8UaaG4waiaahohakmaaBaaajeaq baqcLbmacaaIXaaaleqaaKqzGeGaaG4jaiaaiYcacaaMi8UaaC4CaO WaaSbaaKqaafaajugWaiaaikdaaSqabaqcLbsacaaINaGaaGyxaaaa @4BE5@ with s 1 '=[ s 1 ,, s k ], MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaho hakmaaBaaajeaqbaqcLbmacaaIXaaaleqaaKqzGeGaaG4jaiaaygW7 caaI9aGaaGzaVlaaiUfacaWGZbGcdaWgaaqcbauaaKqzadGaaGymaa WcbeaajugibiaaiYcacqWIMaYscaaISaGaam4CaOWaaSbaaKqaafaa jugWaiaadUgaaSqabaqcLbsacaaIDbGaaGilaaaa@4EA4@ to obtain = ϕ Z (s)=exp[ιs'G'Xβ 1 2 (s'G'Gs) α 2 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabew 9aMPWaaSbaaKqaafaajugWaiaahQfaaSqabaqcLbsacaaMb8UaaGik aiaahohacaaIPaGaaGzaVlaai2dacaaMb8UaciyzaiaacIhacaGGWb GaaG4waiabeM7aPjaayIW7caWHZbGaaG4jaiaahEeacaaINaGaaGza VlaahIfacqaHYoGycqGHsislkmaalaaabaqcLbsacaaIXaaakeaaju gibiaaikdaaaGaaGikaiaahohacaaINaGaaC4raiaaiEcacaWHhbGa aC4CaiaaiMcakmaaCaaaleqajeaqbaqcfa4aaSaaaKqaafaajugWai abeg7aHbqcbauaaKqzadGaaGOmaaaaaaqcLbsacaaIDbaaaa@65CC@ = exp[ι s 1 'β 1 2 ( s 1 '(X'X ) 1 s 1 + s 2 'P s 2 ) α 2 ]. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiGacw gacaGG4bGaaiiCaiaaiUfacqaH5oqAcaaMi8UaaC4CaOWaaSbaaKqa afaajugWaiaaigdaaSqabaqcLbsacaaINaGaeqOSdiMaeyOeI0Icda WcaaqaaKqzGeGaaGymaaGcbaqcLbsacaaIYaaaaiaaiIcacaWHZbGc daWgaaqcbauaaKqzadGaaGymaaWcbeaajugibiaaiEcacaaIOaGaaC iwaiaaiEcacaaMb8UaaCiwaiaaiMcakmaaCaaaleqabaqcLbsacaaM b8UaeyOeI0IaaGymaaaacaWHZbGcdaWgaaqcbauaaKqzadGaaGymaa WcbeaajugibiabgUcaRiaahohakmaaBaaajeaqbaqcLbmacaaIYaaa leqaaKqzGeGaaG4jaiaahcfacaWHZbGcdaWgaaqcbauaaKqzadGaaG OmaaWcbeaajugibiaaiMcakmaaCaaaleqajeaqbaGcdaWcaaqcbaua aKqzadGaeqySdegajeaqbaqcLbmacaaIYaaaaaaajugibiaai2faca aIUaaaaa@6FAD@ The marginal s of β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacuaHYoGyga Weaaaa@3A2B@ and e follow on setting s 2 =0, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaho hakmaaBaaajeaqbaqcLbmacaaIYaaaleqaaKqzGeGaaGzaVlaai2da caaMb8UaaCimaiaaiYcaaaa@423E@ then s 1 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaho hakmaaBaaajeaqbaqcLbmacaaIXaaaleqaaKqzGeGaaGzaVlaai2da caaMb8UaaCimaaaa@4187@ in succession, to give conclusions (ii) and (iii). Conclusion (iv) attributes to Hartman et al.24 through Theorem 1. Specifically, a change of variables uee'e=(nk) S 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaahw hacaaMb8UaeyOKH4QaaGzaVlaahwgacaaMb8UaeyOKH4QaaGzaVlaa hwgacaaINaGaaCyzaiaaygW7caaI9aGaaGzaVlaaiIcacaWGUbGaaG zaVlabgkHiTiaaygW7caWGRbGaaGykaiaadofakmaaCaaaleqajeaq baqcLbmacaaIYaaaaaaa@55E1@ behind the integral on the right of Theorem 1(iii) gives the conditional density for L((nk) S 2 |s), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaatuuDJXwAK1 uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqbaKqzGeGae8NeHWKaaGik aiaaiIcacaWGUbGaaGzaVlabgkHiTiaaygW7caWGRbGaaGykaiaado fakmaaCaaaleqajeaqbaqcLbmacaaIYaaaaKqzGeGaaGiFaiaayIW7 caWGZbGaaGykaiaaiYcaaaa@5452@ namely the scaled chi–squared density h(u;ν,s) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadI gacaaIOaGaamyDaiaaiUdacqaH9oGBcaaISaGaam4CaiaaiMcaaaa@4076@ depending on s so that integrating with respect to dΨ(s;α) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaads gacqqHOoqwcaaIOaGaam4CaiaaiUdacqaHXoqycaaIPaaaaa@4038@ gives conclusion (iv).

Remark 3 That S= σ 2 Diag(X'X,P) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaajo facaaMb8UaaGypaiaaygW7cqaHdpWCkmaaCaaaleqajeaqbaqcLbma caaIYaaaaKqzGeGaamiraiaadMgacaWGHbGaam4zaiaaiIcacaWHyb GaaG4jaiaaygW7caWHybGaaGilaiaahcfacaaIPaaaaa@4CEF@ is block–diagonal in conclusion (i), assures under SαS MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaabo facqaHXoqycaqGtbaaaa@3C4A@ errors that ( β ,e) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaiI cakiqbek7aIzaataqcLbsacaaISaGaaCyzaiaaiMcaaaa@3E5C@ are α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabeg 7aHbaa@3A9E@ –unassociated as in Definition 3, well known to be mutually uncorrelated under second moments.

It remains to reexamine topics in inference under  errors. The following are germane.

Definition 5 An estimator θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacuaH4oqCga Weaaaa@3A40@ for θ k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabeI 7aXjaaygW7cqGHiiIZcaaMb8+efv3ySLgznfgDOjdaryqr1ngBPrgi nfgDObcv39gaiuaacqWFDeIukmaaCaaaleqajeaqbaqcLbmacaWGRb aaaaaa@4CA4@ is said to be linearly median unbiased if and only if the median med(a' θ )=a'θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaab2 gacaqGLbGaaeizaiaaiIcacaWHHbGaaG4jaOGafqiUdeNbambajugi biaaiMcacaaMb8UaaGypaiaaygW7caWHHbGaaG4jaiabeI7aXbaa@4853@ for each a k ; MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaahg gacaaMb8UaeyicI4SaaGzaVprr1ngBPrwtHrhAYaqeguuDJXwAKbst HrhAGq1DVbacfaGae8xhHiLcdaahaaWcbeqcbauaaKqzadGaam4Aaa aajugibiaaiUdaaaa@4D2C@ and to be modal unbiased provided that the mode

