eISSN: 2378-315X BBIJ

Biometrics & Biostatistics International Journal
Research Article
Volume 7 Issue 3

On local polynomial estimation of hazard rates and their derivatives under left truncation and right censoring models

Jiancheng Jiang,1 Lingju Chen,2 Yuze Yuan2
1Department of Mathematics and Statistics, University of North Carolina at Charlotte, USA
2Department of Mathematics, Minjiang University, China
Received: April 28, 2018 | Published: May 15, 2018

Correspondence: Yuze Yuan, Department of Mathematics, Minjiang University, China. Email

Citation: Jiang J, Chen L, Yuan Y. On local polynomial estimation of hazard rates and their derivatives under left truncation and right censoring models. Biom Biostat Int J. 2018;7(3):199‒203. DOI: 10.15406/bbij.2018.07.00209

Abstract

Estimating hazard rate function is an important problem in survival analysis. There are some estimation approaches based on kernel smoothing. However, they suffer from the boundary effects or need high order kernels, which increases the mean squared error. We introduce local polynomial estimators of hazard rates and their derivatives for the left truncation and right censoring models. The estimators have favorable properties similar to those of local polynomial regression estimators. Asymptotic expressions for the mean squared errors (AMSE’s) are obtained. Consistency and joint asymptotic normality of the local polynomial estimators are established. A data-based local bandwidth selection rule is proposed.

Keywords: censoring, data-driven local bandwidth selection, hazard rate, local polynomial, truncation

Introduction

Consider a subject in survival studies. Only if its onset time, i.e. the time origin of its lifetime, passes the beginning of the study, the subject can enter into the study. For the entered individuals, each of them is then followed for a fixed time point. Such subjects are so-called left truncated and right censored. To be specific, let X MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadI faaaa@39DC@  be the lifetime with distribution function (df) F(x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadA eacaaIOaGaamiEaiaaiMcaaaa@3C2C@ , T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaads faaaa@39D8@  the random left truncation time with arbitrary df G(x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadE eacaaIOaGaamiEaiaaiMcaaaa@3C2D@ , and C MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaado eaaaa@39C7@  the random censoring time with arbitrary df L(x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadY eacaaIOaGaamiEaiaaiMcaaaa@3C32@ , where s MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaado haaaa@39F7@  is independent of (T,C) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaiI cacaWGubGaaGilaiaadoeacaaIPaaaaa@3CBB@ . Then the cumulative hazard function of F(x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadA eacaaIOaGaamiEaiaaiMcaaaa@3C2C@  is Λ(x)= 0 x dF(t)/[1F(t)] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabfU 5amjaaiIcacaWG4bGaaGykaiaai2dakmaapedabeqcbauaaKqzadGa aGimaaqcbauaaKqzadGaamiEaaqcLbsacqGHRiI8aiaadsgacaWGgb GaaGikaiaadshacaaIPaGaaG4laiaaiUfacaaIXaGaeyOeI0IaamOr aiaaiIcacaWG0bGaaGykaiaai2faaaa@5057@ . Under the left truncation and right censoring model, one observes (Y,T,δ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaiI cacaWGzbGaaGilaiaadsfacaaISaGaeqiTdqMaaGykaaaa@3F2C@  if YT MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadM facqGHLjYScaWGubaaaa@3C7C@ , where Y=min(X,C) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadM facaaI9aGaciyBaiaacMgacaGGUbGaaGikaiaadIfacaaISaGaam4q aiaaiMcaaaa@4136@  and δ=I(XC) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabes 7aKjaai2dacaWGjbGaaGikaiaadIfacqGHKjYOcaWGdbGaaGykaaaa @40F8@  is an indicator of the censoring status of X MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadI faaaa@39DC@  which takes value one if XC MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadI facqGHKjYOcaWGdbaaaa@3C59@  and zero otherwise. When Y<T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadM facaaI8aGaamivaaaa@3B7C@ , nothing is observed (see for example, Gurler & Wang1). Let the distribution of Y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadM faaaa@39DD@  be W(y) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadE facaaIOaGaamyEaiaaiMcaaaa@3C3E@  and assume that α=P(TY)>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabeg 7aHjaai2dacaWGqbGaaGikaiaadsfacqGHKjYOcaWGzbGaaGykaiaa i6dacaaIWaaaaa@428D@ . Then W ¯ = F ¯ L ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiqadE fagaqeaiaai2daceWGgbGbaebaceWGmbGbaebaaaa@3C86@ , where and throughout the paper for any df. E MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadw eaaaa@39C9@ , E ¯ =1E MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiqadw eagaqeaiaai2dacaaIXaGaeyOeI0Iaamyraaaa@3D1A@  is the corresponding survival function.

The left truncation and right censoring model has been investigated by many authors. Interesting work along the field can be found in Gross & Lai2 and Gurler & Wang1 among others. Several authors have considered the estimation of hazard functions under the left truncation and right censoring model. For examples, Uzunogullari & Wang,3 and Wu & Wells.4

In the present investigation, we study local polynomial (LP for short) estimators of hazard functions and their derivatives based on the i.i.d sample {( Y i , T i , δ i )} i n =1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaiU hacaaIOaGaamywaKqbaoaaBaaajeaqbaqcLbmacaWGPbaajeaqbeaa jugibiaaiYcacaWGubqcfa4aaSbaaKqaafaajugWaiaadMgaaKqaaf qaaKqzGeGaaGilaiabes7aKLqbaoaaBaaajeaqbaqcLbmacaWGPbaa jeaqbeaajugibiaaiMcacaaI9bGcdaqhaaWcbaqcLbmacaWGPbaaje aqbaqcLbmacaWGUbaaaOGaeyypa0JaaGymaaaa@538B@  from (Y,T,δ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaiI cacaWGzbGaaGilaiaadsfacaaISaGaeqiTdqMaaGykaaaa@3F2C@ . Under the left truncation and right censoring model, one observes only those i.i.d pairs ( Y i , T i , δ i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaiI cacaWGzbGcdaWgaaqcbauaaKqzadGaamyAaaWcbeaajugibiaaiYca caWGubGcdaWgaaqcbawaaKqzGdGaamyAaaqcbawabaqcLbsacaaISa GaeqiTdqMcdaWgaaqcbauaaKqzadGaamyAaaWcbeaajugibiaaiMca aaa@494C@  for which Y i T i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadM fakmaaBaaajeaqbaqcLbmacaWGPbaaleqaaKqzGeGaeyyzImRaamiv aOWaaSbaaKqaafaajugWaiaadMgaaSqabaaaaa@4243@ . It is worth pointing out that our estimators inherit some favorable properties from local polynomial regression estimators, in particular, our estimators can reduce the bias according to the degree of the polynomial without increasing the variance and automatically correct the left boundary effect. The point wise asymptotic normality of our estimators enables one to find the asymptotically optimal variable bandwidth choice, and thereafter allows one to develop a data-driven optimal local bandwidth selector by using the ideas of Fan & Gurler.5 We here present a simpler data-driven method for choosing the local bandwidth.

The outline of this paper is as follows. In Section 2, we introduce the LP estimators. In Section 3 we concentrate on the asymptotic properties of the proposed estimators, including point wise strong consistency and joint asymptotic normality. In Section 4, we propose the data-driven local bandwidth selection rule. Technical proofs are given in the Appendix.

Estimation

In order to introduce the LP estimators, we use the following notation from Gurler & Wang1

  1. C(z)=P(TzY|YT)= α 1 P(TzC)[1F(z)] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaado eacaaIOaGaamOEaiaaiMcacaaI9aGaamiuaiaaiIcacaWGubGaeyiz ImQaamOEaiabgsMiJkaadMfacaaI8bGaamywaiabgwMiZkaadsfaca aIPaGaaGypaiabeg7aHPWaaWbaaSqabKqaafaajugWaiabgkHiTiaa igdaaaqcLbsacaWGqbGaaGikaiaadsfacqGHKjYOcaWG6bGaeyizIm Qaam4qaiaaiMcacaaIBbGaaGymaiabgkHiTiaadAeacaaIOaGaamOE aiaaiMcacaaIDbaaaa@5F02@
  2. C n (z)=(n+ 1) 1 i=1 n I( T i z Y i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaado eakmaaBaaajeaqbaqcLbmacaWGUbaaleqaaKqzGeGaaGikaiaadQha caaIPaGaaGypaiaaiIcacaWGUbGaey4kaSIaaGymaiaaiMcajuaGda ahaaqcbauabeaajugWaiabgkHiTiaaigdaaaGcdaaeWaqabKqaafaa jugWaiaadMgacaaI9aGaaGymaaqcbauaaKqzadGaamOBaaqcLbsacq GHris5aiaadMeacaaIOaGaamivaOWaaSbaaKqaafaajugWaiaadMga aSqabaqcLbsacqGHKjYOcaWG6bGaeyizImQaamywaOWaaSbaaKqaaf aajugWaiaadMgaaSqabaqcLbsacaaIPaaaaa@5F67@ , the modified empirical estimator of C(z) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaado eacaaIOaGaamOEaiaaiMcaaaa@3C2B@ .
  3. F n (x)=1 Π Y i x (1 [n C n ( Y i )] 1 ) δ i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadA eakmaaBaaajeaqbaqcLbmacaWGUbaaleqaaKqzGeGaaGikaiaadIha caaIPaGaaGypaiaaigdacqGHsislcqqHGoaukmaaBaaajeaqbaqcLb macaWGzbqcfa4aaSbaaKqaafaajugWaiaadMgaaKqaafqaaKqzadGa eyizImQaamiEaaWcbeaajugibiaaiIcacaaIXaGaeyOeI0IaaG4wai aad6gacaWGdbGcdaWgaaqcbauaaKqzadGaamOBaaWcbeaajugibiaa iIcacaWGzbGcdaWgaaqcbauaaKqzadGaamyAaaWcbeaajugibiaaiM cacaaIDbGcdaahaaWcbeqcbauaaKqzadGaeyOeI0IaaGymaaaajugi biaaiMcakmaaCaaaleqajeaqbaqcLbmacqaH0oazjuaGdaWgaaqcba uaaKqzadGaamyAaaqcbauabaaaaaaa@676E@ , the product-limit estimator of F(x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadA eacaaIOaGaamiEaiaaiMcaaaa@3C2C@  from Tsai.

Let a E =inf{t:E(t)>0} MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadg gakmaaBaaajeaqbaqcLbmacaWGfbaaleqaaKqzGeGaaGypaiGacMga caGGUbGaaiOzaiaaiUhacaWG0bGaaGOoaiaaysW7caWGfbGaaGikai aadshacaaIPaGaaGOpaiaaicdacaaI9baaaa@4A7E@  and b E =inf{t:E(t)=1} MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadk gakmaaBaaajeaqbaqcLbmacaWGfbaaleqaaKqzGeGaaGypaiGacMga caGGUbGaaiOzaiaaiUhacaWG0bGaaGOoaiaaysW7caWGfbGaaGikai aadshacaaIPaGaaGypaiaaigdacaaI9baaaa@4A7F@  denote the left and right endpoints of the support for any d.f. E(x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadw eacaaIOaGaamiEaiaaiMcaaaa@3C2B@ , respectively. Then F(x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadA eacaaIOaGaamiEaiaaiMcaaaa@3C2C@  is identifiable if a G a W MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadg gakmaaBaaajeaqbaqcLbmacaWGhbaaleqaaKqzGeGaeyizImQaamyy aOWaaSbaaKqaafaajugWaiaadEfaaSqabaaaaa@4213@  and b G b W MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadk gakmaaBaaajeaqbaqcLbmacaWGhbaaleqaaKqzGeGaeyizImQaamOy aOWaaSbaaKqaafaajugWaiaadEfaaSqabaaaaa@4215@  (Woodroofe6 and Gurler & Wang1).

Therefore, we assume this condition holds. As in Gurler & Wang,1 for estimating the density function f(x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadA gacaaIOaGaamiEaiaaiMcaaaa@3C4C@ of X MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadI faaaa@39DC@ , we also assume a G a W MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadg gakmaaBaaajeaqbaqcLbmacaWGhbaaleqaaKqzGeGaeyizImQaamyy aOWaaSbaaKqaafaajugWaiaadEfaaSqabaaaaa@4213@ .

Following Gurler & Wang,1 we define W 1 (y)=P(Yy,δ=1|YT) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadE fajuaGdaWgaaqcbauaaKqzadGaaGymaaqcbauabaqcLbsacaaIOaGa amyEaiaaiMcacaaI9aGaamiuaiaaiIcacaWGzbGaeyizImQaamyEai aaiYcacqaH0oazcaaI9aGaaGymaiaaiYhacaWGzbGaeyyzImRaamiv aiaaiMcaaaa@4EEB@  and W 1n (y)= n 1 i=1 n I( Y i y, δ i =1). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadE fakmaaBaaajeaqbaqcLbmacaaIXaGaamOBaaWcbeaajugibiaaiIca caWG5bGaaGykaiaai2dacaWGUbGcdaahaaWcbeqcbauaaKqzadGaey OeI0IaaGymaaaakmaaqadabeqcbauaaKqzadGaamyAaiaai2dacaaI XaaajeaqbaqcLbmacaWGUbaajugibiabggHiLdGaamysaiaaiIcaca WGzbGcdaWgaaqcbauaaKqzGdGaamyAaaWcbeaajugibiabgsMiJkaa dMhacaaISaGaeqiTdqMcdaWgaaqcbauaaKqzadGaamyAaaWcbeaaju gibiaai2dacaaIXaGaaGykaiaai6caaaa@5EE0@  Then d W 1 (y)= α 1 P(TYC)dF(y) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaads gacaWGxbGcdaWgaaqcbauaaKqzadGaaGymaaWcbeaajugibiaaiIca caWG5bGaaGykaiaai2dacqaHXoqykmaaCaaaleqajeaqbaqcLbmacq GHsislcaaIXaaaaKqzGeGaamiuaiaaiIcacaWGubGaeyizImQaamyw aiabgsMiJkaadoeacaaIPaGaamizaiaadAeacaaIOaGaamyEaiaaiM caaaa@52A5@  and

Λ(x)= 0 x d W 1 (y)/C(y). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabfU 5amjaaiIcacaWG4bGaaGykaiaai2dakmaapedabeqcbauaaKqzadGa aGimaaqcbauaaKqzadGaamiEaaqcLbsacqGHRiI8aiaadsgacaWGxb GcdaWgaaqcbauaaKqzadGaaGymaaWcbeaajugibiaaiIcacaWG5bGa aGykaiaai+cacaWGdbGaaGikaiaadMhacaaIPaGaaGOlaaaa@50AB@

By Gurler & Wang,1 the Nelson-Aalen type estimator of Λ(x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabfU 5amjaaiIcacaWG4bGaaGykaaaa@3CD6@  is given by

Λ n (x)= 0 x [ C n (y)] 1 d W 1n (y)= i=1 n I( Y i x, δ i =1) n C n ( Y i ) . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabfU 5amPWaaSbaaKqaafaajugWaiaad6gaaSqabaqcLbsacaaIOaGaamiE aiaaiMcacaaI9aGcdaWdXaqabKqaafaajugWaiaaicdaaKqaafaaju g4aiaadIhaaKqzGeGaey4kIipacaaIBbGaam4qaOWaaSbaaKqaafaa jugWaiaad6gaaSqabaqcLbsacaaIOaGaamyEaiaaiMcacaaIDbGcda ahaaWcbeqcbauaaKqzadGaeyOeI0IaaGymaaaajugibiaadsgacaWG xbGcdaWgaaqcbauaaKqzadGaaGymaiaad6gaaSqabaqcLbsacaaIOa GaamyEaiaaiMcacaaI9aGcdaaeWbqabKqaafaajugWaiaadMgacaaI 9aGaaGymaaqcbauaaKqzadGaamOBaaqcLbsacqGHris5aOWaaSaaae aajugibiaadMeacaaIOaGaamywaOWaaSbaaKqaafaajugWaiaadMga aSqabaqcLbsacqGHKjYOcaWG4bGaaGilaiaaysW7cqaH0oazkmaaBa aajeaqbaqcLbmacaWGPbaaleqaaKqzGeGaaGypaiaaigdacaaIPaaa keaajugibiaad6gacaWGdbGcdaWgaaqcbauaaKqzadGaamOBaaWcbe aajugibiaaiIcacaWGzbGcdaWgaaqcbauaaKqzadGaamyAaaWcbeaa jugibiaaiMcaaaGaaGOlaaaa@8413@       (1)