Definition 6 An estimator θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacuaH4oqCga Weaaaa@3A40@ for θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabeI 7aXbaa@3AB5@ is said to be more concentrated about θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabeI 7aXbaa@3AB5@ than provided that θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacuaH4oqCga Weaaaa@3A40@ P(( θ ^ θ) C 0 )P(( θ ˜ θ) C 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadc facaaIOaGaaGikaOWaaecaaeaajugibiabeI7aXbGccaGLcmaajugi biaaygW7cqGHsislcaaMb8UaeqiUdeNaaGykaiaaygW7cqGHiiIZca aMb8Uaae4qaOWaaSbaaKqaafaajugWaiaaicdaaSqabaqcLbsacaaI PaGaeyyzImRaamiuaiaaiIcacaaIOaGcdaaiaaqaaKqzGeGaeqiUde hakiaawoWaaKqzGeGaaGzaVlabgkHiTiaaygW7cqaH4oqCcaaIPaGa aGzaVlabgIGiolaaygW7caqGdbGcdaWgaaqcbauaaKqzadGaaGimaa WcbeaajugibiaaiMcaaaa@656F@ for every convex set C 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaabo eakmaaBaaajeaqbaqcLbmacaaIWaaaleqaaaaa@3C2D@ in k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaatuuDJXwAK1 uy0HMmaeHbfv3ySLgzG0uy0HgiuD3BaGqbaKqzGeGae8xhHiLcdaah aaWcbeqcbauaaKqzadGaam4Aaaaaaaa@4656@ symmetric under reflection about 0 k . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaahc dacaaMb8UaeyicI4SaaGzaVprr1ngBPrwtHrhAYaqeguuDJXwAKbst HrhAGq1DVbacfaGae8xhHiLcdaahaaWcbeqcbauaaKqzadGaam4Aaa aajugibiaai6caaaa@4CEE@

 Essential properties under SαS MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaabo facqaHXoqycaqGtbaaaa@3C4A@ errors include the following.

Theorem 3 For L(Y)= S n α (Xβ, σ 2 I n ), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaatuuDJXwAK1 uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqbaKqzGeGae8NeHWKaaGik aiaahMfacaaIPaGaaGzaVlaai2dacaaMb8Uaae4uaOWaa0baaKqaaf aajugWaiaad6gaaKqaafaajugWaiabeg7aHbaajugibiaaiIcacaWH ybGaeqOSdiMaaGilaiabeo8aZPWaaWbaaSqabKqaafaajugWaiaaik daaaqcLbsacaWHjbGcdaWgaaqcbauaaKqzadGaamOBaaWcbeaajugi biaaiMcacaaISaaaaa@5ED4@ consider properties of the OLS MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaad+ eacaaMb8UaamitaiaaygW7caWGtbaaaa@3E90@ solutions β = (X'X) 1 X'Y, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacuaHYoGyga WeaKqzGeGaaGypaiaaygW7caaIOaGaaCiwaiaaiEcacaaMb8UaaCiw aiaaiMcakmaaCaaaleqajeaqbaqcLbmacaaMb8UaeyOeI0IaaGymaa aajugibiaahIfacaaINaGaaCywaiaaiYcaaaa@4B07@ and of the equivariant estimators β =δ(Y) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacuaHYoGyga WeaKqzGeGaaGypaiaaygW7cqaH0oazcaaIOaGaaCywaiaaiMcaaaa@40F7@ of Definition 4.

  1. β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiqbek 7aIzaataaaaa@3ABA@ is unbiased for β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabek 7aIbaa@3AA0@ for each {1<α2}; MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaiU hacaaIXaGaaGzaVlaaiYdacaaMb8UaeqySdeMaaGzaVlabgsMiJkaa ygW7caaIYaGaaGyFaiaaiUdaaaa@4789@
  2. β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiqbek 7aIzaataaaaa@3ABA@ is linearly median unbiased for β; MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabek 7aIjaaiUdaaaa@3B65@
  3. β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiqbek 7aIzaataaaaa@3ABA@ is most concentrated about β; MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabek 7aIjaaiUdaaaa@3B65@ among all median–unbiased linear estimators;
  4. β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiqbek 7aIzaataaaaa@3ABA@ is modal unbiased for β; MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabek 7aIjaaiUdaaaa@3B65@
  5. β N MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacuaHYoGyga WeamaaBaaajeaqbaqcLbmacaaMb8UaamOtaaWcbeaaaaa@3E2C@ is consistent for β; MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabek 7aIjaaiUdaaaa@3B65@ in a sequence of identical but dependent experiments { Y i =Xβ+ e i ;i=1,2,,N}; MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaiU hacaWHzbGcdaWgaaqcbauaaKqzadGaaGzaVlaadMgaaSqabaqcLbsa caaMb8UaaGypaiaaygW7caWHybGaeqOSdiMaaGzaVlabgUcaRiaayg W7caGILbGcdaWgaaqcbauaaKqzadGaamyAaaWcbeaajugibiaaiUda caaMi8UaamyAaiaai2dacaaIXaGaaGilaiaaikdacaaISaGaeSOjGS KaaGilaiaad6eacaaI9bGaaG4oaaaa@596F@
  6. The null distribution of F=( β β 0 ) X'X( β β 0 )/ S 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadA eacaaMb8UaaGypaiaaygW7caaIOaGccuaHYoGygaWeaKqzGeGaaGza VlabgkHiTiaaygW7cqaHYoGykmaaBaaajeaqbaqcLbmacaaIWaaale qaaKqzGeGabGykayaafaGaaCiwaiaaiEcacaaMb8UaaCiwaiaaiIca kiqbek7aIzaataqcLbsacaaMb8UaeyOeI0IaaGzaVlabek7aIPWaaS baaKqaafaajugWaiaaicdaaSqabaqcLbsacaaIPaGaaG4laiaadofa kmaaCaaaleqajeaqbaqcLbmacaaIYaaaaaaa@5E4E@ has exactly its normal–theory form; the power increases with increasing λ=(β β 0 ) X'X(β β 0 )/ σ 2 ; MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabeU 7aSjaaygW7caaI9aGaaGzaVlaaiIcacqaHYoGycqGHsislcaaMb8Ua eqOSdiMcdaWgaaqcbauaaKqzadGaaGimaaWcbeaajugibiqaiMcaga qbaiaahIfacaaINaGaaGzaVlaahIfacaaIOaGaeqOSdiMaaGzaVlab gkHiTiaaygW7cqaHYoGykmaaBaaajeaqbaqcLbmacaaIWaaaleqaaK qzGeGaaGykaiaai+cacqaHdpWCkmaaCaaaleqajeaqbaqcLbmacaaI YaaaaKqzGeGaaG4oaaaa@5E86@ and such tests are unbiased;
  7. β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiqbek 7aIzaataaaaa@3ABA@ is most concentrated about β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabek 7aIbaa@3AA0@ among all modal–unbiased linear estimators.

β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiqbek 7aIzaataaaaa@3ABA@ is most concentrated about β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabek 7aIbaa@3AA0@ among all equivariant estimators β =δ(Y). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiqbek 7aIzaataGaaGzaVlaai2dacaaMb8UaeqiTdqMaaGikaiaahMfacaaI PaGaaGOlaaaa@4339@

Proof. Conclusions (i)–(vi) carry over from reference Jensen DR27 without benefit of moments, regardless of membership in the SαS MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaabo facqaHXoqycaqGtbaaaa@3C4A@ class. To consider concentration properties of modal–unbiased estimators, begin with ϕ Y (t)=exp[ιt'Xβ 1 2 (t't) α 2 ], MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabew 9aMPWaaSbaaKqaafaajugWaiaahMfaaSqabaqcLbsacaaMb8UaaGik aiaahshacaaIPaGaaGzaVlaai2dacaaMb8UaciyzaiaacIhacaGGWb GaaG4waiabeM7aPjaayIW7caWH0bGaaG4jaiaahIfacqaHYoGycqGH sislkmaalaaabaqcLbsacaaIXaaakeaajugibiaaikdaaaGaaGikai aahshacaaINaGaaCiDaiaaiMcakmaaCaaaleqajeaqbaqcfa4aaSaa aKqaafaajugWaiabeg7aHbqcbauaaKqzadGaaGOmaaaaaaqcLbsaca aIDbGaaGilaaaa@6129@ and consider β ˜ =L'Y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaaiaaqaaK qzGeGaeqOSdigakiaawoWaaKqzGeGaaGzaVlaai2dacaWHmbGaaG4j aiaahMfaaaa@40B4@ with L'=[(X'X ) 1 X',G'], MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaahY eacaaINaGaaGzaVlaai2dacaaMb8UaaG4waiaaiIcacaWHybGaaG4j aiaaygW7caWHybGaaGykaOWaaWbaaSqabKqaafaajugWaiaaygW7cq GHsislcaaIXaaaaKqzGeGaaCiwaiaaiEcacaaISaGaaC4raiaaiEca caaIDbGaaGilaaaa@4F7D@ so that

ϕ β ˜ (s)=exp[ιs'L'Xβ 1 2 (s'L'Ls) α 2 ]; MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabew 9aMPWaaSbaaSqaaOWaaacaaKqaafaajugWaiabek7aIbWccaGLdmaa aeqaaKqzGeGaaGzaVlaaiIcacaWHZbGaaGykaiaai2dacaaMi8Uaci yzaiaacIhacaGGWbGaaG4waiabeM7aPjaayIW7caWHZbGaaG4jaiaa hYeacaaINaGaaCiwaiabek7aIjabgkHiTOWaaSaaaeaajugibiaaig daaOqaaKqzGeGaaGOmaaaacaaIOaGaaC4CaiaaiEcacaWHmbGaaG4j aiaahYeacaWHZbGaaGykaOWaaWbaaSqabKqaafaajuaGdaWcaaqcba uaaKqzadGaeqySdegajeaqbaqcLbmacaaIYaaaaaaajugibiaai2fa caaI7aaaaa@6528@

s'L'Xβ=s'[(X'X ) 1 X',G']Xβ. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaho hacaaINaGaaCitaiaaiEcacaWHybGaeqOSdiMaaGypaiaayIW7caWH ZbGaaG4jaiaaiUfacaaIOaGaaCiwaiaaiEcacaaMb8UaaCiwaiaaiM cakmaaCaaaleqajeaqbaqcLbmacaaMb8UaeyOeI0IaaGymaaaajugi biaahIfacaaINaGaaGilaiaahEeacaaINaGaaGyxaiaahIfacqaHYo GycaaIUaaaaa@565A@

 That β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa Wdbiqbek7aIzaataaaaa@3A4B@ should have mode at β, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabek 7aIjaaiYcaaaa@3B56@ it is necessary that s'L'Xβ=s'β, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaho hacaaINaGaaCitaiaaiEcacaWHybGaeqOSdiMaaGzaVlaai2dacaaM b8UaaC4CaiaaiEcacqaHYoGycaaISaaaaa@4693@ i.e. G'X=0. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaahE eacaaINaGaaCiwaiaaygW7caaI9aGaaGzaVlaahcdacaaIUaaaaa@40AD@ accordingly, ϕ β (s)=exp[ιs'β 1 2 [s'Ωs] α 2 ], MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabew 9aMPWaaSbaaSqaaabaaaaaaaaapeGafqOSdiMbambaa8aabeaajugi biaaygW7caaIOaGaaC4CaiaaiMcacaaMb8UaaGypaiaaygW7ciGGLb GaaiiEaiaacchacaaIBbGaeqyUdKMaaGjcVlaahohacaaINaGaeqOS diMaeyOeI0IcdaWcaaqaaKqzGeGaaGymaaGcbaqcLbsacaaIYaaaai aaiUfacaWHZbGaaG4jaiabfM6axjaahohacaaIDbGcdaahaaWcbeqc bauaaKqbaoaalaaajeaqbaqcLbmacqaHXoqyaKqaafaajugWaiaaik daaaaaaKqzGeGaaGyxaiaaiYcaaaa@61C9@ with Ω=L'L=[(X'X ) 1 +G'G]. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabfM 6axjaaygW7caaI9aGaaGzaVlaahYeacaaINaGaaCitaiaaygW7caaI 9aGaaGzaVlaaiUfacaaIOaGaaCiwaiaaiEcacaaMb8UaaCiwaiaaiM cakmaaCaaaleqajeaqbaqcLbmacaaMb8UaeyOeI0IaaGymaaaajugi biaaygW7cqGHRaWkcaaMb8UaaC4raiaaiEcacaWHhbGaaGyxaiaai6 caaaa@583B@ Clearly the matrix [L'L (X'X) 1 ]=G'G MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaiU facaWHmbGaaG4jaiaahYeacaaMb8UaeyOeI0IaaGzaVlaaiIcacaWH ybGaaG4jaiaaygW7caWHybGaaGykaOWaaWbaaSqabKqaafaajugWai aaygW7cqGHsislcaaIXaaaaKqzGeGaaGyxaiaaygW7caaI9aGaaGza VlaahEeacaaINaGaaC4raaaa@5225@ is positive semi definite, giving conclusion (vii) from Jensen.28 Conclusion (viii) follows from Theorem 2.7 of Burk et al.29 since  distributions are unimodal from Theorem 1(iv).

Spherical cauchy errors

Spherical multivariate t errors on v degrees of freedom trace to Zellner30 to include Cauchy errors at ν=1, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabe2 7aUjaaygW7caaI9aGaaGzaVlaaigdacaaISaaaaa@4003@ equivalently, at α=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabeg 7aHjaaygW7caaI9aGaaGzaVlaaigdaaaa@3F34@ in the class SαS. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaabo facqaHXoqycaqGtbGaaGOlaaaa@3D02@ Specializing from Theorem 1(ii), the spherical Cauchy chf is ϕ Z (t)=exp[it'd 1 2 (t't) 1 2 ]. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabew 9aMPWaaSbaaKqaafaajugWaiaahQfaaSqabaqcLbsacaaMb8UaaGik aiaahshacaaIPaGaaGzaVlaai2dacaaMb8UaciyzaiaacIhacaGGWb GaaG4waiaadMgacaaMi8UaaCiDaiaaiEcacaGIKbGaaGzaVlabgkHi TiaaygW7kmaalaaabaqcLbsacaaIXaaakeaajugibiaaikdaaaGaaG ikaiaahshacaaINaGaaCiDaiaaiMcakmaaCaaaleqajeaqbaqcfa4a aSaaaKqaafaajugWaiaaigdaaKqaafaajugWaiaaikdaaaaaaKqzGe GaaGyxaiaai6caaaa@6109@ Recast in terms of linear inference, we have the following specialization of Theorems 1 and 2.

Corollary 1 Under the conditions of Theorems and 2, the following properties hold under spherical Cauchy errors.