Gijbel & Wang1 considered the following kernel estimator for the density function f(x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadA gacaaIOaGaamiEaiaaiMcaaaa@3C4C@  of X MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadI faaaa@39DC@ , which is a convolution of the product-limit estimator F n (x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadA eakmaaBaaajeaqbaqcLbmacaWGUbaaleqaaKqzGeGaaGikaiaadIha caaIPaaaaa@3F5C@  with an appropriate kernel function K ν MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadU eakmaaBaaajeaqbaqcLbmacqaH9oGBaSqabaaaaa@3D35@ :

f ^ (ν) (x)= 1 b n ν+1 K ν ( xu b n )d F n (u), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiqadA gagaqcaKqbaoaaCaaajeaqbeqaaKqzadGaaGikaiabe27aUjaaiMca aaqcLbsacaaIOaGaamiEaiaaiMcacaaI9aGcdaWcaaqaaKqzGeGaaG ymaaGcbaqcLbsacaWGIbGcdaqhaaqcbauaaKqzadGaamOBaaqcbaua aKqzadGaeqyVd4Maey4kaSIaaGymaaaaaaGcdaWdbaqabSqabeqaju gibiabgUIiYdGaam4saOWaaSbaaKqaafaajugWaiabe27aUbWcbeaa jugibiaaiIcakmaalaaabaqcLbsacaWG4bGaeyOeI0IaamyDaaGcba qcLbsacaWGIbGcdaWgaaqcbauaaKqzadGaamOBaaWcbeaaaaqcLbsa caaIPaGaamizaiaadAeakmaaBaaajeaqbaqcLbmacaWGUbaaleqaaK qzGeGaaGikaiaadwhacaaIPaGaaGilaaaa@6627@      (2)

where K ν MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadU eakmaaBaaajeaqbaqcLbmacqaH9oGBaSqabaaaaa@3D35@  is a higher order kernel. The method can be also adapted to the case of estimating the hazard function λ(x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabeU 7aSjaaiIcacaWG4bGaaGykaaaa@3D15@  and its derivatives if one uses Λ n (x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabfU 5amPWaaSbaaKqaafaajugWaiaad6gaaSqabaqcLbsacaaIOaGaamiE aiaaiMcaaaa@4006@  instead of F n (x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadA eakmaaBaaajeaqbaqcLboacaWGUbaaleqaaKqzGeGaaGikaiaadIha caaIPaaaaa@3F7C@ :

λ ^ (x)= 1 b n ν+1 K ν ( xu b n )d Λ n (u). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiqbeU 7aSzaajaGaaGikaiaadIhacaaIPaGaaGypaOWaaSaaaeaajugibiaa igdaaOqaaKqzGeGaamOyaKqbaoaaDaaajeaqbaqcLbmacaWGUbaaje aqbaqcLbmacqaH9oGBcqGHRaWkcaaIXaaaaaaakmaapeaabeWcbeqa bKqzGeGaey4kIipacaWGlbGcdaWgaaqcbauaaKqzadGaeqyVd4gale qaaKqzGeGaaGikaOWaaSaaaeaajugibiaadIhacqGHsislcaWG1baa keaajugibiaadkgakmaaBaaajeaqbaqcLbmacaWGUbaaleqaaaaaju gibiaaiMcacaWGKbGaeu4MdWKcdaWgaaqcbauaaKqzadGaamOBaaWc beaajugibiaaiIcacaWG1bGaaGykaiaai6caaaa@624C@     (3)

The estimator is an extension to that of Müller & Wang7,8 where right censoring model is considered. However, for the estimation of derivatives or reduction of bias, the estimator needs higher order kernels, which can lead to a negative hazard rate estimator. The practical advantages of using higher order kernels can be quite small for moderate sample sizes as demonstrated in Marron & Wand.9 When estimating at point x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadI haaaa@39FC@  near a G MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadg gakmaaBaaajeaqbaqcLbmacaWGhbaaleqaaaaa@3C5F@  or b W MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadk gakmaaBaaajeaqbaqcLbmacaWGxbaaleqaaaaa@3C70@ , the effective support [x b n ,x+ b n ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaiU facaWG4bGaeyOeI0IaamOyaOWaaSbaaKqaafaajugWaiaad6gaaSqa baqcLbsacaaISaGaamiEaiabgUcaRiaadkgakmaaBaaajeaqbaqcLb macaWGUbaaleqaaKqzGeGaaGyxaaaa@4778@  of the kernel is not contained in [ a G , b W ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaiU facaWGHbGcdaWgaaqcbauaaKqzadGaam4raaWcbeaajugibiaaiYca caWGIbGcdaWgaaqcbauaaKqzGdGaam4vaaWcbeaajugibiaai2faaa a@4390@ , most kernel estimators in density estimation and regression settings will encounter boundary effects. The estimator (3) suffers from boundary effects near the endpoints of the support of the hazard rates. In the presence of censoring for estimating hazard rate function, Müller & Wang9 solved the problem by employing boundary kernels and a data-adaptive varying bandwidth selection procedure. Hall & Wehrly10 used a geometrical method for removing edge effects from kernel-type nonparametric regression estimators. These boundary correction methods may also be adapted to the estimation approach in (3). Here we introduce a simple and intuitive approach to the problem. Our approach does not need higher order kernels or boundary kernels while automatically correcting the boundary effects. Our idea is similar to that of Jiang & Doksum,11 but their procedure cannot be directly applied to the current setting.

Following Jiang and Doksum,11 we consider the following optimization problem:

min a j 1 b n K( ux b n )[λ(u) j=0 p a j (ux) j ] 2 du. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaGfqbqabK qaafaajugWaiaadggajuaGdaWgaaqcbauaaKqzadGaamOAaaqcbaua baaaleqakeaajugibiGac2gacaGGPbGaaiOBaaaakmaapeaabeWcbe qabKqzGeGaey4kIipakmaalaaabaqcLbsacaaIXaaakeaajugibiaa dkgakmaaBaaajeaqbaqcLbmacaWGUbaaleqaaaaajugibiaadUeaca aIOaGcdaWcaaqaaKqzGeGaamyDaiabgkHiTiaadIhaaOqaaKqzGeGa amOyaOWaaSbaaKqaafaajugWaiaad6gaaSqabaaaaKqzGeGaaGykai aaiUfacqaH7oaBcaaIOaGaamyDaiaaiMcacqGHsislkmaaqahabeqc bauaaKqzadGaamOAaiaai2dacaaIWaaajeaqbaqcLbmacaWGWbaaju gibiabggHiLdGaamyyaOWaaSbaaKqaafaajugWaiaadQgaaSqabaqc LbsacaaIOaGaamyDaiabgkHiTiaadIhacaaIPaGcdaahaaWcbeqcba uaaKqzadGaamOAaaaajugibiaai2fakmaaCaaaleqajeaqbaqcLbma caaIYaaaaKqzGeGaamizaiaadwhacaaIUaaaaa@7623@         (4)

By Taylor expansion, the solution of the optimization problem, a * (x) ( a 0 * ,, a p * ) T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaahg gajuaGdaahaaqcbauabeaajugWaiaaiQcaaaqcLbsacaaIOaGaamiE aiaaiMcacqGHHjIUcaaIOaGaamyyaKqbaoaaDaaajeaqbaqcLbmaca aIWaaajeaqbaqcLbmacaaIQaaaaKqzGeGaaGilaiabl+UimjaaiYca caWGHbqcfa4aa0baaKqaafaajugWaiaadchaaKqaafaajugWaiaaiQ caaaqcLbsacaaIPaGcdaahaaWcbeqcbauaaKqzadGaamivaaaaaaa@5607@ , will estimate a(x) (λ(x),, λ (p) (x)/p!) T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaahg gacaaIOaGaamiEaiaaiMcacqGHHjIUcaaIOaGaeq4UdWMaaGikaiaa dIhacaaIPaGaaGilaiabl+UimjaaiYcacqaH7oaBkmaaCaaaleqaje aqbaqcLbmacaaIOaGaamiCaiaaiMcaaaqcLbsacaaIOaGaamiEaiaa iMcacaaIVaGaamiCaiaaigcacaaIPaGcdaahaaWcbeqcbauaaKqzad Gaamivaaaaaaa@5478@ . Since Λ n (x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabfU 5amPWaaSbaaKqaafaajugWaiaad6gaaSqabaqcLbsacaaIOaGaamiE aiaaiMcaaaa@4006@  in (1) is the empirical estimator of Λ(x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabfU 5amjaaiIcacaWG4bGaaGykaaaa@3CD6@ , we define the following generalized empirical hazard rate as the generalized derivative of Λ n (x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabfU 5amPWaaSbaaKqaafaajugWaiaad6gaaSqabaqcLbsacaaIOaGaamiE aiaaiMcaaaa@4006@ :

λ n (x)= i=1 n D(x Y i ) δ i n C n ( Y i ) , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabeU 7aSPWaaSbaaKqaafaajugWaiaad6gaaSqabaqcLbsacaaIOaGaamiE aiaaiMcacaaI9aGcdaaeWbqabKqaafaajugWaiaadMgacaaI9aGaaG ymaaqcbauaaKqzadGaamOBaaqcLbsacqGHris5aOWaaSaaaeaajugi biaadseacaaIOaGaamiEaiabgkHiTiaadMfakmaaBaaajeaqbaqcLb macaWGPbaaleqaaKqzGeGaaGykaiabes7aKPWaaSbaaKqaafaajugW aiaadMgaaSqabaaakeaajugibiaad6gacaWGdbGcdaWgaaqcbauaaK qzadGaamOBaaWcbeaajugibiaaiIcacaWGzbGcdaWgaaqcbauaaKqz adGaamyAaaWcbeaajugibiaaiMcaaaGaaGilaaaa@62D6@                   (5)

where D(x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaads eacaaIOaGaamiEaiaaiMcaaaa@3C2A@  is the Dirac function with the following property:

g(u)D(ux)du=g(x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWdbaqabS qabeqajugibiabgUIiYdGaam4zaiaaiIcacaWG1bGaaGykaiaadsea caaIOaGaamyDaiabgkHiTiaadIhacaaIPaGaamizaiaadwhacaaI9a Gaam4zaiaaiIcacaWG4bGaaGykaaaa@4943@

For any integral function g(x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadE gacaaIOaGaamiEaiaaiMcaaaa@3C4D@ . Then 0 x λ n (t)dt= Λ n (x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWdXaqabK qaafaajugWaiaaicdaaKqaafaajugWaiaadIhaaKqzGeGaey4kIipa cqaH7oaBkmaaBaaajeaqbaqcLbmacaWGUbaaleqaaKqzGeGaaGikai aadshacaaIPaGaamizaiaadshacaaI9aGaeu4MdWKcdaWgaaqcbaua aKqzadGaamOBaaWcbeaajugibiaaiIcacaWG4bGaaGykaaaa@50AA@ , which is why we call λ n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabeU 7aSPWaaSbaaKqaafaajugWaiaad6gaaSqabaaaaa@3D54@  the generalized empirical hazard rate. Replacing λ(x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabeU 7aSjaaiIcacaWG4bGaaGykaaaa@3D15@  in (4) by λ n (x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabeU 7aSPWaaSbaaKqaafaajugWaiaad6gaaSqabaqcLbsacaaIOaGaamiE aiaaiMcaaaa@4045@ , we obtain that

a ^ (x) ( a ^ 0 ,, a ^ p ) T =arg min a j 1 b n K( ux b n )[ λ n (u) j=0 p a j (ux) j ] 2 du. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiqahg gagaqcaiaaiIcacaWG4bGaaGykaiabggMi6kaaiIcaceWGHbGbaKaa juaGdaWgaaqcbauaaKqzadGaaGimaaqcbauabaqcLbsacaaISaGaeS 47IWKaaGilaiqadggagaqcaOWaaSbaaKqaafaajugWaiaadchaaSqa baqcLbsacaaIPaGcdaahaaWcbeqcbauaaKqzadGaamivaaaajugibi aai2daciGGHbGaaiOCaiaacEgakmaawafabeqcbauaaKqzadGaamyy aKqbaoaaBaaajeaqbaqcLbmacaWGQbaajeaqbeaaaSqabOqaaKqzGe GaciyBaiaacMgacaGGUbaaaOWaa8qaaeqaleqabeqcLbsacqGHRiI8 aOWaaSaaaeaajugibiaaigdaaOqaaKqzGeGaamOyaOWaaSbaaKqaaf aajugWaiaad6gaaSqabaaaaKqzGeGaam4saiaaiIcakmaalaaabaqc LbsacaWG1bGaeyOeI0IaamiEaaGcbaqcLbsacaWGIbGcdaWgaaqcba uaaKqzadGaamOBaaWcbeaaaaqcLbsacaaIPaGaaG4waiabeU7aSPWa aSbaaKqaafaajugWaiaad6gaaSqabaqcLbsacaaIOaGaamyDaiaaiM cacqGHsislkmaaqahabeqcbauaaKqzadGaamOAaiaai2dacaaIWaaa jeaqbaqcLbmacaWGWbaajugibiabggHiLdGaamyyaOWaaSbaaKqaaf aajugWaiaadQgaaSqabaqcLbsacaaIOaGaamyDaiabgkHiTiaadIha caaIPaGcdaahaaWcbeqcbauaaKqzadGaamOAaaaajugibiaai2fakm aaCaaaleqajeaqbaqcLbmacaaIYaaaaKqzGeGaamizaiaadwhacaaI Uaaaaa@934E@         (6)

Then a ^ (x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiqahg gagaqcaiaaiIcacaWG4bGaaGykaaaa@3C5B@  is the LP estimator of a(x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaahg gacaaIOaGaamiEaiaaiMcaaaa@3C4B@ . Jones12 considered a locally linear estimator and established its link with the generalized jackknife boundary correction for p=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadc hacaaI9aGaaGymaaaa@3B76@  by using the ideas of Lejeune & Sarda13 in local linear fitting to the empirical distribution F n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadA eakmaaBaaajeaqbaqcLbmacaWGUbaaleqaaaaa@3C6B@ . Here we study the local polynomial estimation of hazard functions and their derivatives, a(x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaahg gacaaIOaGaamiEaiaaiMcaaaa@3C4B@ , under the left truncation and right censoring model. Obviously, our method can be used for the complete data case, which corresponds to “no truncation and no censoring”.