  1. The spherical Cauchy density on n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaatuuDJXwAK1 uy0HMmaeHbfv3ySLgzG0uy0HgiuD3BaGqbaKqzGeGae8xhHiLcdaah aaWcbeqcbauaaKqzadGaamOBaaaaaaa@4659@ at α=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabeg 7aHjaaygW7caaI9aGaaGzaVlaaigdaaaa@3F34@ is
  2. f n 1 (z;δ, I n )= 0 g n (z;δ, s 2 I n )dΨ(s;1) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadA gakmaaDaaajeaqbaqcLbmacaWGUbaajeaqbaqcLbmacaaIXaaaaKqz GeGaaGikaiaahQhacaaI7aGaeqiTdqMaaGilaiaahMeakmaaBaaale aajugibiaad6gaaSqabaqcLbsacaaIPaGaaGypaiaayIW7kmaapeda beqcbauaaKqzadGaaGimaaqcbauaaKqzadGaeyOhIukajugibiabgU IiYdGaam4zaOWaaSbaaKqaafaajugWaiaad6gaaSqabaqcLbsacaaI OaGaaCOEaiaaiUdacqaH0oazcaaISaGaam4CaOWaaWbaaSqabKqaaf aajugWaiabgkHiTiaaikdaaaqcLbsacaWHjbGcdaWgaaqcbauaaKqz adGaamOBaaWcbeaajugibiaaiMcacaaMi8UaamizaiabfI6azjaaiI cacaWGZbGaaG4oaiaaigdacaaIPaaaaa@6C78@

    =c(n) [ 1+(zδ ) (zδ) ] n+1 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaai2 dacaaMi8UaaGjcVlaadogacaaIOaGaamOBaiaaiMcacaaMi8UcdaWa daqaaKqzGeGaaGymaiabgUcaRiaaiIcacaWH6bGaaGzaVlabgkHiTi aaygW7cqaH0oazceaIPaGbauaacaaIOaGaaCOEaiaaygW7cqGHsisl caaMb8UaeqiTdqMaaGykaaGccaGLBbGaayzxaaWaaWbaaSqabKqaaf aajugWaiabgkHiTKqbaoaalaaajeaqbaqcLbmacaWGUbGaey4kaSIa aGymaaqcbauaaKqzadGaaGOmaaaaaaaaaa@5F7F@

    c(n)=Γ( n+1 2 )/ π n+1 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaado gacaaIOaGaamOBaiaaiMcacaaI9aGaaGjcVlaayIW7cqqHtoWrcaaI OaGcdaWcaaqaaKqzGeGaamOBaiaaygW7cqGHRaWkcaaMb8UaaGymaa GcbaqcLbsacaaIYaaaaiaaiMcacaaIVaGaeqiWdaNcdaahaaWcbeqc bauaaKqbaoaalaaajeaqbaqcLbmacaWGUbGaaGzaVlabgUcaRiaayg W7caaIXaaajeaqbaqcLbmacaaIYaaaaaaaaaa@577C@

    where dΨ(s;1)= e s 2 2 /(2π ) 1 2 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaads gacqqHOoqwcaaIOaGaam4CaiaaiUdacaaIXaGaaGykaiaaygW7caaI 9aGaaGzaVlaadwgakmaaCaaaleqajeaqbaqcLbmacqGHsisljuaGda WcaaqcbauaaKqzadGaam4CaKqbaoaaCaaajeaqbeqaaKqzadGaaGOm aaaaaKqaafaajugWaiaaikdaaaaaaKqzGeGaaG4laiaaiIcacaaIYa GaeqiWdaNaaGykaOWaaWbaaSqabKqaafaajuaGdaWcaaqcbauaaKqz adGaaGymaaqcbauaaKqzadGaaGOmaaaaaaqcLbsacaaISaaaaa@5ACC@ the mixing χ(s;1) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabeE 8aJjaaiIcacaWGZbGaaG4oaiaaigdacaaIPaaaaa@3E93@ density.

  3. The elliptical Cauchy density for β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa Wdbiqbek7aIzaataaaaa@3A4B@ on k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaatuuDJXwAK1 uy0HMmaeHbfv3ySLgzG0uy0HgiuD3BaGqbaKqzGeGae8xhHiLcdaah aaWcbeqcbauaaKqzadGaam4Aaaaaaaa@4656@ is

f k 1 ( β ;β,X'X)=c(k) [ 1+( β β ) X'X( β β) ] k+1 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadA gakmaaDaaajeaqbaqcLbmacaWGRbaajeaqbaqcLbmacaaIXaaaaKqz GeGaaGikaOaeaaaaaaaaa8qacuaHYoGygaWeaKqzGeWdaiaaiUdacq aHYoGycaaISaGaaCiwaiaaiEcacaaMb8UaaCiwaiaaiMcacaaMb8Ua aGypaiaaygW7caWGJbGaaGikaiaadUgacaaIPaGcdaWadaqaaKqzGe GaaGymaiabgUcaRiaaiIcak8qacuaHYoGygaWeaKqzGeWdaiaaygW7 cqGHsislcaaMb8UaeqOSdiMabGykayaafaGaaCiwaiaaiEcacaaMb8 UaaCiwaiaaiIcak8qacuaHYoGygaWeaKqzGeWdaiaaygW7cqGHsisl caaMb8UaeqOSdiMaaGykaaGccaGLBbGaayzxaaWaaWbaaSqabKqaaf aajugWaiabgkHiTKqbaoaalaaajeaqbaqcLbmacaWGRbGaey4kaSIa aGymaaqcbauaaKqzadGaaGOmaaaaaaaaaa@75F1@ (3)

Proof. The multivariate t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaads haaaa@39F8@ distribution on n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaatuuDJXwAK1 uy0HMmaeHbfv3ySLgzG0uy0HgiuD3BaGqbaKqzGeGae8xhHiLcdaah aaWcbeqcbauaaKqzadGaamOBaaaaaaa@4659@ is that of { T i = Y i /S;1in} MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaiU hacaWGubGcdaWgaaqcbauaaKqzadGaamyAaaWcbeaajugibiaaygW7 caaI9aGaaGzaVlaadMfakmaaBaaajeaqbaqcLbmacaaMb8UaamyAaa Wcbeaajugibiaai+cacaWGtbGaaG4oaiaayIW7caaIXaGaaGzaVlab gsMiJkaaygW7caWGPbGaaGzaVlabgsMiJkaaygW7caWGUbGaaGyFaa aa@5892@ from L(Y)= N n (δ, σ 2 I n ), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaatuuDJXwAK1 uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqbaKqzGeGae8NeHWKaaGik aiaahMfacaaIPaGaaGzaVlaai2dacaaMb8UaaeOtaOWaaSbaaKqaaf aajugWaiaad6gaaSqabaqcLbsacaaIOaGaeqiTdqMaaGilaiabeo8a ZPWaaWbaaSqabKqaafaajugWaiaaikdaaaqcLbsacaWHjbGcdaWgaa qcbauaaKqzadGaamOBaaWcbeaajugibiaaiMcacaaISaaaaa@5AE5@ with S MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaado faaaa@39D7@ as a sample standard deviation on ν MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabe2 7aUbaa@3AB7@ degrees of freedom, known to be spherical Cauchy at ν=1. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabe2 7aUjaaygW7caaI9aGaaGzaVlaaigdacaaIUaaaaa@4005@ This gives conclusion (i) on specializing the conventional multivariate  density. Conclusion (ii) follows directly on specializing Theorem 2(ii) at α=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabeg 7aHjaaygW7caaI9aGaaGzaVlaaigdaaaa@3F34@