Taking the derivative with respect to a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadg gaaaa@39E5@ ’s of the integral in (6), we obtain the LP estimator a ^ (x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiqahg gagaqcaiaaiIcacaWG4bGaaGykaaaa@3C5B@  as the solution to the linear equations: for l=0,,p MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiablo riSjaai2dacaaIWaGaaGilaiablAciljaaiYcacaWGWbaaaa@3F34@ ,

i=1 n 1 b n K( Y i x b n )( Y i x ) l δ i n C n ( Y i ) = i=0 p a i u0 (ux) i+l 1 b n K( ux b n )du. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaaeWbqabK qaafaajugWaiaadMgacaaI9aGaaGymaaqcbauaaKqzadGaamOBaaqc LbsacqGHris5aOWaaSaaaeaajugibiaaigdaaOqaaKqzGeGaamOyaO WaaSbaaKqaafaajugWaiaad6gaaSqabaaaaKqzGeGaam4saiaaiIca kmaalaaabaqcLbsacaWGzbGcdaWgaaqcbauaaKqzadGaamyAaaWcbe aajugibiabgkHiTiaadIhaaOqaaKqzGeGaamOyaOWaaSbaaKqaafaa jugWaiaad6gaaSqabaaaaKqzGeGaaGykaiaaiIcacaWGzbGcdaWgaa qcbauaaKqzadGaamyAaaWcbeaajugibiabgkHiTiaadIhacaaIPaGc daahaaWcbeqaaKqzGeGaeS4eHWgaaOWaaSaaaeaajugibiabes7aKP WaaSbaaKqaafaajugWaiaadMgaaSqabaaakeaajugibiaad6gacaWG dbGcdaWgaaqcbauaaKqzadGaamOBaaWcbeaajugibiaaiIcacaWGzb GcdaWgaaqcbauaaKqzadGaamyAaaWcbeaajugibiaaiMcaaaGaaGyp aOWaaabCaeqajeaqbaqcLbmacaWGPbGaaGypaiaaicdaaKqaafaaju gWaiaadchaaKqzGeGaeyyeIuoacaWGHbGcdaWgaaqcbauaaKqzadGa amyAaaWcbeaakmaapebabeWcbaqcLbsacaWG1bGaeyyzImRaaGimaa WcbeqcLbsacqGHRiI8aiaaiIcacaWG1bGaeyOeI0IaamiEaiaaiMca kmaaCaaaleqajeaqbaqcLbmacaWGPbGaey4kaSIaeS4eHWgaaOWaaS aaaeaajugibiaaigdaaOqaaKqzGeGaamOyaOWaaSbaaKqaafaajugW aiaad6gaaSqabaaaaKqzGeGaam4saiaaiIcakmaalaaabaqcLbsaca WG1bGaeyOeI0IaamiEaaGcbaqcLbsacaWGIbGcdaWgaaqcbauaaKqz adGaamOBaaWcbeaaaaqcLbsacaaIPaGaamizaiaadwhacaaIUaaaaa@9E63@        (7)

It follows that the LP estimator at any point x 0 ( a G , b W ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadI hakmaaBaaajeaqbaqcLbmacaaIWaaaleqaaKqzGeGaeyicI4SaaGik aiaadggakmaaBaaajeaqbaqcLbmacaWGhbaaleqaaKqzGeGaaGilai aadkgakmaaBaaajeaqbaqcLbmacaWGxbaaleqaaKqzGeGaaGykaaaa @4881@  has the following closed form:

B a ^ ( x 0 )= S 1 S n ( x 0 ), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaahk eaceWHHbGbaKaacaaIOaGaamiEaOWaaSbaaKqaafaajugWaiaaicda aSqabaqcLbsacaaIPaGaaGypaiaahofakmaaCaaaleqabaqcLbsacq GHsislcaaIXaaaaiaahofakmaaBaaajeaqbaqcLbmacaWGUbaaleqa aKqzGeGaaGikaiaadIhakmaaBaaajeaqbaqcLbmacaaIWaaaleqaaK qzGeGaaGykaiaaiYcaaaa@4E49@          (8)

where S n ( x 0 )=( S n0 ( x 0 ),, S np ( x 0 )) T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaho fakmaaBaaajeaqbaqcLbmacaWGUbaaleqaaKqzGeGaaGikaiaadIha kmaaBaaajeaqbaqcLbmacaaIWaaaleqaaKqzGeGaaGykaiaai2daca aIOaGaam4uaKqbaoaaBaaajeaqbaqcLbmacaWGUbGaaGimaaqcbaua baqcLbsacaaIOaGaamiEaOWaaSbaaKqaafaajugWaiaaicdaaSqaba qcLbsacaaIPaGaaGilaiabl+UimjaaiYcacaWGtbGcdaWgaaqcbaua aKqzadGaamOBaiaadchaaSqabaqcLbsacaaIOaGaamiEaOWaaSbaaK qaafaajugWaiaaicdaaSqabaqcLbsacaaIPaGaaGykaOWaaWbaaSqa bKqaafaajugWaiaadsfaaaaaaa@5FA6@ , and

S nl ( x 0 )= i=1 n 1 b n K( Y i x 0 b n )( Y i x 0 b n ) l δ i n C n ( Y i ) . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaado fakmaaBaaajeaqbaqcLbmacaWGUbGaeS4eHWgaleqaaKqzGeGaaGik aiaadIhakmaaBaaajeaqbaqcLbmacaaIWaaaleqaaKqzGeGaaGykai aai2dakmaaqahabeqcbauaaKqzadGaamyAaiaai2dacaaIXaaajeaq baqcLbmacaWGUbaajugibiabggHiLdGcdaWcaaqaaKqzGeGaaGymaa GcbaqcLbsacaWGIbGcdaWgaaqcbauaaKqzadGaamOBaaWcbeaaaaqc LbsacaWGlbGaaGikaOWaaSaaaeaajugibiaadMfakmaaBaaajeaqba qcLbmacaWGPbaaleqaaKqzGeGaeyOeI0IaamiEaOWaaSbaaKqaafaa jugWaiaaicdaaSqabaaakeaajugibiaadkgakmaaBaaajeaqbaqcLb macaWGUbaaleqaaaaajugibiaaiMcacaaIOaGcdaWcaaqaaKqzGeGa amywaOWaaSbaaKqaafaajugWaiaadMgaaSqabaqcLbsacqGHsislca WG4bGcdaWgaaqcbauaaKqzadGaaGimaaWcbeaaaOqaaKqzGeGaamOy aOWaaSbaaKqaafaajugWaiaad6gaaSqabaaaaKqzGeGaaGykaOWaaW baaSqabKqaafaajugWaiabloriSbaakmaalaaabaqcLbsacqaH0oaz kmaaBaaajeaqbaqcLbmacaWGPbaaleqaaaGcbaqcLbsacaWGUbGaam 4qaOWaaSbaaKqaafaajugWaiaad6gaaSqabaqcLbsacaaIOaGaamyw aOWaaSbaaKqaafaajugWaiaadMgaaSqabaqcLbsacaaIPaaaaiaai6 caaaa@85F9@         (9)

When p=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadc hacaaI9aGaaGimaaaa@3B75@  and s 0 =1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaado hakmaaBaaajeaqbaqcLbmacaaIWaaaleqaaKqzGeGaaGypaiaaigda aaa@3E70@ , a ^ 0 = i=1 n 1 b n K( Y i x 0 b n ) δ i n C n ( Y i ) = 1 b n K( x 0 u b n )d Λ n (u) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiqadg gagaqcaOWaaSbaaKqaafaajugWaiaaicdaaSqabaqcLbsacaaI9aGc daaeWaqabKqaafaajugWaiaadMgacaaI9aGaaGymaaqcbauaaKqzad GaamOBaaqcLbsacqGHris5aOWaaSaaaeaajugibiaaigdaaOqaaKqz GeGaamOyaOWaaSbaaKqaafaajugWaiaad6gaaSqabaaaaKqzGeGaam 4saiaaiIcakmaalaaabaqcLbsacaWGzbGcdaWgaaqcbauaaKqzadGa amyAaaWcbeaajugibiabgkHiTiaadIhakmaaBaaajeaqbaqcLbmaca aIWaaaleqaaaGcbaqcLbsacaWGIbGcdaWgaaqcbauaaKqzadGaamOB aaWcbeaaaaqcLbsacaaIPaGcdaWcaaqaaKqzGeGaeqiTdqMcdaWgaa qcbauaaKqzadGaamyAaaWcbeaaaOqaaKqzGeGaamOBaiaadoeakmaa BaaajeaqbaqcLbmacaWGUbaaleqaaKqzGeGaaGikaiaadMfajuaGda WgaaqcbauaaKqzadGaamyAaaqcbauabaqcLbsacaaIPaaaaiaai2da kmaalaaabaqcLbsacaaIXaaakeaajugibiaadkgakmaaBaaajeaqba qcLbmacaWGUbaaleqaaaaakmaapeaabeWcbeqabKqzGeGaey4kIipa caWGlbGaaGikaOWaaSaaaeaajugibiaadIhakmaaBaaajeaqbaqcLb macaaIWaaaleqaaKqzGeGaeyOeI0IaamyDaaGcbaqcLbsacaWGIbGc daWgaaqcbauaaKqzadGaamOBaaWcbeaaaaqcLbsacaaIPaGaamizai abfU5amPWaaSbaaKqaafaajugWaiaad6gaaSqabaqcLbsacaaIOaGa amyDaiaaiMcaaaa@8B50@ , which is the same as the kernel estimator of λ( x 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabeU 7aSjaaiIcacaWG4bGcdaWgaaqcbauaaKqzadGaaGimaaWcbeaajugi biaaiMcaaaa@400C@  in (3). However, this equivalence does not hold for boundary points.

We will show in next section that the LP estimator shares nice properties with the local polynomial regression estimator, in particular, the estimator will keep its convergence rate up to the left boundary point, i.e. the estimator automatically corrects the left boundary effect, which contrasts with the results for other hazard rate estimators.

Asymptotic properties

In this section, we will establish the consistency and joint asymptotic normality of the local polynomial estimators. To this end, we introduce some regularity conditions. For a given point x 0 ( a G , b W ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadI hakmaaBaaajeaqbaqcLboacaaIWaaaleqaaKqzGeGaeyicI4SaaGik aiaadggakmaaBaaajeaqbaqcLbmacaWGhbaaleqaaKqzGeGaaGilai aadkgakmaaBaaajeaqbaqcLbmacaWGxbaaleqaaKqzGeGaaGykaaaa @48A1@ , the following notations and assumptions are needed.

The hazard rate function λ(x)= Λ (x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabeU 7aSjaaiIcacaWG4bGaaGykaiaai2dacuqHBoatgaqbaiaaiIcacaWG 4bGaaGykaaaa@41BF@  has a continuous (p+1)th MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaiI cacaWGWbGaey4kaSIaaGymaiaaiMcacaWG0bGaamiAaaaa@3EDC@  derivative at the point x 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadI hakmaaBaaajeaqbaqcLbmacaaIWaaaleqaaaaa@3C64@ .

The sequence of bandwidths b n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadk gakmaaBaaajeaqbaqcLbmacaWGUbaaleqaaaaa@3C87@  tends to zero such that n b n + MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaad6 gacaWGIbGcdaWgaaqcbauaaKqzadGaamOBaaWcbeaajugibiabgkzi UkabgUcaRiabg6HiLcaa@4249@  as n+ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaad6 gacqGHsgIRcqGHRaWkcqGHEisPaaa@3E32@ . Let B=diag(1, b n ,, b n p ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaahk eacaaI9aGaaGjcVlaadsgacaWGPbGaamyyaiaadEgacaaMi8UaaGik aiaaigdacaaISaGaamOyaOWaaSbaaKqaafaajugWaiaad6gaaSqaba qcLbsacaaISaGaeS47IWKaaGilaiaadkgajuaGdaqhaaqcbauaaKqz adGaamOBaaqcbauaaKqzadGaamiCaaaajugibiaaiMcaaaa@52A1@ .

C(x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaado eacaaIOaGaamiEaiaaiMcaaaa@3C29@  is continuous at the point x 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadI hakmaaBaaajeaqbaqcLbmacaaIWaaaleqaaaaa@3C64@ .

The kernel function K(x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadU eacaaIOaGaamiEaiaaiMcaaaa@3C31@  is a continuous function of bounded variation and with bounded support [1,1] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaiU facqGHsislcaaIXaGaaGilaiaaigdacaaIDbaaaa@3DE4@ , say. Let s l = 1 1 K(u) u l du MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaado hakmaaBaaajeaqbaqcLbmacqWItecBaSqabaqcLbsacaaI9aGcdaWd XaqabKqaafaajugWaiabgkHiTiaaigdaaKqaafaajugWaiaaigdaaK qzGeGaey4kIipacaWGlbGaaGikaiaadwhacaaIPaGaamyDaOWaaWba aSqabKqaafaajugWaiabloriSbaajugibiaadsgacaWG1baaaa@4FA5@ , v l = 1 1 u l K 2 (u)du MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadA hakmaaBaaajeaqbaqcLbmacqWItecBaSqabaqcLbsacaaI9aGcdaWd XaqabKqaafaajugWaiabgkHiTiaaigdaaKqaafaajugWaiaaigdaaK qzGeGaey4kIipacaWG1bGcdaahaaWcbeqcbauaaKqzadGaeS4eHWga aKqzGeGaam4saOWaaWbaaSqabKqaafaajugWaiaaikdaaaqcLbsaca aIOaGaamyDaiaaiMcacaWGKbGaamyDaaaa@52A2@ , for l0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiablo riSjabgwMiZkaaicdaaaa@3CB0@ , c p =( s p+1 ,, s 2p+1 ) T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaho gakmaaBaaajeaqbaqcLbmacaWGWbaaleqaaKqzGeGaaGypaiaaiIca caWGZbGcdaWgaaqcbauaaKqzadGaamiCaiabgUcaRiaaigdaaSqaba qcLbsacaaISaGaeS47IWKaaGilaiaadohakmaaBaaajeaqbaqcLbma caaIYaGaamiCaiabgUcaRiaaigdaaSqabaqcLbsacaaIPaGcdaahaa WcbeqcbauaaKqzadGaamivaaaaaaa@5175@ , S=( s i+j ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaho facaaI9aGaaGikaiaadohakmaaBaaajeaqbaqcLbmacaWGPbGaey4k aSIaamOAaaWcbeaajugibiaaiMcaaaa@41FB@  and V * =( v i+j ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaahA fakmaaCaaaleqajeaqbaqcLbmacaaIQaaaaKqzGeGaaGypaiaaiIca caWG2bGcdaWgaaqcbauaaKqzadGaamyAaiabgUcaRiaadQgaaSqaba qcLbsacaaIPaaaaa@44F3@   (for0ip; MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaiI cacaaMi8UaaGjbVlaadAgacaWGVbGaamOCaiaayIW7caaMe8UaaGim aiabgsMiJkaadMgacqGHKjYOcaWGWbGaaG4oaaaa@498F@   0jp) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaic dacqGHKjYOcaWGQbGaeyizImQaamiCaiaaiMcaaaa@3FBA@  be (p+1)×(p+1) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaiI cacaWGWbGaey4kaSIaaGymaiaaiMcacqGHxdaTcaaIOaGaamiCaiab gUcaRiaaigdacaaIPaaaaa@4304@  matrices.

Theorem 3.1 Under conditions (A1) – (A4),

B( a ^ ( x 0 )a( x 0 )) a.s. 0,n. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaahk eacaaIOaGabCyyayaajaGaaGikaiaadIhakmaaBaaajeaqbaqcLbma caaIWaaaleqaaKqzGeGaaGykaiabgkHiTiaahggacaaIOaGaamiEaO WaaSbaaKqaafaajugWaiaaicdaaSqabaqcLbsacaaIPaGaaGykaOWa aCbiaeaajugibiabgkziUcWcbeqcbauaaKqzadGaamyyaiaai6caca WGZbGaaGOlaaaajugibiaaicdacaaISaGaaGzbVlaad6gacqGHsgIR cqGHEisPcaaIUaaaaa@58DD@

Theorem 3.2 Under conditions (A1) – (A4),

n b n [B( a ^ ( x 0 )a( x 0 )) λ (p+1) ( x 0 ) b n p+1 (p+1)! S 1 c p (1+ o p (1))] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaGcaaqaaK qzGeGaamOBaiaadkgakmaaBaaajeaqbaqcLbmacaWGUbaaleqaaaqa baqcLbsacaaIBbGaaCOqaiaaiIcaceWHHbGbaKaacaaIOaGaamiEaO WaaSbaaKqaafaajugWaiaaicdaaSqabaqcLbsacaaIPaGaeyOeI0Ia aCyyaiaaiIcacaWG4bGcdaWgaaqcbauaaKqzadGaaGimaaWcbeaaju gibiaaiMcacaaIPaGaeyOeI0IcdaWcaaqaaKqzGeGaeq4UdWMcdaah aaWcbeqcbauaaKqzadGaaGikaiaadchacqGHRaWkcaaIXaGaaGykaa aajugibiaaiIcacaWG4bGcdaWgaaqcbauaaKqzadGaaGimaaWcbeaa jugibiaaiMcacaWGIbqcfa4aa0baaKqaafaajugWaiaad6gaaKqaaf aajugWaiaadchacqGHRaWkcaaIXaaaaaGcbaqcLbsacaaIOaGaamiC aiabgUcaRiaaigdacaaIPaGaaGyiaaaacaWHtbGcdaahaaWcbeqcba uaaKqzadGaeyOeI0IaaGymaaaajugibiaahogakmaaBaaajeaqbaqc LbmacaWGWbaaleqaaKqzGeGaaGikaiaaigdacqGHRaWkcaWGVbGcda WgaaqcbauaaKqzadGaamiCaaWcbeaajugibiaaiIcacaaIXaGaaGyk aiaaiMcacaaIDbaaaa@7DDF@

L N(0, S 1 V * S 1 λ( x 0 ) C 1 ( x 0 )). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWfGaqaaK qzGeGaeyOKH4kaleqajeaqbaqcLbmacaWGmbaaaKqzGeGaamOtaiaa iIcacaaIWaGaaGilaiaahofakmaaCaaaleqajeaqbaqcLbmacqGHsi slcaaIXaaaaKqzGeGaaCOvaOWaaWbaaSqabKqaafaajugWaiaaiQca aaqcLbsacaWHtbGcdaahaaWcbeqcbauaaKqzadGaeyOeI0IaaGymaa aajugibiabeU7aSjaaiIcacaWG4bGcdaWgaaqcbauaaKqzadGaaGim aaWcbeaajugibiaaiMcacaWGdbGcdaahaaWcbeqcbauaaKqzadGaey OeI0IaaGymaaaajugibiaaiIcacaWG4bGcdaWgaaqcbauaaKqzadGa aGimaaWcbeaajugibiaaiMcacaaIPaGaaGOlaaaa@60D6@         (10)

Remark 3.1

When estimating a hazard rate which is a polynomial of order p MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadc haaaa@39F4@  on an interval, the finite sample bias of the LP estimators on the interval is zero (see the proof of Theorem 3.2). This contrasts with the methods of Müller and Wang7,8 based on higher order kernels, for which the respective zero bias only holds true asymptotically.

Remark 3.2

When there is no truncation, we take T=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaads facaaI9aGaaGimaaaa@3B59@  and G(x)=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadE eacaaIOaGaamiEaiaaiMcacaaI9aGaaGymaaaa@3DAF@  over the support of F(x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadA eacaaIOaGaamiEaiaaiMcaaaa@3C2C@ , then C(x)= α 1 P(Cx) F ¯ (x)= W ¯ (x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaado eacaaIOaGaamiEaiaaiMcacaaI9aGaeqySdeMcdaahaaWcbeqcbaua aKqzadGaeyOeI0IaaGymaaaajugibiaadcfacaaIOaGaam4qaiabgw MiZkaadIhacaaIPaGabmOrayaaraGaaGikaiaadIhacaaIPaGaaGyp aiqadEfagaqeaiaaiIcacaWG4bGaaGykaaaa@4F9C@ , and the asymptotic normality for interior point x 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadI hakmaaBaaajeaqbaqcLbmacaaIWaaaleqaaaaa@3C64@  is the same as in Müller & Wang.7,8 When there is no censoring, we take the censoring variable C=+ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaado eacaaI9aGaey4kaSIaeyOhIukaaa@3CE1@  and L(x)=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadY eacaaIOaGaamiEaiaaiMcacaaI9aGaaGimaaaa@3DB3@  over the support of F(x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadA eacaaIOaGaamiEaiaaiMcaaaa@3C2C@ , then C(x)=P(TxX|YT) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaado eacaaIOaGaamiEaiaaiMcacaaI9aGaamiuaiaaiIcacaWGubGaeyiz ImQaamiEaiabgsMiJkaadIfacaaI8bGaamywaiabgwMiZkaadsfaca aIPaaaaa@49CA@ , and the asymptotic normality for interior point x 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadI hakmaaBaaajeaqbaqcLbmacaaIWaaaleqaaaaa@3C64@  is the same as in Gürler & Wang.1

AMS E k ( b n , x 0 )= b n 2(p+1k) [ e k T S 1 c p λ (p+1) ( x 0 ) (p+1)! ] 2 + 1 n b n 2k+1 e k T S 1 V * S 1 e k λ( x 0 ) C( x 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadg eacaWGnbGaam4uaiaadweakmaaBaaajeaqbaqcLbmacaWGRbaaleqa aKqzGeGaaGikaiaadkgakmaaBaaajeaqbaqcLbmacaWGUbaaleqaaK qzGeGaaGilaiaadIhakmaaBaaajeaqbaqcLbmacaaIWaaaleqaaKqz GeGaaGykaiaai2dacaWGIbWcdaqhaaqcbauaaKqzGcGaamOBaaqcba uaaKqzGcGaaGOmaiaaiIcacaWGWbGaey4kaSIaaGymaiabgkHiTiaa dUgacaaIPaaaaKqzGeGaaG4waiaahwgakmaaDaaajeaqbaqcLbmaca WGRbaajeaqbaqcLbmacaWGubaaaKqzGeGaaC4uaOWaaWbaaSqabKqa afaajugWaiabgkHiTiaaigdaaaqcLbsacaWHJbGcdaWgaaqcbauaaK qzadGaamiCaaWcbeaakmaalaaabaqcLbsacqaH7oaBkmaaCaaaleqa jeaqbaqcLbmacaaIOaGaamiCaiabgUcaRiaaigdacaaIPaaaaKqzGe GaaGikaiaadIhakmaaBaaajeaqbaqcLbmacaaIWaaaleqaaKqzGeGa aGykaaGcbaqcLbsacaaIOaGaamiCaiabgUcaRiaaigdacaaIPaGaaG yiaaaacaaIDbGcdaahaaWcbeqcbauaaKqzadGaaGOmaaaajugibiab gUcaROWaaSaaaeaajugibiaaigdaaOqaaKqzGeGaamOBaiaadkgakm aaDaaajeaqbaqcLbmacaWGUbaajeaqbaqcLbmacaaIYaGaam4Aaiab gUcaRiaaigdaaaaaaKqzGeGaaCyzaOWaa0baaKqaafaajugWaiaadU gaaKqaafaajugWaiaadsfaaaqcLbsacaWHtbGcdaahaaWcbeqcbaua aKqzadGaeyOeI0IaaGymaaaajugibiaahAfakmaaCaaaleqajeaqba qcLbmacaaIQaaaaKqzGeGaaC4uaOWaaWbaaSqabKqaafaajugWaiab gkHiTiaaigdaaaqcLbsacaWHLbGcdaWgaaqcbauaaKqzadGaam4Aaa WcbeaakmaalaaabaqcLbsacqaH7oaBcaaIOaGaamiEaOWaaSbaaKqa afaajugWaiaaicdaaSqabaqcLbsacaaIPaaakeaajugibiaadoeaca aIOaGaamiEaOWaaSbaaKqaafaajugWaiaaicdaaSqabaqcLbsacaaI Paaaaaaa@AF1B@       (11)

where e k =(0,,1, ,0) T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaahw gakmaaBaaajeaqbaqcLbmacaWGRbaaleqaaKqzGeGaaGypaiaaiIca caaIWaGaaGilaiabl+UimjaaiYcacaaIXaGaaGilaiabl+UimjaaiY cacaaIWaGaaGykaOWaaWbaaSqabKqaafaajugWaiaadsfaaaaaaa@4AB1@  has one in the k+1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadU gacqGHRaWkcaaIXaaaaa@3B8C@ th component and zeros in the others. Therefore, the optimal local bandwidth for estimating the k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadU gaaaa@39EF@ th derivative of λ(x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabeU 7aSjaaiIcacaWG4bGaaGykaaaa@3D15@  at x 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadI hakmaaBaaajeaqbaqcLbmacaaIWaaajeaybeaaaaa@3CC3@ , in the sense of minimizing AMS E k ( b n , x 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadg eacaWGnbGaam4uaiaadweakmaaBaaajeaqbaqcLbmacaWGRbaaleqa aKqzGeGaaGikaiaadkgakmaaBaaajeaqbaqcLbmacaWGUbaaleqaaK qzGeGaaGilaiaadIhakmaaBaaajeaqbaqcLbmacaaIWaaaleqaaKqz GeGaaGykaaaa@498C@ , is

b k,opt ( x 0 )= n 1 2p+3 ( [(p+1)!] 2 e k T S 1 V * S 1 e k λ( x 0 )/C( x 0 ) 2(p+1k)[ λ (p+1) ( x 0 )] 2 ( e k T S 1 c p ) 2 ) 1/(2p+3) . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadk gakmaaBaaajeaqbaqcLbmacaWGRbGaaGilaiaad+gacaWGWbGaamiD aaWcbeaajugibiaaiIcacaWG4bGcdaWgaaqcbauaaKqzadGaaGimaa WcbeaajugibiaaiMcacaaI9aGaamOBaOWaaWbaaSqabKqaafaajugW aiabgkHiTKqbaoaalaaajeaqbaqcLbmacaaIXaaajeaqbaqcLbmaca aIYaGaamiCaiabgUcaRiaaiodaaaaaaOWaaeWaaeaadaWcaaqaaKqz GeGaaG4waiaaiIcacaWGWbGaey4kaSIaaGymaiaaiMcacaaIHaGaaG yxaOWaaWbaaSqabKqaafaajugWaiaaikdaaaqcLbsacaWHLbGcdaqh aaqcbauaaKqzadGaam4AaaqcbauaaKqzadGaamivaaaajugibiaaho fakmaaCaaaleqajeaqbaqcLbmacqGHsislcaaIXaaaaKqzGeGaaCOv aOWaaWbaaSqabKqaafaajugWaiaaiQcaaaqcLbsacaWHtbGcdaahaa WcbeqcbauaaKqzadGaeyOeI0IaaGymaaaajugibiaahwgakmaaBaaa jeaqbaqcLbmacaWGRbaaleqaaKqzGeGaeq4UdWMaaGikaiaadIhakm aaBaaajeaqbaqcLbmacaaIWaaaleqaaKqzGeGaaGykaiaai+cacaWG dbGaaGikaiaadIhakmaaBaaajeaqbaqcLbmacaaIWaaaleqaaKqzGe GaaGykaaGcbaqcLbsacaaIYaGaaGikaiaadchacqGHRaWkcaaIXaGa eyOeI0Iaam4AaiaaiMcacaaIBbGaeq4UdWMcdaahaaWcbeqcbauaaK qzadGaaGikaiaadchacqGHRaWkcaaIXaGaaGykaaaajugibiaaiIca caWG4bGcdaWgaaqcbauaaKqzadGaaGimaaWcbeaajugibiaaiMcaca aIDbGcdaahaaWcbeqcbauaaKqzadGaaGOmaaaajugibiaaiIcacaWH LbGcdaqhaaqcbauaaKqzadGaam4AaaqcbauaaKqzadGaamivaaaaju gibiaahofakmaaCaaaleqajeaqbaqcLbmacqGHsislcaaIXaaaaKqz GeGaaC4yaOWaaSbaaKqaafaajugWaiaadchaaSqabaqcLbsacaaIPa GcdaahaaWcbeqcbauaaKqzadGaaGOmaaaaaaaakiaawIcacaGLPaaa daahaaWcbeqcbauaaKqzadGaaGymaiaai+cacaaIOaGaaGOmaiaadc hacqGHRaWkcaaIZaGaaGykaaaajugibiaai6caaaa@B8A8@          (12)

Theorem 3.3

Consider the left edge effect on the estimator. Assume that we estimate a(x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaahg gacaaIOaGaamiEaiaaiMcaaaa@3C4B@  at x n =d b n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadI hakmaaBaaajeaqbaqcLbmacaWGUbaaleqaaKqzGeGaaGypaiaadsga caWGIbGcdaWgaaqcbauaaKqzadGaamOBaaWcbeaaaaa@4264@  in the left boundary region for some positive constant d[0,1] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaads gacqGHiiIZcaaIBbGaaGimaiaaiYcacaaIXaGaaGyxaaaa@3F63@ . Then similar to (8), a ^ ( x n ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiqahg gagaqcaiaaiIcacaWG4bGcdaWgaaqcbauaaKqzadGaamOBaaWcbeaa jugibiaaiMcaaaa@3F8B@  in (6) has the following closed form, for p=0,1,2, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadc hacaaI9aGaaGimaiaaiYcacaaIXaGaaGilaiaaikdacaaISaGaeS47 IWeaaa@40FC@ ,

B a ^ ( x n )= S d 1 S n ( x n ), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaahk eaceWHHbGbaKaacaaIOaGaamiEaOWaaSbaaKqaafaajugWaiaad6ga aSqabaqcLbsacaaIPaGaaGypaiaahofakmaaDaaajeaqbaqcLbmaca WGKbaajeaqbaqcLbmacqGHsislcaaIXaaaaKqzGeGaaC4uaOWaaSba aKqaafaajugWaiaad6gaaSqabaqcLbsacaaIOaGaamiEaOWaaSbaaK qaafaajugWaiaad6gaaSqabaqcLbsacaaIPaGaaGilaaaa@5289@          (13)

where S d MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaho fakmaaBaaajeaqbaqcLbmacaWGKbaaleqaaaaa@3C72@  defined as S MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaho faaaa@39DB@  but with s i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaado hakmaaBaaajeaqbaqcLbmacaWGPbaaleqaaaaa@3C93@  replaced by s i,d = d 1 u i K(u)du MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaado hakmaaBaaajeaqbaqcLbmacaWGPbGaaGilaiaadsgaaSqabaqcLbsa caaI9aGcdaWdXaqabKqaafaajugWaiabgkHiTiaadsgaaKqaafaaju gWaiaaigdaaKqzGeGaey4kIipacaWG1bGcdaahaaWcbeqcbauaaKqz GdGaamyAaaaajugibiaadUeacaaIOaGaamyDaiaaiMcacaWGKbGaam yDaaaa@510C@ . Let v i,d = d 1 u i K 2 (u)du MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadA hakmaaBaaajeaqbaqcLbmacaWGPbGaaGilaiaadsgaaSqabaqcLbsa caaI9aGcdaWdXaqabSqaaKqzGeGaeyOeI0IaamizaaqcbauaaKqzad GaaGymaaqcLbsacqGHRiI8aiaadwhakmaaCaaaleqajeaqbaqcLbma caWGPbaaaKqzGeGaam4saOWaaWbaaSqabKqaafaajugWaiaaikdaaa qcLbsacaaIOaGaamyDaiaaiMcacaWGKbGaamyDaaaa@530B@ . Then the joint asymptotic normality (10) continues to hold with c p MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaho gakmaaBaaajeaqbaqcLbmacaWGWbaaleqaaaaa@3C8E@ , S MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaho faaaa@39DB@  and V * MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeGabaaI=NqzGe GaaCOvaOWaaWbaaSqabKqaafaajugWaiaaiQcaaaaaaa@3DCB@  replaced by c p,d MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaho gakmaaBaaajeaqbaqcLbmacaWGWbGaaGilaiaadsgaaSqabaaaaa@3E2D@ , S d MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaho fakmaaBaaajeaqbaqcLbmacaWGKbaaleqaaaaa@3C72@  and V d * MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaahA fajuaGdaqhaaqcbauaaKqzadGaamizaaqcbauaaKqzadGaaGOkaaaa aaa@3F1B@ , respectively, where c p,d =( s p+1,d ,, s 2p+1,d ) T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaho gakmaaBaaajeaqbaqcLbmacaWGWbGaaGilaiaadsgaaSqabaqcLbsa caaI9aGaaGikaiaadohakmaaBaaajeaqbaqcLbmacaWGWbGaey4kaS IaaGymaiaaiYcacaWGKbaaleqaaKqzGeGaaGilaiabl+UimjaaiYca caWGZbGcdaWgaaqcbauaaKqzadGaaGOmaiaadchacqGHRaWkcaaIXa GaaGilaiaadsgaaSqabaqcLbsacaaIPaGcdaahaaWcbeqcbauaaKqz adGaamivaaaaaaa@5652@  and V d * =( v i+j,d ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaahA fajuaGdaqhaaqcbauaaKqzadGaamizaaqcbauaaKqzadGaaGOkaaaa jugibiaai2dacaaIOaGaamODaOWaaSbaaKqaafaajugWaiaadMgacq GHRaWkcaWGQbGaaGilaiaadsgaaSqabaqcLbsacaaIPaaaaa@496C@  is (p+1)×(p+1) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaiI cacaWGWbGaey4kaSIaaGymaiaaiMcacqGHxdaTcaaIOaGaamiCaiab gUcaRiaaigdacaaIPaaaaa@4304@  matrices.

This property of our estimator in Theorem 3.3 is similar to that of local polynomial regression estimation, which is not shared by other kernel estimators of hazard rates (Hess et al.14) The LP estimators are automatically boundary adaptive in the sense of Fan & Gurler.15 Note that the above property holds even for p=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadc hacaaI9aGaaGimaaaa@3B75@ , which contrasts with the cases of local polynomial regression.