Case study

The viability ( Y i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaiI cacaWGzbGcdaWgaaqcbauaaKqzadGaaGzaVlaadMgaaSqabaqcLbsa caaIPaaaaa@3FF7@ for each of n=13 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaad6 gacaaMb8UaaGypaiaaygW7caaIXaGaaG4maaaa@3F45@ biological specimens was recorded after storage under additives X i1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadI fakmaaBaaajeaqbaqcLbmacaWGPbGaaGymaaWcbeaaaaa@3D33@ and X i2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadI fakmaaBaaajeaqbaqcLbmacaWGPbGaaGOmaaWcbeaaaaa@3D34@ as listed in Table 1;31 Walpole RE & Myers RH.31 The model is { Y i = β 0 + β 1 X i1 + β 2 X i2 + ε i }, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaiU hacaWGzbGcdaWgaaqcbauaaKqzadGaaGzaVlaadMgaaSqabaqcLbsa caaMb8UaaGypaiaaygW7cqaHYoGykmaaBaaajeaqbaqcLbmacaaIWa aaleqaaKqzGeGaaGzaVlabgUcaRiaaygW7cqaHYoGykmaaBaaajeaq baqcLbmacaaIXaaaleqaaKqzGeGaamiwaOWaaSbaaKqaafaajugWai aadMgacaaIXaaaleqaaKqzGeGaaGzaVlabgUcaRiaaygW7cqaHYoGy kmaaBaaajeaqbaqcLbmacaaIYaaaleqaaKqzGeGaamiwaOWaaSbaaK qaafaajugWaiaadMgacaaIYaaaleqaaKqzGeGaaGzaVlabgUcaRiaa ygW7cqaH1oqzkmaaBaaajeaqbaqcLbmacaWGPbaaleqaaKqzGeGaaG yFaiaaiYcaaaa@6D35@ where the errors are taken to be spherical Cauchy. The conventional OLS MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaad+ eacaaMb8UaamitaiaaygW7caWGtbaaaa@3E90@ solutions are β 0 =36.094, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacuaHYoGyga WeamaaBaaaleaacaaIWaaabeaajugibiaaygW7caaI9aGaaGzaVlaa iodacaaI2aGaaGOlaiaaicdacaaI5aGaaGinaiaaiYcaaaa@44A1@ β 1 ^ =1.031, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaqiaaqaaK qzGeGaeqOSdiMcdaWgaaqcbauaaKqzadGaaGymaaWcbeaaaOGaayPa daqcLbsacaaMb8UaaGypaiaaygW7caaIXaGaaGOlaiaaicdacaaIZa GaaGymaiaaiYcaaaa@469A@ β 2 =1.870, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiqbek 7aIzaataGcdaWgaaWcbaGaaGOmaaqabaqcLbsacaaMb8UaaGypaiaa ygW7cqGHsislcaaIXaGaaGOlaiaaiIdacaaI3aGaaGimaiaaiYcaaa a@4569@ as elements of β =[ β 0 , β 1 , β 2 ] . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiqbek 7aIzaataGaaGypaiaaygW7caaIBbGafqOSdiMbambakmaaBaaaleaa caaIWaaabeaajugibiaaiYcacuaHYoGygaWeaOWaaSbaaSqaaiaaig daaeqaaKqzGeGaaGilaiqbek7aIzaataGcdaWgaaWcbaGaaGOmaaqa baqcLbsaceaIDbGbauaacaaIUaaaaa@4AB8@ The matrix X'X, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaahI facaaINaGaaGzaVlaahIfacaaISaaaaa@3DB2@ its inverse (X'X) 1 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaiI cacaWHybGaaG4jaiaaygW7caWHybGaaGykaOWaaWbaaSqabKqaafaa jugWaiaaygW7cqGHsislcaaIXaaaaKqzGeGaaGilaaaa@4487@ and the transition of the latter into its α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabeg 7aHbaa@3A9E@ -association form of Definition 3 are given respectively by

[ 13 59.43 81.82 59.43 394.7255 360.6621 81.82 360.6621 576.7264 ] 1 =[ 1.0114 0.0494 0.1126 0.0494 0.0083 0.0018 0.1126 0.0018 0.0166 ][ 1 0.5392 0.8690 0.5392 1 0.1533 0.8690 0.1533 1 ]. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWadaqaaK qzGeqbaeqabmWaaaWcbaqcLbsacaaIXaGaaG4maaWcbaqcLbsacaaI 1aGaaGyoaiaai6cacaaI0aGaaG4maaWcbaqcLbsacaaI4aGaaGymai aai6cacaaI4aGaaGOmaaWcbaqcLbsacaaI1aGaaGyoaiaai6cacaaI 0aGaaG4maaWcbaqcLbsacaaIZaGaaGyoaiaaisdacaaIUaGaaG4nai aaikdacaaI1aGaaGynaaWcbaqcLbsacaaIZaGaaGOnaiaaicdacaaI UaGaaGOnaiaaiAdacaaIYaGaaGymaaWcbaqcLbsacaaI4aGaaGymai aai6cacaaI4aGaaGOmaaWcbaqcLbsacaaIZaGaaGOnaiaaicdacaaI UaGaaGOnaiaaiAdacaaIYaGaaGymaaWcbaqcLbsacaaI1aGaaG4nai aaiAdacaaIUaGaaG4naiaaikdacaaI2aGaaGinaaaaaOGaay5waiaa w2faamaaCaaaleqajeaqbaqcLbmacqGHsislcaaIXaaaaKqzGeGaaG zaVlaai2dacaaMb8UcdaWadaqaaKqzGeqbaeqabmWaaaWcbaqcLbsa caaMe8UaaGjbVlaayIW7caaIXaGaaGOlaiaaicdacaaIXaGaaGymai aaisdaaSqaaKqzGeGaeyOeI0IaaGimaiaai6cacaaIWaGaaGinaiaa iMdacaaI0aaaleaajugibiabgkHiTiaaicdacaaIUaGaaGymaiaaig dacaaIYaGaaGOnaaWcbaqcLbsacqGHsislcaaIWaGaaGOlaiaaicda caaI0aGaaGyoaiaaisdaaSqaaKqzGeGaaGjbVlaaysW7caaMi8UaaG imaiaai6cacaaIWaGaaGimaiaaiIdacaaIZaaaleaajugibiaaysW7 caaMe8UaaGjcVlaaicdacaaIUaGaaGimaiaaicdacaaIXaGaaGioaa WcbaqcLbsacqGHsislcaaIWaGaaGOlaiaaigdacaaIXaGaaGOmaiaa iAdaaSqaaKqzGeGaaGjbVlaaysW7caaMi8UaaGimaiaai6cacaaIWa GaaGimaiaaigdacaaI4aaaleaajugibiaaysW7caaMe8UaaGjcVlaa icdacaaIUaGaaGimaiaaigdacaaI2aGaaGOnaaaaaOGaay5waiaaw2 faaKqzGeGaaGzaVlabgkziUkaaygW7kmaadmaabaqcLbsafaqabeWa daaaleaajugibiaaysW7caaMe8UaaGjcVlaaigdaaSqaaKqzGeGaey OeI0IaaGimaiaai6cacaaI1aGaaG4maiaaiMdacaaIYaaaleaajugi biabgkHiTiaaicdacaaIUaGaaGioaiaaiAdacaaI5aGaaGimaaWcba qcLbsacqGHsislcaaIWaGaaGOlaiaaiwdacaaIZaGaaGyoaiaaikda aSqaaKqzGeGaaGymaaWcbaqcLbsacaaMe8UaaGjbVlaaicdacaaIUa GaaGymaiaaiwdacaaIZaGaaG4maaWcbaqcLbsacqGHsislcaaIWaGa aGOlaiaaiIdacaaI2aGaaGyoaiaaicdaaSqaaKqzGeGaaGjbVlaays W7caaIWaGaaGOlaiaaigdacaaI1aGaaG4maiaaiodaaSqaaKqzGeGa aGjbVlaaysW7caaMi8UaaGymaaaaaOGaay5waiaaw2faaKqzGeGaaG Olaaaa@F9C1@

 The following properties are evident.