Remark 3.3 For a finite sample, one may encounter right boundary effects when estimating near T. A good method for dealing with the problem is to use the following estimator similar to the estimator in (13):

B a ^ ( x 0 )= S q 1 S n ( x 0 ), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaahk eaceWHHbGbaKaacaaIOaGaamiEaOWaaSbaaKqaafaajugWaiaaicda aSqabaqcLbsacaaIPaGaaGypaiaahofakmaaDaaajeaqbaqcLbmaca WGXbaajeaqbaqcLbmacqGHsislcaaIXaaaaKqzGeGaaC4uaOWaaSba aKqaafaajugWaiaad6gaaSqabaqcLbsacaaIOaGaamiEaOWaaSbaaK qaafaajugWaiaaicdaaSqabaqcLbsacaaIPaGaaGilaaaa@5224@           (14)

where x 0 =Tq b n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadI hakmaaBaaajeaqbaqcLbmacaaIWaaaleqaaKqzGeGaaGypaiaadsfa cqGHsislcaWGXbGaamOyaOWaaSbaaKqaafaajugWaiaad6gaaSqaba aaaa@43FE@  for q[0,1] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadg hacqGHiiIZcaaIBbGaaGimaiaaiYcacaaIXaGaaGyxaaaa@3F70@ , and S q MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaho fakmaaBaaajeaqbaqcLbmacaWGXbaaleqaaaaa@3C7F@  defined as S MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaho faaaa@39DB@  but with s i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaado hakmaaBaaajeaqbaqcLbmacaWGPbaaleqaaaaa@3C93@  replaced by s i,q = 1 q u i K(u)du MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaado hakmaaBaaajeaqbaqcLbmacaWGPbGaaGilaiaadghaaSqabaqcLbsa caaI9aGcdaWdXaqabKqaafaajugWaiabgkHiTiaaigdaaKqaafaaju gWaiaadghaaKqzGeGaey4kIipacaWG1bqcfa4aaWbaaSqabKqaafaa jugWaiaadMgaaaqcLbsacaWGlbGaaGikaiaadwhacaaIPaGaamizai aadwhaaaa@518A@ .

Data-driven local bandwidth choice

The proposed estimators depend on the band width b n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadk gakmaaBaaajeaqbaqcLbmacaWGUbaaleqaaaaa@3C87@ . It is important to develop a local bandwidth choice for estimating hazard functions and their derivatives, especially when one would like to have a data-driven approach to bandwidth choice in practice. For hazard rate estimation, Patil16 considered least squares cross-validation bandwidth selection, and Gonz lez-Manteiga et al.17 studied smoothed bootstrap selection of the global bandwidth. Müller & Wang,7 Hess et al.14 and Jiang & Doksum11 studied the local bandwidth choice for estimating hazard rates under right censoring. Here we extend the data-driven local bandwidth choice of Jiang & Doksum11 to the left truncation and right censoring model.

From the proof of Theorem 3.2, we see that the exact bias of the estimator a ^ k ( x 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiqadg gagaqcaOWaaSbaaKqaafaajug4aiaadUgaaSqabaqcLbsacaaIOaGa amiEaOWaaSbaaKqaafaajugWaiaaicdaaSqabaqcLbsacaaIPaaaaa@429B@  is

B k ( b n , x 0 )= e k T ( S 1 =9 β n ( x 0 )Ba( x 0 )), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadk eakmaaBaaajeaqbaqcLbmacaWGRbaaleqaaKqzGeGaaGikaiaadkga kmaaBaaajeaqbaqcLbmacaWGUbaaleqaaKqzGeGaaGilaiaadIhakm aaBaaajeaqbaqcLbmacaaIWaaaleqaaKqzGeGaaGykaiaai2dacaWH Lbqcfa4aa0baaKqaafaajugWaiaadUgaaKqaafaajugWaiaadsfaaa qcLbsacaaIOaGaaC4uaOWaaWbaaSqabKqaafaajugWaiabgkHiTiaa igdaaaqcLbsacaaI9aGaaGyoaiabek7aIPWaaSbaaKqaafaajugWai aad6gaaSqabaqcLbsacaaIOaGaamiEaOWaaSbaaKqaafaajugWaiaa icdaaSqabaqcLbsacaaIPaGaeyOeI0IaaCOqaiaahggacaaIOaGaam iEaOWaaSbaaKqaafaajug4aiaaicdaaSqabaqcLbsacaaIPaGaaGyk aiaaiYcaaaa@6972@               (15)

where =9 β n ( x 0 )=( β n0 ( x 0 ), β np ( x 0 )) T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaai2 dacaaI5aGaeqOSdiMcdaWgaaqcbauaaKqzadGaamOBaaWcbeaajugi biaaiIcacaWG4bGcdaWgaaqcbauaaKqzadGaaGimaaWcbeaajugibi aaiMcacaaI9aGaaGikaiabek7aIPWaaSbaaKqaafaajugWaiaad6ga caaIWaaaleqaaKqzGeGaaGikaiaadIhakmaaBaaajeaqbaqcLbmaca aIWaaaleqaaKqzGeGaaGykaiabl+UimjaaiYcacqaHYoGykmaaBaaa jeaqbaqcLbmacaWGUbGaamiCaaWcbeaajugibiaaiIcacaWG4bGcda WgaaqcbauaaKqzadGaaGimaaWcbeaajugibiaaiMcacaaIPaGcdaah aaWcbeqcbauaaKqzadGaamivaaaaaaa@620E@  and β nk ( x 0 )= K(t) t k λ( x 0 + b n t)dt. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabek 7aIPWaaSbaaKqaafaajugWaiaad6gacaWGRbaaleqaaKqzGeGaaGik aiaadIhakmaaBaaajeaqbaqcLbmacaaIWaaaleqaaKqzGeGaaGykai aai2dakmaapeaabeWcbeqabKqzGeGaey4kIipacaWGlbGaaGikaiaa dshacaaIPaGaamiDaOWaaWbaaSqabKqaafaajugWaiaadUgaaaqcLb sacqaH7oaBcaaIOaGaamiEaOWaaSbaaKqaafaajugWaiaaicdaaSqa baqcLbsacqGHRaWkcaWGIbGcdaWgaaqcbauaaKqzadGaamOBaaWcbe aajugibiaadshacaaIPaGaamizaiaadshacaaIUaaaaa@5E56@  The asymptotic variance of a ^ k ( x 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiqadg gagaqcaOWaaSbaaKqaafaajugWaiaadUgaaSqabaqcLbsacaaIOaGa amiEaOWaaSbaaKqaafaajugWaiaaicdaaSqabaqcLbsacaaIPaaaaa@427B@  is

V k ( b n , x 0 )= 1 n b n e k T S 1 V ˜ S 1 e k , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadA fakmaaBaaajeaqbaqcLbmacaWGRbaaleqaaKqzGeGaaGikaiaadkga kmaaBaaajeaqbaqcLbmacaWGUbaaleqaaKqzGeGaaGilaiaadIhakm aaBaaajeaqbaqcLbmacaaIWaaaleqaaKqzGeGaaGykaiaai2dakmaa laaabaqcLbsacaaIXaaakeaajugibiaad6gacaWGIbGcdaWgaaqcba uaaKqzadGaamOBaaWcbeaaaaqcLbsacaWHLbqcfa4aa0baaKqaafaa jugWaiaadUgaaKqaafaajugWaiaadsfaaaqcLbsacaWHtbGcdaahaa WcbeqcbauaaKqzadGaeyOeI0IaaGymaaaajugibiqahAfagaacaiaa hofakmaaCaaaleqajeaqbaqcLbmacqGHsislcaaIXaaaaKqzGeGaaC yzaOWaaSbaaKqaafaajug4aiaadUgaaSqabaqcLbsacaaISaaaaa@6544@           (16)

where V ˜ =( v ˜ ij ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiqahA fagaacaiaai2dacaaIOaGabmODayaaiaGcdaWgaaqcbauaaKqzadGa amyAaiaadQgaaSqabaqcLbsacaaIPaaaaa@413D@ and v ˜ ij = [ K 2 (t) t i+j λ( x 0 + b n t)/C( x 0 + b n t)]dt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiqadA hagaacaOWaaSbaaKqaafaajugWaiaadMgacaWGQbaaleqaaKqzGeGa aGypaOWaa8qaaeqaleqabeqcLbsacqGHRiI8aiaaiUfacaWGlbGcda ahaaWcbeqcbauaaKqzadGaaGOmaaaajugibiaaiIcacaWG0bGaaGyk aiaadshakmaaCaaaleqajeaqbaqcLbmacaWGPbGaey4kaSIaamOAaa aajugibiabeU7aSjaaiIcacaWG4bGcdaWgaaqcbauaaKqzadGaaGim aaWcbeaajugibiabgUcaRiaadkgakmaaBaaajeaqbaqcLbmacaWGUb aaleqaaKqzGeGaamiDaiaaiMcacaaIVaGaam4qaiaaiIcacaWG4bGc daWgaaqcbauaaKqzGcGaaGimaaWcbeaajugibiabgUcaRiaadkgakm aaBaaajeaqbaqcLbmacaWGUbaaleqaaKqzGeGaamiDaiaaiMcacaaI DbGaamizaiaadshaaaa@6AE9@ . Then we propose to estimate the AMSE of λ ^ k ( x 0 )=k! a ^ k ( x 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiqbeU 7aSzaajaGcdaWgaaqcbauaaKqzadGaam4AaaWcbeaajugibiaaiIca caWG4bGcdaWgaaqcbauaaKqzadGaaGimaaWcbeaajugibiaaiMcaca aI9aGaam4AaiaaigcaceWGHbGbaKaakmaaBaaajeaqbaqcLbmacaWG RbaaleqaaKqzGeGaaGikaiaadIhakmaaBaaajeaqbaqcLbmacaaIWa aaleqaaKqzGeGaaGykaaaa@4F27@  via

"0362AMS E k ( b n , x 0 )= B ^ k 2 ( b n , x 0 )+ V ^ k ( b n , x 0 ), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaik cacaaIWaGaaG4maiaaiAdacaaIYaGaamyqaiaad2eacaWGtbGaamyr aOWaaSbaaKqaafaajugOaiaadUgaaSqabaqcLbsacaaIOaGaamOyaO WaaSbaaKqaafaajugWaiaad6gaaSqabaqcLbsacaaISaGaamiEaOWa aSbaaKqaafaajugWaiaaicdaaSqabaqcLbsacaaIPaGaaGypaiqadk eagaqcaOWaa0baaKqaafaajugWaiaadUgaaKqaafaajugWaiaaikda aaqcLbsacaaIOaGaamOyaOWaaSbaaKqaafaajugWaiaad6gaaSqaba qcLbsacaaISaGaamiEaOWaaSbaaKqaafaajugWaiaaicdaaSqabaqc LbsacaaIPaGaey4kaSIabmOvayaajaGcdaWgaaqcbauaaKqzadGaam 4AaaWcbeaajugibiaaiIcacaWGIbGcdaWgaaqcbauaaKqzadGaamOB aaWcbeaajugibiaaiYcacaWG4bGcdaWgaaqcbauaaKqzadGaaGimaa WcbeaajugibiaaiMcacaaISaaaaa@6DFC@         (17)

where B ^ k ( b n , x 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiqadk eagaqcaOWaaSbaaKqaafaajugWaiaadUgaaSqabaqcLbsacaaIOaGa amOyaOWaaSbaaKqaafaajugWaiaad6gaaSqabaqcLbsacaaISaGaam iEaOWaaSbaaKqaafaajugWaiaaicdaaSqabaqcLbsacaaIPaaaaa@4729@  and V ^ k ( b n , x 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiqadA fagaqcaOWaaSbaaKqaafaajugWaiaadUgaaSqabaqcLbsacaaIOaGa amOyaOWaaSbaaKqaafaajugWaiaad6gaaSqabaqcLbsacaaISaGaam iEaOWaaSbaaKqaafaajugWaiaaicdaaSqabaqcLbsacaaIPaaaaa@473D@  are defined similarly to B k ( b n , x 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadk eakmaaBaaajeaqbaqcLbmacaWGRbaaleqaaKqzGeGaaGikaiaadkga kmaaBaaajeaqbaqcLbmacaWGUbaaleqaaKqzGeGaaGilaiaadIhakm aaBaaajeaqbaqcLbmacaaIWaaaleqaaKqzGeGaaGykaaaa@4719@  and V k ( b n , x 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadA fakmaaBaaajeaqbaqcLbmacaWGRbaaleqaaKqzGeGaaGikaiaadkga kmaaBaaajeaqbaqcLbmacaWGUbaaleqaaKqzGeGaaGilaiaadIhakm aaBaaajeaqbaqcLbmacaaIWaaaleqaaKqzGeGaaGykaaaa@472D@  but with λ(x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabeU 7aSjaaiIcacaWG4bGaaGykaaaa@3D15@  replaced by a pilot estimator and C(x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaado eacaaIOaGaamiEaiaaiMcaaaa@3C29@  replaced by its empirical estimator C n (x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaado eajuaGdaWgaaqcbauaaKqzadGaamOBaaqcbauabaqcLbsacaaIOaGa amiEaiaaiMcaaaa@401C@ . We define

b ^ k,opt ( x 0 )=arg min b "0362AMS E k (b, x 0 ). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiqadk gagaqcaOWaaSbaaKqaafaajugWaiaadUgacaaISaGaam4Baiaadcha caWG0baaleqaaKqzGeGaaGikaiaadIhakmaaBaaajeaqbaqcLbmaca aIWaaaleqaaKqzGeGaaGykaiaai2daciGGHbGaaiOCaiaacEgakmaa wafabeqcbauaaKqzadGaamOyaaWcbeGcbaqcLbsaciGGTbGaaiyAai aac6gaaaqcLbmacaaIIaGaaGimaiaaiodacaaI2aGaaGOmaiaadgea caWGnbGaam4uaiaadweakmaaBaaajeaqbaqcLboacaWGRbaaleqaaK qzGeGaaGikaiaadkgacaaISaGaamiEaOWaaSbaaKqaafaajugWaiaa icdaaSqabaqcLbsacaaIPaGaaGOlaaaa@62F3@

Then, for estimating λ(x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabeU 7aSjaaiIcacaWG4bGaaGykaaaa@3D15@  when p=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadc hacaaI9aGaaGymaaaa@3B76@ , the following algorithm similar to Jiang and Doksum11 can be used for estimating λ(x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabeU 7aSjaaiIcacaWG4bGaaGykaaaa@3D15@ .

Algorithm for estimating λ(x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabeU 7aSjaaiIcacaWG4bGaaGykaaaa@3D15@ :

Step 1: Pilot estimators of λ(x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabeU 7aSjaaiIcacaWG4bGaaGykaaaa@3D15@ : Choose a kernel, such as the Epanechnikov kernel, and an initial global bandwidth b 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadk gakmaaBaaajeaqbaqcLbmacaaIWaaaleqaaaaa@3C4E@ . The choice of the initial bandwidth depends on the specific case. Assume the data are available on [0,T] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaiU facaaIWaGaaGilaiaadsfacaaIDbaaaa@3D14@ , then a possible value for b 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadk gakmaaBaaajeaqbaqcLbmacaaIWaaaleqaaaaa@3C4E@  is T/(8 n u 1 5 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaads facaaIVaGaaGikaiaaiIdacaWGUbqcfa4aa0baaKqaafaajugWaiaa dwhaaKqaafaajuaGdaWcaaqcbauaaKqzadGaaGymaaqcbauaaKqzad GaaGynaaaaaaqcLbsacaaIPaaaaa@46AE@  as recommended by Muller & Wang,8 where n u MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaad6 gakmaaBaaajeaqbaqcLbmacaWG1baaleqaaaaa@3C9A@  is the number of uncensored observations. The pilot estimators λ ^ (x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiqbeU 7aSzaajaGaaGikaiaadIhacaaIPaaaaa@3D25@  of λ(x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabeU 7aSjaaiIcacaWG4bGaaGykaaaa@3D15@  are obtained by using b n (x) b 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadk gakmaaBaaajeaqbaqcLbmacaWGUbaaleqaaKqzGeGaaGikaiaadIha caaIPaGaeyyyIORaamOyaOWaaSbaaKqaafaajugWaiaaicdaaSqaba aaaa@4490@  and our estimators (8), (13) and (14).

Step 2: Minimizing of "0362AMSE( b n ,x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaik cacaaIWaGaaG4maiaaiAdacaaIYaGaamyqaiaad2eacaWGtbGaamyr aiaaiIcacaWGIbGcdaWgaaqcbauaaKqzadGaamOBaaWcbeaajugibi aaiYcacaWG4bGaaGykaaaa@4707@ : Choose an equispaced grid of m1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaad2 gacaaIXaaaaa@3AAC@  points x ˜ i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiqadI hagaacaOWaaSbaaKqaafaajugWaiaadMgaaSqabaaaaa@3CA7@ , i=1,,m1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadM gacaaI9aGaaGymaiaaiYcacqWIVlctcaaISaGaamyBaiaaigdaaaa@4076@  between 0 and T. For each of the grid points x ˜ i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiqadI hagaacaOWaaSbaaKqaafaajugWaiaadMgaaSqabaaaaa@3CA7@  compute "0362AMSE( b n , x ˜ i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaik cacaaIWaGaaG4maiaaiAdacaaIYaGaamyqaiaad2eacaWGtbGaamyr aiaaiIcacaWGIbGcdaWgaaqcbauaaKqzadGaamOBaaWcbeaajugibi aaiYcaceWG4bGbaGaakmaaBaaajeaqbaqcLbmacaWGPbaaleqaaKqz GeGaaGykaaaa@4A41@  in (17) with k=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadU gacaaI9aGaaGimaaaa@3B70@  and obtain its minimizes b ˜ ( x ˜ i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiqadk gagaacaiaaiIcaceWG4bGbaGaakmaaBaaajeaqbaqcLbmacaWGPbaa leqaaKqzGeGaaGykaaaa@3F91@  on the interval [ b 0 /4,4 b 0 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaiU facaWGIbGcdaWgaaqcbauaaKqzadGaaGimaaWcbeaajugibiaai+ca caaI0aGaaGilaiaaisdacaWGIbGcdaWgaaqcbauaaKqzadGaaGimaa Wcbeaajugibiaai2faaaa@4572@ , say. The minimization of "0362AMSE( b n , x ˜ i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaik cacaaIWaGaaG4maiaaiAdacaaIYaGaamyqaiaad2eacaWGtbGaamyr aiaaiIcacaWGIbGcdaWgaaqcbauaaKqzadGaamOBaaWcbeaajugibi aaiYcaceWG4bGbaGaakmaaBaaajeaqbaqcLbmacaWGPbaaleqaaKqz GeGaaGykaaaa@4A41@  may be computed via Discretisation.

Step 3: Smoothing bandwidths: Choose another equispaced grid of m2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaad2 gacaaIYaaaaa@3AAD@  points x r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadI hakmaaBaaajeaqbaqcLbmacaWGYbaaleqaaaaa@3CA1@ , r=1,,m2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadk hacaaI9aGaaGymaiaaiYcacqWIVlctcaaISaGaamyBaiaaikdaaaa@4080@ , over the interval [0,T] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaiU facaaIWaGaaGilaiaadsfacaaIDbaaaa@3D14@  on which the final hazard estimator is desired. Running local linear smoother by employing global bandwidth b ˜ 0 = b 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiqadk gagaacaOWaaSbaaKqaafaajugWaiaaicdaaSqabaqcLbsacaaI9aGa amOyaOWaaSbaaKqaafaajugWaiaaicdaaSqabaaaaa@4102@  or 3 2 b 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWcaaqaaK qzGeGaaG4maaGcbaqcLbsacaaIYaaaaiaadkgakmaaBaaajeaqbaqc LbmacaaIWaaaleqaaaaa@3E70@ :

b ^ ( x r )= i=1 m1 w i b ˜ ( x ˜ i )/ i=1 m1 w i , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiqadk gagaqcaiaaiIcacaWG4bGcdaWgaaqcbauaaKqzadGaamOCaaWcbeaa jugibiaaiMcacaaI9aGcdaaeWbqabKqaafaajugWaiaadMgacaaI9a GaaGymaaqcbauaaKqzGdGaamyBaiaaigdaaKqzGeGaeyyeIuoacaWG 3bGcdaWgaaqcbauaaKqzadGaamyAaaWcbeaajugibiqadkgagaacai aaiIcaceWG4bGbaGaajuaGdaWgaaqcbauaaKqzadGaamyAaaqcbaua baqcLbsacaaIPaGaaG4laOWaaabCaeqajeaqbaqcLbmacaWGPbGaaG ypaiaaigdaaKqaafaajugWaiaad2gacaaIXaaajugibiabggHiLdGa am4DaOWaaSbaaKqaafaajugWaiaadMgaaSqabaqcLbsacaaISaaaaa@6527@

where w i =K( x ˜ i x r b ˜ 0 )[ M n,2 M n,1 ( x ˜ i x r )] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadE hakmaaBaaajeaqbaqcLbmacaWGPbaaleqaaKqzGeGaaGypaiaadUea caaIOaGcdaWcaaqaaKqzGeGabmiEayaaiaGcdaWgaaqcbauaaKqzad GaamyAaaWcbeaajugibiabgkHiTiaadIhakmaaBaaajeaqbaqcLbma caWGYbaaleqaaaGcbaqcLbsaceWGIbGbaGaakmaaBaaajeaqbaqcLb macaaIWaaaleqaaaaajugibiaaiMcacaaIBbGaamytaOWaaSbaaKqa afaajugWaiaad6gacaaISaGaaGOmaaWcbeaajugibiabgkHiTiaad2 eakmaaBaaajeaqbaqcLbmacaWGUbGaaGilaiaaigdaaSqabaqcLbsa caaIOaGabmiEayaaiaGcdaWgaaqcbauaaKqzadGaamyAaaWcbeaaju gibiabgkHiTiaadIhakmaaBaaajeaqbaqcLbmacaWGYbaaleqaaKqz GeGaaGykaiaai2faaaa@6671@  and M n,j = i=1 m1 K(( x ˜ i x r )/ b ˜ 0 )( x ˜ i x r ) j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaad2 eakmaaBaaajeaqbaqcLbmacaWGUbGaaGilaiaadQgaaSqabaqcLbsa caaI9aGcdaaeWaqabKqaafaajugWaiaadMgacaaI9aGaaGymaaqcba uaaKqzadGaamyBaiaaigdaaKqzGeGaeyyeIuoacaWGlbGaaGikaiaa iIcaceWG4bGbaGaakmaaBaaajeaqbaqcLbmacaWGPbaaleqaaKqzGe GaeyOeI0IaamiEaOWaaSbaaKqaafaajugWaiaadkhaaSqabaqcLbsa caaIPaGaaG4laiqadkgagaacaOWaaSbaaKqaafaajugWaiaaicdaaS qabaqcLbsacaaIPaGaaGikaiqadIhagaacaOWaaSbaaKqaafaajugW aiaadMgaaSqabaqcLbsacqGHsislcaWG4bGcdaWgaaqcbauaaKqzad GaamOCaaWcbeaajugibiaaiMcakmaaCaaaleqajeaqbaqcLbmacaWG Qbaaaaaa@67EB@ , for j=1,2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadQ gacaaI9aGaaGymaiaaiYcacaaIYaaaaa@3CE2@ .

Step 4: Final hazard function estimators: Using (8), (13) and (14), obtain the estimators λ ^ ( x r ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiqbeU 7aSzaajaGaaGikaiaadIhakmaaBaaajeaqbaqcLbmacaWGYbaaleqa aKqzGeGaaGykaaaa@4059@  by employing the bandwidth b ^ ( x r ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiqadk gagaqcaiaaiIcacaWG4bqcfa4aaSbaaKqaafaajugWaiaadkhaaKqa afqaaKqzGeGaaGykaaaa@404F@ , for r=1,,m2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadk hacaaI9aGaaGymaiaaiYcacqWIVlctcaaISaGaamyBaiaaikdaaaa@4080@ .

The above algorithm can be repeated by using the estimators λ ^ ( x r ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiqbeU 7aSzaajaGaaGikaiaadIhakmaaBaaajeaqbaqcLbmacaWGYbaaleqa aKqzGeGaaGykaaaa@4059@  in Step 4 as pilot estimators in Step 1 and running Step 2 – Step 4 again. The pilot estimators of λ(x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabeU 7aSjaaiIcacaWG4bGaaGykaaaa@3D15@  in Step 1 can also be obtained via maximum likelihood if one has a plausible parametric model in mind.18

Appendix proofs of theorems

Proofs of theorems 3.1 and 3.2

Using (1) and (9), we obtain that

  S nl ( x 0 )= 1 b n K( u x 0 b n )( u x 0 b n ) l d Λ n (u) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaado fakmaaBaaajeaqbaqcLbmacaWGUbGaeS4eHWgaleqaaKqzGeGaaGik aiaadIhakmaaBaaajeaqbaqcLbmacaaIWaaaleqaaKqzGeGaaGykai aai2dakmaapeaabeWcbeqabKqzGeGaey4kIipakmaalaaabaqcLbsa caaIXaaakeaajugibiaadkgakmaaBaaajeaqbaqcLbmacaWGUbaale qaaaaajugibiaadUeacaaIOaGcdaWcaaqaaKqzGeGaamyDaiabgkHi TiaadIhakmaaBaaajeaqbaqcLbmacaaIWaaaleqaaaGcbaqcLbsaca WGIbGcdaWgaaqcbauaaKqzGdGaamOBaaWcbeaaaaqcLbsacaaIPaGa aGikaOWaaSaaaeaajugibiaadwhacqGHsislcaWG4bGcdaWgaaqcba uaaKqzadGaaGimaaWcbeaaaOqaaKqzGeGaamOyaOWaaSbaaKqaafaa jugWaiaad6gaaSqabaaaaKqzGeGaaGykaOWaaWbaaSqabKqaafaaju gWaiabloriSbaajugibiaadsgacqqHBoatkmaaBaaajeaqbaqcLbma caWGUbaaleqaaKqzGeGaaGikaiaadwhacaaIPaaaaa@7154@

= 1 b n K( u x 0 b n )( u x 0 b n ) l dΛ(u)+ 1 b n K( u x 0 b n )( u x 0 b n ) l d[ Λ n (u)Λ(u)] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaai2 dakmaapeaabeWcbeqabKqzGeGaey4kIipakmaalaaabaqcLbsacaaI XaaakeaajugibiaadkgakmaaBaaajeaqbaqcLbmacaWGUbaaleqaaa aajugibiaadUeacaaIOaGcdaWcaaqaaKqzGeGaamyDaiabgkHiTiaa dIhakmaaBaaajeaqbaqcLbmacaaIWaaaleqaaaGcbaqcLbsacaWGIb GcdaWgaaqcbauaaKqzadGaamOBaaWcbeaaaaqcLbsacaaIPaGaaGik aOWaaSaaaeaajugibiaadwhacqGHsislcaWG4bGcdaWgaaqcbauaaK qzadGaaGimaaWcbeaaaOqaaKqzGeGaamOyaOWaaSbaaKqaafaajugW aiaad6gaaSqabaaaaKqzGeGaaGykaOWaaWbaaSqabeaajugibiablo riSbaacaWGKbGaeu4MdWKaaGikaiaadwhacaaIPaGaey4kaSIcdaWd baqabSqabeqajugibiabgUIiYdGcdaWcaaqaaKqzGeGaaGymaaGcba qcLbsacaWGIbGcdaWgaaqcbauaaKqzadGaamOBaaWcbeaaaaqcLbsa caWGlbGaaGikaOWaaSaaaeaajugibiaadwhacqGHsislcaWG4bGcda WgaaqcbauaaKqzadGaaGimaaWcbeaaaOqaaKqzGeGaamOyaOWaaSba aKqaafaajugWaiaad6gaaSqabaaaaKqzGeGaaGykaiaaiIcakmaala aabaqcLbsacaWG1bGaeyOeI0IaamiEaOWaaSbaaKqaafaajugWaiaa icdaaSqabaaakeaajugibiaadkgakmaaBaaajeaqbaqcLbmacaWGUb aaleqaaaaajugibiaaiMcakmaaCaaaleqajeaqbaqcLbmacqWItecB aaqcLbsacaWGKbGaaG4waiabfU5amPWaaSbaaKqaafaajugWaiaad6 gaaSqabaqcLbsacaaIOaGaamyDaiaaiMcacqGHsislcqqHBoatcaaI OaGaamyDaiaaiMcacaaIDbaaaa@9645@

β nl ( x 0 )+ γ nl . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabgg Mi6kabek7aIPWaaSbaaKqaafaajugWaiaad6gacqWItecBaSqabaqc LbsacaaIOaGaamiEaOWaaSbaaKqaafaajugWaiaaicdaaSqabaqcLb sacaaIPaGaey4kaSIaeq4SdCMcdaWgaaqcbauaaKqzadGaamOBaiab loriSbWcbeaajugibiaai6caaaa@4DC5@                       (18)

Let β n ( x 0 )=( β n0 ,, β np ) T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabek 7aIPWaaSbaaKqaafaajugWaiaad6gaaSqabaqcLbsacaaIOaGaamiE aOWaaSbaaKqaafaajugWaiaaicdaaSqabaqcLbsacaaIPaGaaGypai aaiIcacqaHYoGykmaaBaaajeaqbaqcLbmacaWGUbGaaGimaaWcbeaa jugibiaaiYcacqWIVlctcaaISaGaeqOSdiMcdaWgaaqcbauaaKqzad GaamOBaiaadchaaSqabaqcLbsacaaIPaGcdaahaaWcbeqcbauaaKqz adGaamivaaaaaaa@5688@  and γ n ( x 0 )=( γ n0 ,, γ np ) T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabeo 7aNPWaaSbaaKqaafaajugWaiaad6gaaSqabaqcLbsacaaIOaGaamiE aOWaaSbaaKqaafaajugWaiaaicdaaSqabaqcLbsacaaIPaGaaGypai aaiIcacqaHZoWzkmaaBaaajeaqbaqcLbmacaWGUbGaaGimaaWcbeaa jugibiaaiYcacqWIVlctcaaISaGaeq4SdCMcdaWgaaqcbauaaKqzad GaamOBaiaadchaaSqabaqcLbsacaaIPaGcdaahaaWcbeqcbauaaKqz adGaamivaaaaaaa@569A@ . Then S n ( x 0 )= β n ( x 0 )+ γ n ( x 0 ). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaho fakmaaBaaajeaqbaqcLbmacaWGUbaaleqaaKqzGeGaaGikaiaadIha kmaaBaaajeaqbaqcLbmacaaIWaaaleqaaKqzGeGaaGykaiaai2dacq aHYoGykmaaBaaajeaqbaqcLbmacaWGUbaaleqaaKqzGeGaaGikaiaa dIhakmaaBaaajeaqbaqcLbmacaaIWaaaleqaaKqzGeGaaGykaiabgU caRiabeo7aNPWaaSbaaKqaafaajugWaiaad6gaaSqabaqcLbsacaaI OaGaamiEaOWaaSbaaKqaafaajugWaiaaicdaaSqabaqcLbsacaaIPa GaaGOlaaaa@591F@  By (8), we have

B[ a ^ ( x 0 )a( x 0 )]=[ S 1 β n ( x 0 )Ba( x 0 )]+ S 1 γ n ( x 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaahk eacaaIBbGabCyyayaajaGaaGikaiaadIhakmaaBaaajeaqbaqcLbma caaIWaaaleqaaKqzGeGaaGykaiabgkHiTiaahggacaaIOaGaamiEaO WaaSbaaKqaafaajugWaiaaicdaaSqabaqcLbsacaaIPaGaaGyxaiaa i2dacaaIBbGaaC4uaKqbaoaaCaaajeaqbeqaaKqzadGaeyOeI0IaaG ymaaaajugibiabek7aIPWaaSbaaSqaaKqzGeGaamOBaaWcbeaajugi biaaiIcacaWG4bGcdaWgaaqcbauaaKqzadGaaGimaaWcbeaajugibi aaiMcacqGHsislcaWHcbGaaCyyaiaaiIcacaWG4bGcdaWgaaqcbaua aKqzadGaaGimaaWcbeaajugibiaaiMcacaaIDbGaey4kaSIaaC4uaO WaaWbaaSqabKqaafaajugWaiabgkHiTiaaigdaaaqcLbsacqaHZoWz kmaaBaaaleaajugibiaad6gaaSqabaqcLbsacaaIOaGaamiEaOWaaS baaKqaafaajugWaiaaicdaaSqabaqcLbsacaaIPaaaaa@7124@

α n ( x 0 )+ S 1 γ n ( x 0 ). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabgg Mi6kabeg7aHPWaaSbaaKqaafaajugWaiaad6gaaSqabaqcLbsacaaI OaGaamiEaOWaaSbaaKqaafaajugWaiaaicdaaSqabaqcLbsacaaIPa Gaey4kaSIaaC4uaOWaaWbaaSqabKqaafaajugWaiabgkHiTiaaigda aaqcLbsacqaHZoWzkmaaBaaajeaqbaqcLbmacaWGUbaaleqaaKqzGe GaaGikaiaadIhakmaaBaaajeaqbaqcLbmacaaIWaaaleqaaKqzGeGa aGykaiaai6caaaa@557C@                   (19)

We will show that α n ( x 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabeg 7aHPWaaSbaaKqaafaajugWaiaad6gaaSqabaqcLbsacaaIOaGaamiE aOWaaSbaaKqaafaajugWaiaaicdaaSqabaqcLbsacaaIPaaaaa@4327@  contributes to the bias term, and S 1 γ n ( x 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaho fakmaaCaaaleqajeaqbaqcLbmacqGHsislcaaIXaaaaKqzGeGaeq4S dCMcdaWgaaqcbauaaKqzadGaamOBaaWcbeaajugibiaaiIcacaWG4b GcdaWgaaqcbauaaKqzadGaaGimaaWcbeaajugibiaaiMcaaaa@47F1@  the variance term of our estimator.