  1. The elliptical Cauchy density for β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacuaHYoGyga Weaaaa@3A2B@ is given by equation (3) with k=3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadU gacaaMb8UaaGypaiaaygW7caaIZaaaaa@3E87@ and X'X MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaahI facaaINaGaaGzaVlaahIfaaaa@3CFC@ as listed for these data.
  2. The solution β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacuaHYoGyga Weaaaa@3A2B@ is both linear median–unbiased and modal–unbiased, and among all such estimators is most concentrated about β. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabek 7aIjaac6caaaa@3B52@
  3. The normal–theory confidence set {β( β β ) X'X( β β) S 2 c γ } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaiU hacqaHYoGycaaMb8UaeyicI4SaaGzaVlaaiIcakiqbek7aIzaataqc LbsacaaMb8UaeyOeI0IaaGzaVlabek7aIjqaiMcagaqbaiaahIfaca aINaGaaGzaVlaahIfacaaIOaGccuaHYoGygaWeaKqzGeGaaGzaVlab gkHiTiaaygW7cqaHYoGycaaIPaGaaGzaVlabgsMiJkaaygW7caWGtb GcdaahaaWcbeqcbauaaKqzadGaaGOmaaaajugibiaadogakmaaBaaa jeaqbaqcLbmacqaHZoWzaSqabaqcLbsacaaI9baaaa@656A@ holds exactly with confidence coefficient 1γ=0.95, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaig dacaaMb8UaeyOeI0IaaGzaVlabeo7aNjaaygW7caaI9aGaaGzaVlaa icdacaaIUaGaaGyoaiaaiwdacaaISaaaaa@46E7@ where S 2 =4.001 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaado fakmaaCaaaleqajeaqbaqcLbmacaaIYaaaaKqzGeGaaGzaVlaai2da caaMb8UaaGinaiaai6cacaaIWaGaaGimaiaaigdaaaa@4451@ is the residual mean square on ν=10 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabe2 7aUjaaygW7caaI9aGaaGzaVlaaigdacaaIWaaaaa@4007@ degrees of freedom, and c γ =3.71 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaado gakmaaBaaajeaqbaqcLbmacqaHZoWzaSqabaqcLbsacaaMb8UaaGyp aiaaygW7caaIZaGaaGOlaiaaiEdacaaIXaaaaa@4497@ is the upper 0.95 percentile for F(3,10,0). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadA eacaaIOaGaaG4maiaaiYcacaaIXaGaaGimaiaaiYcacaaIWaGaaGyk aiaai6caaaa@403F@
  4. As correlations are undefined, elements of the α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabeg 7aHbaa@3A9E@ –association matrix nonetheless do serve to quantify the degrees of association among [ β 0 , β 1 , β 2 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaiU fakiqbek7aIzaataWaaSbaaSqaaiaaicdaaeqaaKqzGeGaaGilaOGa fqOSdiMbambadaWgaaWcbaGaaGymaaqabaqcLbsacaaISaGccuaHYo GygaWeamaaBaaaleaacaaIYaaabeaajugibiaai2faaaa@45E8@ as in Definition 3, on taking α=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabeg 7aHjaaygW7caaI9aGaaGzaVlaaigdaaaa@3F34@ in Lemma 1.
  5. In particular, β 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacuaHYoGyga WeamaaBaaaleaacaaIWaaabeaaaaa@3B11@ is negatively associated with ( β 1 , β 2 ), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaiI cakiqbek7aIzaataWaaSbaaSqaaiaaigdaaeqaaKqzGeGaaGilaOGa fqOSdiMbambadaWgaaWcbaGaaGOmaaqabaqcLbsacaaIPaGaaGilaa aa@4247@ whereas ( β 1 , β 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaiI cakiqbek7aIzaataWaaSbaaSqaaiaaigdaaeqaaKqzGeGaaGilaOGa fqOSdiMbambadaWgaaWcbaGaaGOmaaqabaqcLbsacaaIPaaaaa@4191@ are themselves positively associated (Table 1).

Y i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadM fajuaGdaWgaaWcbaqcLbmacaaMb8UaamyAaaWcbeaaaaa@3E48@

X i1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadI fajuaGdaWgaaWcbaqcLbmacaWGPbGaaGymaaWcbeaaaaa@3D78@

X i2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadI fajuaGdaWgaaqcbauaaKqzadGaamyAaiaaikdaaKqaafqaaaaa@3DF7@

Y i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadM fajuaGdaWgaaqcbauaaKqzadGaaGzaVlaadMgaaKqaafqaaaaa@3EC6@

X i1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadI fakmaaBaaajeaqbaqcLbmacaWGPbGaaGymaaWcbeaaaaa@3D33@

X i2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeGabaamjKqzGe GaamiwaKqbaoaaBaaajeaqbaqcLbmacaWGPbGaaGOmaaqcbauabaaa aa@3E33@

0.5

1.74

3.3

31.2

6.32

5.42

0.9

6.22

8.41

38.4

10.52

4.63

0.4

1.19

11.6

26.7

1.22

5.85

0.4

4.1

6.62

25.9

6.32

8.72

0

4.08

4.42

25.2

4.15

7.6

0.7

10.15

4.83

35.7

1.72

3.12

0.5

1.7

5.3

 

 

 

Table 1 The viability ( Y i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaiI cacaWGzbqcfa4aaSbaaKqaafaajugWaiaaygW7caWGPbaajeaqbeaa jugibiaaiMcaaaa@40BA@ of n=13 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaad6 gacaaMb8UaaGypaiaaygW7caaIXaGaaG4maaaa@3F45@ biological specimens after storage under additives X i1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadI fajuaGdaWgaaqcbauaaKqzadGaamyAaiaaigdaaKqaafqaaaaa@3DF6@ and X i2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadI fajuaGdaWgaaqcbauaaKqzadGaamyAaiaaikdaaKqaafqaaaaa@3DF7@

Summary and discussion

This study offers further insight into the class SαS MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaabo facqaHXoqycaqGtbaaaa@3C4A@ comprising the spherical α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabeg 7aHbaa@3A9E@ –stable laws as limit distributions under conditions for central limit theory. In addition to their essential properties, expanded here to include representations for density functions, this study focuses on models of type {Y=Xβ+e} MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaiU hacaWHzbGaaGzaVlaai2dacaaMb8UaaCiwaiabek7aIjaaygW7cqGH RaWkcaaMb8UaaOyzaiaai2haaaa@4731@ when devoid of moments undergirding the classical theory. Recall that normal–theory procedures routinely are applied in practice as large–sample approximations in distributions attracted to Gaussian laws. Specifically, Berry–Esséen bounds on rates of convergence to Gaussian limits are given Jensen,32,33 with special reference to linear models in Jensen.34,35 Results here validate corresponding large–sample approximations for distributions attracted to  laws as cited in references.17–21 Of similar importance are rates of convergence to stable limits as in Paulauskas.22 By showing that many standard properties carry over in essence under significantly weakened assumptions, this study gives further credence to the widely and correctly held view that Gauss–Markov estimation and normal theory inferences extend considerably beyond the confines of the classical theory.