  1. By (18), a change of variable for integration, and the Taylor expansion, we get
  2. β nl ( x 0 )= K(t) t l λ( x 0 + b n t)dt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabek 7aIPWaaSbaaKqaafaajugWaiaad6gacqWItecBaSqabaqcLbsacaaI OaGaamiEaOWaaSbaaKqaafaajugWaiaaicdaaSqabaqcLbsacaaIPa GaaGypaOWaa8qaaeqaleqabeqcLbsacqGHRiI8aiaadUeacaaIOaGa amiDaiaaiMcacaWG0bGcdaahaaWcbeqcbauaaKqzadGaeS4eHWgaaK qzGeGaeq4UdWMaaGikaiaadIhakmaaBaaajeaqbaqcLbmacaaIWaaa leqaaKqzGeGaey4kaSIaamOyaOWaaSbaaKqaafaajugWaiaad6gaaS qabaqcLbsacaWG0bGaaGykaiaadsgacaWG0baaaa@5E20@

    = j=0 p b n j j! λ (j) ( x 0 ) s l+j + b n p+1 (p+1)! λ (p+1) ( x 0 ) s l+p+1 (1+ o p (1)). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaai2 dakmaaqahabeqcbauaaKqzadGaamOAaiaai2dacaaIWaaajeaqbaqc LbmacaWGWbaajugibiabggHiLdGcdaWcaaqaaKqzGeGaamOyaKqbao aaDaaajeaqbaqcLbmacaWGUbaajeaqbaqcLbmacaWGQbaaaaGcbaqc LbsacaWGQbGaaGyiaaaacqaH7oaBjuaGdaahaaqcbauabeaajugWai aaiIcacaWGQbGaaGykaaaajugibiaaiIcacaWG4bGcdaWgaaqcbaua aKqzadGaaGimaaWcbeaajugibiaaiMcacaWGZbqcfa4aaSbaaKqaaf aajugWaiabloriSjabgUcaRiaadQgaaKqaafqaaKqzGeGaey4kaSIc daWcaaqaaKqzGeGaamOyaKqbaoaaDaaajeaqbaqcLbmacaWGUbaaje aqbaqcLbmacaWGWbGaey4kaSIaaGymaaaaaOqaaKqzGeGaaGikaiaa dchacqGHRaWkcaaIXaGaaGykaiaaigcaaaGaeq4UdWwcfa4aaWbaaK qaafqabaqcLbmacaaIOaGaamiCaiabgUcaRiaaigdacaaIPaaaaKqz GeGaaGikaiaadIhakmaaBaaajeaqbaqcLbmacaaIWaaaleqaaKqzGe GaaGykaKqzadGaam4CaOWaaSbaaSqaaKqzGeGaeS4eHWMaey4kaSIa amiCaiabgUcaRiaaigdaaSqabaqcLbmacaaIOaGaaGymaiabgUcaRi aad+gajuaGdaWgaaqcbauaaKqzadGaamiCaaqcbauabaqcLbmacaaI OaGaaGymaiaaiMcacaaIPaqcLbsacaaIUaaaaa@907D@

    Then

    α n ( x 0 )= b n p+1 (p+1)! λ (p+1) ( x 0 ) S 1 c p (1+o(1)). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabeg 7aHPWaaSbaaKqaafaajugWaiaad6gaaSqabaqcLbsacaaIOaGaamiE aOWaaSbaaKqaafaajugWaiaaicdaaSqabaqcLbsacaaIPaGaaGypaO WaaSaaaeaajugibiaadkgajuaGdaqhaaqcbauaaKqzadGaamOBaaqc bauaaKqzadGaamiCaiabgUcaRiaaigdaaaaakeaajugibiaaiIcaca WGWbGaey4kaSIaaGymaiaaiMcacaaIHaaaaiabeU7aSPWaaWbaaSqa bKqaafaajugWaiaaiIcacaWGWbGaey4kaSIaaGymaiaaiMcaaaqcLb sacaaIOaGaamiEaOWaaSbaaSqaaKqzGeGaaGimaaWcbeaajugibiaa iMcacaWHtbGcdaahaaWcbeqcbauaaKqzadGaeyOeI0IaaGymaaaaju gibiaahogakmaaBaaaleaajugibiaadchaaSqabaqcLbmacaaIOaGa aGymaiabgUcaRiaad+gacaaIOaGaaGymaiaaiMcacaaIPaqcLbsaca aIUaaaaa@6E40@              (20)

    In particular, if λ(x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabeU 7aSjaaiIcacaWG4bGaaGykaaaa@3D15@  is a polynomial up to order p in a neighborhood of x 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadI hakmaaBaaajeaqbaqcLbmacaaIWaaaleqaaaaa@3C64@ , then the exact bias α n ( x 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabeg 7aHPWaaSbaaKqaafaajugWaiaad6gaaSqabaqcLbsacaaIOaGaamiE aOWaaSbaaKqaafaajugWaiaaicdaaSqabaqcLbsacaaIPaaaaa@4327@  of the LP estimator is zero.

  3. For a G < a W MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadg gakmaaBaaajeaqbaqcLbmacaWGhbaaleqaaKqzGeGaaGipaiaadgga kmaaBaaajeaqbaqcLbmacaWGxbaaleqaaaaa@4124@ , by Gurler & Wang,1 we have

Λ n (x)Λ(x)= 1 n i=1 n ξ( Y i , T i , δ i ,x)+ r n (x), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabfU 5amPWaaSbaaKqaafaajugOaiaad6gaaSqabaqcLbsacaaIOaGaamiE aiaaiMcacqGHsislcqqHBoatcaaIOaGaamiEaiaaiMcacaaI9aGcda WcaaqaaKqzGeGaaGymaaGcbaqcLbsacaWGUbaaaOWaaabCaeqajeaq baqcLbmacaWGPbGaaGypaiaaigdaaKqaafaajugWaiaad6gaaKqzGe GaeyyeIuoacqaH+oaEcaaIOaGaamywaOWaaSbaaKqaafaajugWaiaa dMgaaSqabaqcLbsacaaISaGaamivaOWaaSbaaKqaafaajugWaiaadM gaaSqabaqcLbsacaaISaGaeqiTdqMcdaWgaaqcbauaaKqzadGaamyA aaWcbeaajugibiaaiYcacaWG4bGaaGykaiabgUcaRiaadkhakmaaBa aajeaqbaqcLbmacaWGUbaaleqaaKqzGeGaaGikaiaadIhacaaIPaGa aGilaaaa@6CBD@              (21)

where for x0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadI hacqGHLjYScaaIWaaaaa@3C7C@ , any b< b W MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadk gacaaI8aGaamOyaOWaaSbaaKqaafaajugWaiaadEfaaSqabaaaaa@3E1D@  and δ=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabes 7aKjaai2dacaaIXaaaaa@3C26@  or 0, sup 0xb | r n (x)|=O( logn n ),a.s. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaqfqaqabK qaafaajug4aiaaicdacqGHKjYOcaWG4bGaeyizImQaamOyaaWcbeGc baqcLbsaciGGZbGaaiyDaiaacchaaaGaaGiFaiaadkhakmaaBaaaje aqbaqcLbmacaWGUbaaleqaaKqzGeGaaGikaiaadIhacaaIPaGaaGiF aiaai2dacaWGpbGaaGikaOWaaSaaaeaajugibiGacYgacaGGVbGaai 4zaiaad6gaaOqaaKqzGeGaamOBaaaacaaIPaGaaGilaiaaysW7caaM e8Uaamyyaiaai6cacaWGZbGaaGOlaaaa@5C73@  and

ξ(y,t,δ,x)=I(yx,δ=1)/C(y) 0 x I(tuy)[C(u )] 2 d W 1 (u), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabe6 7a4jaaiIcacaWG5bGaaGilaiaadshacaaISaGaeqiTdqMaaGilaiaa dIhacaaIPaGaaGypaiaadMeacaaIOaGaamyEaiabgsMiJkaadIhaca aISaGaeqiTdqMaaGypaiaaigdacaaIPaGaaG4laiaadoeacaaIOaGa amyEaiaaiMcacqGHsislkmaapedabeqcbauaaKqzadGaaGimaaqcba uaaKqzadGaamiEaaqcLbsacqGHRiI8aiaadMeacaaIOaGaamiDaiab gsMiJkaadwhacqGHKjYOcaWG5bGaaGykaiaaiUfacaWGdbGaaGikai aadwhacaaIPaGaaGyxaOWaaWbaaSqabKqaafaajugWaiabgkHiTiaa ikdaaaqcLbsacaWGKbGaam4vaOWaaSbaaKqaafaajugWaiaaigdaaS qabaqcLbsacaaIOaGaamyDaiaaiMcacaaISaaaaa@72C6@

which satisfies that Eξ( X i , δ i ,x)=0, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadw eacqaH+oaEcaaIOaGaamiwaOWaaSbaaKqaafaajugWaiaadMgaaSqa baqcLbsacaaISaGaeqiTdqMcdaWgaaqcbauaaKqzadGaamyAaaWcbe aajugibiaaiYcacaWG4bGaaGykaiaai2dacaaIWaGaaGilaaaa@4A69@  and Cov(ξ( Y i , T i , δ i ,s),ξ( Y i , T i , δ i ,t))=g(min(s,t)), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeGabaaFfKqzGe Gaam4qaiaad+gacaWG2bGaaGikaiabe67a4jaaiIcacaWGzbGcdaWg aaqcbauaaKqzadGaamyAaaWcbeaajugibiaaiYcacaWGubGcdaWgaa qcbauaaKqzadGaamyAaaWcbeaajugibiaaiYcacqaH0oazkmaaBaaa jeaqbaqcLbmacaWGPbaaleqaaKqzGeGaaGilaiaadohacaaIPaGaaG ilaiabe67a4jaaiIcacaWGzbGcdaWgaaqcbauaaKqzadGaamyAaaWc beaajugibiaaiYcacaWGubGcdaWgaaqcbauaaKqzadGaamyAaaWcbe aajugibiaaiYcacqaH0oazkmaaBaaajeaqbaqcLboacaWGPbaaleqa aKqzGeGaaGilaiaadshacaaIPaGaaGykaiaai2dacaWGNbGaaGikai Gac2gacaGGPbGaaiOBaiaaiIcacaWGZbGaaGilaiaadshacaaIPaGa aGykaiaaiYcaaaa@6F60@  where g(x)= 0 x [C(u)] 2 d W 1 (u). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadE gacaaIOaGaamiEaiaaiMcacaaI9aGcdaWdXaqabKqaafaajugWaiaa icdaaKqaafaajugWaiaadIhaaKqzGeGaey4kIipacaaIBbGaam4qai aaiIcacaWG1bGaaGykaiaai2fakmaaCaaaleqajeaqbaqcLbmacqGH sislcaaIYaaaaKqzGeGaamizaiaadEfajuaGdaWgaaqcbauaaKqzad GaaGymaaqcbauabaqcLbsacaaIOaGaamyDaiaaiMcacaaIUaaaaa@55D7@  Let K l (t)=K(t) t l MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadU eakmaaBaaajeaqbaqcLbkacqWItecBaSqabaqcLbsacaaIOaGaamiD aiaaiMcacaaI9aGaam4saiaaiIcacaWG0bGaaGykaiaadshakmaaCa aaleqajeaqbaqcLbmacqWItecBaaaaaa@4749@ . Using the definition of γ nl MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabeo 7aNPWaaSbaaKqaafaajugWaiaad6gacqWItecBaSqabaaaaa@3E78@ , (21) and integration by parts, we obtain the following almost surely representation of γ nl ( x 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabeo 7aNPWaaSbaaKqaafaajugWaiaad6gacqWItecBaSqabaqcLbsacaaI OaGaamiEaOWaaSbaaKqaafaajugWaiaaicdaaSqabaqcLbsacaaIPa aaaa@4460@ :

γ nl ( x 0 )= σ nl ( x 0 )+ e nl ( x 0 ), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabeo 7aNPWaaSbaaKqaafaajugWaiaad6gacqWItecBaSqabaqcLbsacaaI OaGaamiEaOWaaSbaaKqaafaajugWaiaaicdaaSqabaqcLbsacaaIPa GaaGypaiabeo8aZPWaaSbaaKqaafaajug4aiaad6gacqWItecBaSqa baqcLbsacaaIOaGaamiEaOWaaSbaaKqaafaajugWaiaaicdaaSqaba qcLbsacaaIPaGaey4kaSIaamyzaOWaaSbaaKqaafaajugWaiaad6ga cqWItecBaSqabaqcLbsacaaIOaGaamiEaOWaaSbaaKqaafaajugWai aaicdaaSqabaqcLbsacaaIPaGaaGilaaaa@5D00@          (22)

where

σ nl ( x 0 )= n 1 i=1 n 1 b n K( u x 0 b n )( u x 0 b n ) l dξ( Y i , T i , δ i ,u) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabeo 8aZPWaaSbaaKqaafaajugWaiaad6gacqWItecBaSqabaqcLbsacaaI OaGaamiEaOWaaSbaaKqaafaajugWaiaaicdaaSqabaqcLbsacaaIPa GaaGypaiaad6gakmaaCaaaleqajeaqbaqcLbmacqGHsislcaaIXaaa aOWaaabCaeqajeaqbaqcLbmacaWGPbGaaGypaiaaigdaaKqaafaaju gWaiaad6gaaKqzGeGaeyyeIuoakmaapeaabeWcbeqabKqzGeGaey4k IipakmaalaaabaqcLbsacaaIXaaakeaajugibiaadkgakmaaBaaaje aqbaqcLbmacaWGUbaaleqaaaaajugibiaadUeacaaIOaGcdaWcaaqa aKqzGeGaamyDaiabgkHiTiaadIhakmaaBaaajeaqbaqcLbmacaaIWa aaleqaaaGcbaqcLbsacaWGIbGcdaWgaaqcbauaaKqzadGaamOBaaWc beaaaaqcLbsacaaIPaGaaGikaOWaaSaaaeaajugibiaadwhacqGHsi slcaWG4bGcdaWgaaqcbauaaKqzadGaaGimaaWcbeaaaOqaaKqzGeGa amOyaOWaaSbaaKqaafaajugWaiaad6gaaSqabaaaaKqzGeGaaGykaO WaaWbaaSqabKqaafaajugWaiabloriSbaajugibiaadsgacqaH+oaE caaIOaGaamywaOWaaSbaaKqaafaajugWaiaadMgaaSqabaqcLbsaca aISaGaamivaOWaaSbaaKqaafaajugWaiaadMgaaSqabaqcLbsacaaI SaGaeqiTdqMcdaWgaaqcbauaaKqzadGaamyAaaWcbeaajugibiaaiY cacaWG1bGaaGykaaaa@8B9D@