A appendix

The preceding study has developed exclusively around spherically dependent SαS MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaabo facqaHXoqycaqGtbaaaa@3C4A@ errors, as alternative to iid MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaabM gacaaMb8UaaeyAaiaaygW7caqGKbaaaa@3ED2@ stable errors. This choice is prompted by discrepancies encountered in the simplest case { Z i Z i +δ;i=1,2,,N} MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaiU hacaWGAbGcdaWgaaqcbauaaKqzadGaamyAaaWcbeaajugibiaaygW7 cqGHsgIRcaaMb8UaamOwaOWaaSbaaKqaafaajugWaiaadMgaaSqaba qcLbsacaaMb8Uaey4kaSIaaGzaVlabes7aKjaaiUdacaWGPbGaaGza Vlaai2dacaaMb8UaaGymaiaaiYcacaaIYaGaaGilaiablAciljaaiY cacaWGobGaaGyFaaaa@58D7@ with common location parameter. Essential details from Jensen14 may be summarized as follows. To distinguish the disparate properties of iid MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaabM gacaaMb8UaaeyAaiaaygW7caqGKbaaaa@3ED2@ vs spherical SαS MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaabo facqaHXoqycaqGtbaaaa@3C4A@ models, sequences ={ Z 1 , Z 2 , Z 3 ,} MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaatuuDJXwAK1 uy0HMmaeHbfv3ySLgzG0uy0HgiuD3BaGqbaKqzGeGae8hjHOLaaGza Vlaai2dacaaMb8UaaG4EaiaadQfakmaaBaaajeaqbaqcLbmacaaIXa aaleqaaKqzGeGaaGilaiaadQfakmaaBaaajeaqbaqcLbmacaaIYaaa leqaaKqzGeGaaGilaiaadQfakmaaBaaajeaqbaqcLbmacaaIZaaale qaaKqzGeGaaGilaiablAciljaai2haaaa@5879@ are fundamental in order to take limits. Of significance is that averages of SαS MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaqGtbGaeq ySdeMaae4uaaaa@3BBB@ sequences with α<2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabeg 7aHjaaygW7caaI8aGaaGzaVlaaikdaaaa@3F34@ may be inconsistent for iid MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaabM gacaaMb8UaaeyAaiaaygW7caqGKbaaaa@3ED2@ sequences but consistent under SαS MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaabo facqaHXoqycaqGtbaaaa@3C4A@ symmetry. Accordingly, let L ( Z N )=liminfL( Z N ). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaatuuDJXwAK1 uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqbaKqzGeGae8NeHWKcdaWg aaWcbaqcLbsacqGHEisPaSqabaqcLbsacaaIOaGaamOwaOWaaSbaaK qaafaajugWaiaad6eaaSqabaqcLbsacaaIPaGaaGzaVlaai2dacaaM b8UaciiBaiaacMgacaGGTbGaaiyAaiaac6gacaGGMbGae8NeHWKaaG ikaiaadQfakmaaBaaajeaqbaqcLbmacaWGobaaleqaaKqzGeGaaGyk aiaai6caaaa@5C35@ Essentials follow.

Lemma 2 Given ={ Z 1 , Z 2 , Z 3 ,}, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaatuuDJXwAK1 uy0HMmaeHbfv3ySLgzG0uy0HgiuD3BaGqbaKqzGeGae8hjHOLaaGza Vlaai2dacaaMb8UaaG4EaiaadQfakmaaBaaajeaqbaqcLbmacaaIXa aaleqaaKqzGeGaaGilaiaadQfakmaaBaaajeaqbaqcLbmacaaIYaaa leqaaKqzGeGaaGilaiaadQfakmaaBaaajeaqbaqcLbmacaaIZaaale qaaKqzGeGaaGilaiablAciljaai2hacaaISaaaaa@592F@ consider the case that Z'=[ Z 1 ,, Z N ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaahQ facaaINaGaaGzaVlaai2dacaaMb8UaaG4waiaayIW7caWGAbGcdaWg aaqcbauaaKqzadGaaGymaaWcbeaajugibiaaiYcacqWIMaYscaaISa GaamOwaOWaaSbaaKqaafaajugWaiaad6eaaSqabaqcLbsacaaMi8Ua aGyxaaaa@4DB0@ either are iid MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaabM gacaaMb8UaaeyAaiaaygW7caqGKbaaaa@3ED2@ S 1 α (δ,1), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaabo fakmaaDaaajeaybaqcLbmacaaIXaaajeaqbaqcLbmacqaHXoqyaaqc LbsacaaIOaGaeqiTdqMaaGilaiaayIW7caaIXaGaaGykaiaaiYcaaa a@46BC@ with chf ϕ Z i (t)=exp{ιtδ|t | α }, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabew 9aMPWaaSbaaKqaafaajugWaiaadQfajuaGdaWgaaqcbauaaKqzadGa amyAaaqcbauabaaaleqaaKqzGeGaaGzaVlaaiIcacaWG0bGaaGykai aaygW7caaI9aGaaGzaVlGacwgacaGG4bGaaiiCaiaaiUhacqaH5oqA caaMi8UaamiDaiaayIW7cqaH0oazcqGHsislcaaI8bGaamiDaiaaiY hakmaaCaaaleqajeaqbaqcLbmacqaHXoqyaaqcLbsacaaI9bGaaGil aaaa@5DE0@ or are SαS MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaabo facqaHXoqycaqGtbaaaa@3C4A@ on N MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaatuuDJXwAK1 uy0HMmaeHbfv3ySLgzG0uy0HgiuD3BaGqbaKqzGeGae8xhHiLcdaah aaWcbeqcbauaaKqzadGaamOtaaaaaaa@4639@ with chf ϕ Z (t)=exp{ιδt' 1 N (t't) α 2 }. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabew 9aMPWaaSbaaKqaafaajugWaiaahQfaaSqabaqcLbsacaaMb8UaaGik aiaahshacaaIPaGaaGzaVlaai2dacaaMb8UaciyzaiaacIhacaGGWb GaaG4EaiabeM7aPjaayIW7cqaH0oazcaWH0bGaaG4jaiaahgdakmaa BaaajeaqbaqcLbmacaaMb8UaamOtaaWcbeaajugibiaaygW7cqGHsi slcaaMb8UaaGikaiaahshacaaINaGaaCiDaiaaiMcakmaaCaaaleqa jeaqbaqcfa4aaSaaaKqaafaajugWaiabeg7aHbqcbauaaKqzadGaaG OmaaaaaaqcLbsacaaI9bGaaGOlaaaa@663E@ Let S N =( Z 1 ++ Z N ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaado fakmaaBaaajeaqbaqcLbmacaWGobaaleqaaKqzGeGaaGzaVlaai2da caaMb8UaaGikaiaadQfakmaaBaaajeaqbaqcLbmacaaIXaaaleqaaK qzGeGaey4kaSIaeSOjGSKaey4kaSIaamOwaOWaaSbaaKqaafaajugW aiaad6eaaSqabaqcLbsacaaIPaaaaa@4CD3@ and Z ¯ N = N 1 S N , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaqdaaqaaK qzGeGaamOwaaaakmaaBaaaleaakmaaBaaajeaqbaqcLbmacaWGobaa leqaaaqabaqcLbsacaaMb8UaaGypaiaaygW7caWGobGcdaahaaWcbe qcbauaaKqzadGaeyOeI0IaaGymaaaajugibiaadofakmaaBaaajeaq baqcLbmacaWGobaaleqaaKqzGeGaaGilaaaa@4A67@ and consider the standardized variables U N = N 1 2 ( Z ¯ N δ). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadw fakmaaBaaajeaqbaqcLbmacaWGobaaleqaaKqzGeGaaGzaVlaai2da caaMb8UaamOtaOWaaWbaaSqabKqaafaajuaGdaWcaaqcbauaaKqzad GaaGymaaqcbauaaKqzadGaaGOmaaaaaaqcLbsacaaIOaGcdaqdaaqa aKqzGeGaamOwaaaakmaaBaaaleaakmaaBaaajeaqbaqcLbmacaWGob aaleqaaaqabaqcLbsacaaMb8UaeyOeI0IaaGzaVlabes7aKjaaiMca caaIUaaaaa@543E@

  1. Consistent and inconsistent properties of for Z ¯ N MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaqdaaqaaK qzGeGaamOwaaaakmaaBaaaleaakmaaBaaajiaybaqcLbmacaWGobaa meqaaaWcbeaaaaa@3CD3@ sequences are as follow.
  2.  For 0<α<1: MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaic dacaaMb8UaaGipaiaaygW7cqaHXoqycaaMb8UaaGipaiaaygW7caaI XaGaaGOoaaaa@448B@ ϕ Z ¯ N (t)= e ιtδ N ε |t | α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeGabaaokKqzGe Gaeqy1dywcfa4aaSbaaKqaafaajuaGdaqdaaqcbauaaKqzadGaamOw aaaajuaGdaWgaaqcbauaaKqbaoaaBaaajeaqbaqcLbmacaWGobaaje aqbeaaaeqaaaqabaqcLbsacaaMb8UaaGikaiaadshacaaIPaGaaGza Vlaai2dacaaMb8UaamyzaOWaaWbaaSqabKqaafaajugWaiabeM7aPj aadshacqaH0oazcaaMb8UaeyOeI0IaaGzaVlaad6eajuaGdaahaaqc bauabeaajugWaiabew7aLbaacaaI8bGaamiDaiaaiYhajuaGdaahaa qcbauabeaajugWaiabeg7aHbaaaaaaaa@61DF@ for ε>0, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabew 7aLjaaygW7caaI+aGaaGzaVlaaicdacaaISaaaaa@3FF2@ so that Z ¯ N MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaqdaaqaaK qzGeGaamOwaaaakmaaBaaaleaakmaaBaaajiaybaqcLbmacaWGobaa meqaaaWcbeaaaaa@3CD3@ is inconsistent for δ. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabes 7aKjaai6caaaa@3B5C@

     For α=1, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabeg 7aHjaaygW7caaI9aGaaGzaVlaaigdacaaISaaaaa@3FEA@ ϕ Z ¯ N (t)= e ιtδ|t | α ϕ Z i (t), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabew 9aMPWaaSbaaKqaafaajuaGdaqdaaqcbauaaKqzadGaamOwaaaajuaG daWgaaqcbauaaKqbaoaaBaaajeaqbaqcLbmacaWGobaajeaqbeaaae qaaaWcbeaajugibiaaygW7caaIOaGaamiDaiaaiMcacaaMb8UaaGyp aiaaygW7caWGLbGcdaahaaWcbeqcbauaaKqzadGaeqyUdKMaamiDai abes7aKjaaygW7cqGHsislcaaMb8UaaGiFaiaadshacaaI8bqcfa4a aWbaaKqaafqabaqcLbmacqaHXoqyaaaaaKqzGeGaaGzaVlabggMi6k aaygW7cqaHvpGzkmaaBaaajeaqbaqcLbmacaWGAbqcfa4aaSbaaKqa afaajugWaiaadMgaaKqaafqaaaWcbeaajugibiaaygW7caaIOaGaam iDaiaaiMcacaaISaaaaa@6ECC@ so that Z ¯ N MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaqdaaqaaK qzGeGaamOwaaaakmaaBaaaleaakmaaBaaajiaybaqcLbmacaWGobaa meqaaaWcbeaaaaa@3CD3@ is inconsistent for

     For 1<α2, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaig dacaaMb8UaaGipaiaaygW7cqaHXoqycaaMb8UaeyizImQaaGzaVlaa ikdacaaISaaaaa@456E@ ϕ Z ¯ N (t)= e ιtδ N ε |t | α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabew 9aMPWaaSbaaKqaafaajuaGdaqdaaqcbauaaKqzadGaamOwaaaajuaG daWgaaqcbauaaKqbaoaaBaaajeaqbaqcLbmacaWGobaajeaqbeaaae qaaaWcbeaajugibiaaygW7caaIOaGaamiDaiaaiMcacaaMb8UaaGyp aiaaygW7caWGLbGcdaahaaWcbeqcbauaaKqzadGaeqyUdKMaamiDai abes7aKjaaygW7cqGHsislcaaMb8UaamOtaKqbaoaaCaaajeaqbeqa aKqzadGaeyOeI0IaeqyTdugaaiaaiYhacaWG0bGaaGiFaKqbaoaaCa aajeaqbeqaaKqzadGaeqySdegaaaaaaaa@620E@ for ε>0, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabew 7aLjaaygW7caaI+aGaaGzaVlaaicdacaaISaaaaa@3FF2@ so that Z ¯ N MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaqdaaqaaK qzGeGaamOwaaaakmaaBaaaleaakmaaBaaajeaqbaqcLbmacaWGobaa leqaaaqabaaaaa@3CA6@ is consistent for δ. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabes 7aKjaai6caaaa@3B5C@

  3. For SaS MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaado facaWGHbGaam4uaaaa@3B95@ sequences Z ¯ N MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaqdaaqaaK qzGeGaamOwaaaakmaaBaaaleaakmaaBaaajiaybaqcLbmacaWGobaa meqaaaWcbeaaaaa@3CD3@ is consistent for δ. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabes 7aKjaai6caaaa@3B5C@ for every 0<α2. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaic dacaaMb8UaaGipaiaaygW7cqaHXoqycaaMb8UaeyizImQaaGzaVlaa ikdacaaIUaaaaa@456F@
  4. For iid sequences with 0<α<2, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaic dacaaMb8UaaGipaiaaygW7cqaHXoqycaaMb8UaaGipaiaaygW7caaI YaGaaGilaaaa@447E@ L ( U N ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaatuuDJXwAK1 uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqbaKqzGeGae8NeHWKcdaWg aaqcbauaaKqzadGaeyOhIukaleqaaKqzGeGaaGikaiaadwfakmaaBa aajeaqbaqcLbmacaWGobaaleqaaKqzGeGaaGykaaaa@4C8A@ diverges to an improper distribution.
  5. For SαS MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaabo facqaHXoqycaqGtbaaaa@3C4A@ sequences liminfL( U N )L( Z i ), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiGacY gacaGGPbGaaiyBaiaacMgacaGGUbGaaiOzamrr1ngBPrwtHrhAXaqe guuDJXwAKbstHrhAG8KBLbacfaGae8NeHWKaaGikaiaadwfakmaaBa aajeaqbaqcLbmacaWGobaaleqaaKqzGeGaaGykaiaaygW7cqGHHjIU caaMb8Uae8NeHWKaaGikaiaadQfakmaaBaaajeaqbaqcLbmacaWGPb aaleqaaKqzGeGaaGykaiaaiYcaaaa@5A7B@ the limit being identical to each component.

Acknowledgement

None.

Conflict of interest

Authors declare that there is no conflict of interest.

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