=(n b n ) 1 i=1 n ξ( Y i , T i , δ i , x 0 + b n t)d K l (t) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaai2 dacaaIOaGaamOBaiaadkgakmaaBaaajeaqbaqcLbmacaWGUbaaleqa aKqzGeGaaGykaOWaaWbaaSqabKqaafaajugWaiabgkHiTiaaigdaaa GcdaaeWbqabKqaafaajugWaiaadMgacaaI9aGaaGymaaqcbauaaKqz adGaamOBaaqcLbsacqGHris5aOWaa8qaaeqaleqabeqcLbsacqGHRi I8aiabe67a4jaaiIcacaWGzbGcdaWgaaqcbauaaKqzadGaamyAaaWc beaajugibiaaiYcacaWGubGcdaWgaaqcbauaaKqzadGaamyAaaWcbe aajugibiaaiYcacqaH0oazkmaaBaaajeaqbaqcLbmacaWGPbaaleqa aKqzGeGaaGilaiaadIhakmaaBaaajeaqbaqcLbmacaaIWaaaleqaaK qzGeGaey4kaSIaamOyaOWaaSbaaKqaafaajugWaiaad6gaaSqabaqc LbsacaWG0bGaaGykaiaadsgacaWGlbGcdaWgaaqcbauaaKqzadGaeS 4eHWgaleqaaKqzGeGaaGikaiaadshacaaIPaaaaa@72BD@

is the stochastic component of S nl ( x 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaado fakmaaBaaajeaqbaqcLbmacaWGUbGaeS4eHWgaleqaaKqzGeGaaGik aiaadIhakmaaBaaajeaqbaqcLbmacaaIWaaaleqaaKqzGeGaaGykaa aa@4391@  and contributes to the variance of our estimator, and e nl ( x 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadw gakmaaBaaajeaqbaqcLbmacaWGUbGaeS4eHWgaleqaaKqzGeGaaGik aiaadIhakmaaBaaajeaqbaqcLbmacaaIWaaaleqaaKqzGeGaaGykaa aa@43A3@  is the negligible error of the approximation which satisfies

sup 0 x 0 b | e nl ( x 0 )|=O( logn n ),a.s. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaGfqbqabK qaafaajugWaiaaicdacqGHKjYOcaWG4bqcfa4aaSbaaKqaafaajugW aiaaicdaaKqaafqaaKqzadGaeyizImQaamOyaaWcbeGcbaqcLbsaci GGZbGaaiyDaiaacchaaaGaaGiFaiaadwgakmaaBaaajeaqbaqcLbma caWGUbGaeS4eHWgaleqaaKqzGeGaaGikaiaadIhakmaaBaaajeaqba qcLbmacaaIWaaaleqaaKqzGeGaaGykaiaaiYhacaaI9aGaam4taiaa iIcakmaalaaabaqcLbsaciGGSbGaai4BaiaacEgacaWGUbaakeaaju gibiaad6gaaaGaaGykaiaaiYcacaaMe8UaaGjbVlaadggacaaIUaGa am4Caiaai6caaaa@6508@               (23)

for 0lp MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaic dacqGHKjYOcqWItecBcqGHKjYOcaWGWbaaaa@3F49@ . Note that E( σ nl ( x 0 ))=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadw eacaaIOaGaeq4WdmNcdaWgaaqcbauaaKqzadGaamOBaiabloriSbWc beaajugibiaaiIcacaWG4bGcdaWgaaqcbauaaKqzadGaaGimaaWcbe aajugibiaaiMcacaaIPaGaaGypaiaaicdaaaa@482C@  and

Cov( σ nl ( x 0 ), σ nm ( x 0 ))= 1 n b n 1 b n g(min( x 0 + b n u, x 0 + b n v))d K l (u)d K m (v) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaado eacaWGVbGaamODaiaaiIcacqaHdpWCkmaaBaaajeaqbaqcLbmacaWG UbGaeS4eHWgaleqaaKqzGeGaaGikaiaadIhakmaaBaaajeaqbaqcLb macaaIWaaaleqaaKqzGeGaaGykaiaaiYcacqaHdpWCkmaaBaaajeaq baqcLbmacaWGUbGaamyBaaWcbeaajugibiaaiIcacaWG4bGcdaWgaa qcbauaaKqzadGaaGimaaWcbeaajugibiaaiMcacaaIPaGaaGypaOWa aSaaaeaajugibiaaigdaaOqaaKqzGeGaamOBaiaadkgakmaaBaaaje aqbaqcLbmacaWGUbaaleqaaaaakmaapeaabeWcbeqabKqzGeGaey4k IipakmaapeaabeWcbeqabKqzGeGaey4kIipakmaalaaabaqcLbsaca aIXaaakeaajugibiaadkgakmaaBaaajeaqbaqcLbmacaWGUbaaleqa aaaajugibiaadEgacaaIOaGaciyBaiaacMgacaGGUbGaaGikaiaadI hakmaaBaaajeaqbaqcLbmacaaIWaaaleqaaKqzGeGaey4kaSIaamOy aOWaaSbaaKqaafaajugOaiaad6gaaSqabaqcLbsacaWG1bGaaGilai aadIhakmaaBaaajeaqbaqcLbmacaaIWaaaleqaaKqzGeGaey4kaSIa amOyaOWaaSbaaKqaafaajugWaiaad6gaaSqabaqcLbsacaWG2bGaaG ykaiaaiMcacaWGKbGaam4saOWaaSbaaKqaafaajugWaiabloriSbWc beaajugibiaaiIcacaWG1bGaaGykaiaadsgacaWGlbGcdaWgaaqcba uaaKqzadGaamyBaaWcbeaajugibiaaiIcacaWG2bGaaGykaaaa@90AA@

= 1 n b n 2 + x 0 + b n v K l ( u x 0 b n )dg(u)d K m (v) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaai2 dakmaalaaabaqcLbsacaaIXaaakeaajugibiaad6gacaWGIbqcfa4a a0baaKqaafaajugWaiaad6gaaKqaafaajugWaiaaikdaaaaaaOWaa8 qmaeqajeaqbaqcLbmacqGHsislcqGHEisPaKqaafaajugWaiabgUca Riabg6HiLcqcLbsacqGHRiI8aOWaa8qmaeqaleaajugibiabgkHiTi abg6HiLcWcbaqcLbsacaWG4bGcdaWgaaqcbauaaKqzadGaaGimaaWc beaajugibiabgUcaRiaadkgakmaaBaaajeaqbaqcLbmacaWGUbaale qaaKqzGeGaamODaaGaey4kIipacaWGlbGcdaWgaaqcbauaaKqzadGa eS4eHWgaleqaaKqzGeGaaGikaOWaaSaaaeaajugibiaadwhacqGHsi slcaWG4bGcdaWgaaqcbauaaKqzadGaaGimaaWcbeaaaOqaaKqzGeGa amOyaOWaaSbaaKqaafaajugWaiaad6gaaSqabaaaaKqzGeGaaGykai aadsgacaWGNbGaaGikaiaadwhacaaIPaGaamizaiaadUeakmaaBaaa jeaqbaqcLboacaWGTbaaleqaaKqzGeGaaGikaiaadAhacaaIPaaaaa@78B2@

= 1 n b n λ( x 0 ) C( x 0 ) v l+m (1+o(1)). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaai2 dakmaalaaabaqcLbsacaaIXaaakeaajugibiaad6gacaWGIbGcdaWg aaqcbauaaKqzadGaamOBaaWcbeaaaaGcdaWcaaqaaKqzGeGaeq4UdW MaaGikaiaadIhakmaaBaaajeaqbaqcLbmacaaIWaaaleqaaKqzGeGa aGykaaGcbaqcLbsacaWGdbGaaGikaiaadIhakmaaBaaajeaqbaqcLb macaaIWaaaleqaaKqzGeGaaGykaaaacaWG2bGcdaWgaaqcbauaaKqz adGaeS4eHWMaey4kaSIaamyBaaWcbeaajugibiaaiIcacaaIXaGaey 4kaSIaam4BaiaaiIcacaaIXaGaaGykaiaaiMcacaaIUaaaaa@5BB9@         (24)

Let σ n ( x 0 )=( σ n0 ,, σ np ) T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabeo 8aZPWaaSbaaKqaafaajugWaiaad6gaaSqabaqcLbsacaaIOaGaamiE aOWaaSbaaKqaafaajugWaiaaicdaaSqabaqcLbsacaaIPaGaaGypai aaiIcacqaHdpWCkmaaBaaajeaqbaqcLbkacaWGUbGaaGimaaWcbeaa jugibiaaiYcacqWIVlctcaaISaGaeq4WdmNcdaWgaaqcbauaaKqzad GaamOBaiaadchaaSqabaqcLbsacaaIPaGcdaahaaWcbeqcbauaaKqz adGaamivaaaaaaa@56CE@  and e n ( x 0 )=( e n0 ,, e np ) T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaahw gakmaaBaaajeaqbaqcLbmacaWGUbaaleqaaKqzGeGaaGikaiaadIha kmaaBaaajeaqbaqcLbmacaaIWaaaleqaaKqzGeGaaGykaiaai2daca aIOaGaamyzaOWaaSbaaKqaafaajugWaiaad6gacaaIWaaaleqaaKqz GeGaaGilaiabl+UimjaaiYcacaWGLbGcdaWgaaqcbauaaKqzadGaam OBaiaadchaaSqabaqcLbsacaaIPaGcdaahaaWcbeqcbauaaKqzGdGa amivaaaaaaa@5487@ . Then by (22) and (24)

n b n S 1 γ n ( x 0 )= n b n S 1 σ n ( x 0 )+ n b n S 1 e n ( x 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaGcaaqaaK qzGeGaamOBaiaadkgakmaaBaaajeaqbaqcLboacaWGUbaaleqaaaqa baqcLbsacaWHtbGcdaahaaWcbeqcbauaaKqzadGaeyOeI0IaaGymaa aajugibiabeo7aNPWaaSbaaSqaaKqzGeGaamOBaaWcbeaajugibiaa iIcacaWG4bGcdaWgaaqcbauaaKqzGdGaaGimaaWcbeaajugibiaaiM cacaaI9aGcdaGcaaqaaKqzGeGaamOBaiaadkgakmaaBaaaleaajugi biaad6gaaSqabaaabeaajugibiaahofakmaaCaaaleqajeaqbaqcLb macqGHsislcaaIXaaaaKqzGeGaeq4WdmNcdaWgaaqcbauaaKqzadGa amOBaaWcbeaajugibiaaiIcacaWG4bGcdaWgaaqcbauaaKqzadGaaG imaaWcbeaajugibiaaiMcacqGHRaWkkmaakaaabaqcLbsacaWGUbGa amOyaOWaaSbaaSqaaKqzGeGaamOBaaWcbeaaaeqaaKqzGeGaaC4uaO WaaWbaaSqabKqaafaajug4aiabgkHiTiaaigdaaaqcLbsacaWHLbGc daWgaaWcbaqcLbsacaWGUbaaleqaaKqzGeGaaGikaiaadIhakmaaBa aajeaqbaqcLbmacaaIWaaaleqaaKqzGeGaaGykaaaa@7449@               (25)

and Cov( σ n ( x 0 ), σ n ( x 0 ))= 1 n b n λ( x 0 ) C( x 0 ) V * (1+o(1)). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaado eacaWGVbGaamODaiaaiIcacqaHdpWCkmaaBaaajeaqbaqcLbmacaWG UbaaleqaaKqzGeGaaGikaiaadIhakmaaBaaajeaqbaqcLbkacaaIWa aaleqaaKqzGeGaaGykaiaaiYcacqaHdpWCkmaaBaaajeaqbaqcLbma caWGUbaaleqaaKqzGeGaaGikaiaadIhakmaaBaaajeaqbaqcLbmaca aIWaaaleqaaKqzGeGaaGykaiaaiMcacaaI9aGcdaWcaaqaaKqzGeGa aGymaaGcbaqcLbsacaWGUbGaamOyaOWaaSbaaSqaaKqzGeGaamOBaa WcbeaaaaGcdaWcaaqaaKqzGeGaeq4UdWMaaGikaiaadIhakmaaBaaa jeaqbaqcLbmacaaIWaaaleqaaKqzGeGaaGykaaGcbaqcLbsacaWGdb GaaGikaiaadIhakmaaBaaajeaqbaqcLbmacaaIWaaaleqaaKqzGeGa aGykaaaacaWHwbGcdaahaaWcbeqcbauaaKqzadGaaGOkaaaajugibi aaiIcacaaIXaGaey4kaSIaam4BaiaaiIcacaaIXaGaaGykaiaaiMca caaIUaaaaa@71B9@  By the central limit theorem, we get

n b n σ n ( x 0 ) L N(0, λ( x 0 ) C( x 0 ) V * ). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaGcaaqaaK qzGeGaamOBaiaadkgakmaaBaaajeaqbaqcLbmacaWGUbaaleqaaaqa baqcLbsacqaHdpWCkmaaBaaajeaqbaqcLbmacaWGUbaaleqaaKqzGe GaaGikaiaadIhakmaaBaaajeaqbaqcLbmacaaIWaaaleqaaKqzGeGa aGykaOWaaCbiaeaajugibiabgkziUcWcbeqcbauaaKqzadGaamitaa aajugibiaad6eacaaIOaGaaGimaiaaiYcakmaalaaabaqcLbsacqaH 7oaBcaaIOaGaamiEaOWaaSbaaKqaafaajugWaiaaicdaaSqabaqcLb sacaaIPaaakeaajugibiaadoeacaaIOaGaamiEaOWaaSbaaKqaafaa jugWaiaaicdaaSqabaqcLbsacaaIPaaaaiaahAfakmaaCaaaleqaje aqbaqcLbmacaaIQaaaaKqzGeGaaGykaiaai6caaaa@64AD@          (26)

By (23), we know

sup 0 x 0 b | S 1 e n ( x 0 )|=O( logn n ),a.s. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaGfqbqabK qaafaajugOaiaaicdacqGHKjYOcaWG4bWcdaWgaaqcbauaaKqzGcGa aGimaaqcbauabaqcLbkacqGHKjYOcaWGIbaaleqakeaajugibiGaco hacaGG1bGaaiiCaaaacaaI8bGaaC4uaOWaaWbaaSqabKqaafaajugO aiabgkHiTiaaigdaaaqcLbsacaWHLbGcdaWgaaqcbauaaKqzadGaam OBaaWcbeaajugibiaaiIcacaWG4bGcdaWgaaqcbauaaKqzadGaaGim aaWcbeaajugibiaaiMcacaaI8bGaaGypaiaad+eacaaIOaGcdaWcaa qaaKqzGeGaciiBaiaac+gacaGGNbGaamOBaaGcbaqcLbsacaWGUbaa aiaaiMcacaaISaGaaGjbVlaaysW7caWGHbGaaGOlaiaadohacaaIUa aaaa@679A@       (27)

Then by (25), (26) and (27)

n b n S 1 γ n ( x 0 ) L N(0, λ( x 0 ) C( x 0 ) S 1 V * S 1 ). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaGcaaqaaK qzGeGaamOBaiaadkgakmaaBaaajeaqbaqcLbmacaWGUbaaleqaaaqa baqcLbsacaWHtbGcdaahaaWcbeqcbauaaKqzadGaeyOeI0IaaGymaa aajugibiabeo7aNPWaaSbaaKqaafaajugWaiaad6gaaSqabaqcLbsa caaIOaGaamiEaOWaaSbaaKqaafaajugWaiaaicdaaSqabaqcLbsaca aIPaGcdaWfGaqaaKqzGeGaeyOKH4kaleqabaqcLbmacaWGmbaaaKqz GeGaamOtaiaaiIcacaaIWaGaaGilaOWaaSaaaeaajugibiabeU7aSj aaiIcacaWG4bGcdaWgaaqcbauaaKqzadGaaGimaaWcbeaajugibiaa iMcaaOqaaKqzGeGaam4qaiaaiIcacaWG4bGcdaWgaaqcbauaaKqzad GaaGimaaWcbeaajugibiaaiMcaaaGaaC4uaOWaaWbaaSqabKqaafaa jug4aiabgkHiTiaaigdaaaqcLbsacaWHwbGcdaahaaWcbeqcbauaaK qzGdGaaGOkaaaajugibiaahofakmaaCaaaleqajeaqbaqcLbmacqGH sislcaaIXaaaaKqzGeGaaGykaiaai6caaaa@72CD@                   (28)

Combination of (19), (20) and (28) completes the proof of Theorem 3.2.

Note that σ nl ( x 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabeo 8aZPWaaSbaaKqaafaajug4aiaad6gacqWItecBaSqabaqcLbsacaaI OaGaamiEaOWaaSbaaKqaafaajugWaiaaicdaaSqabaqcLbsacaaIPa aaaa@449C@ , for l=0,,p MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiablo riSjaai2dacaaIWaGaaGilaiabl+UimjaaiYcacaWGWbaaaa@4000@ , are i.i.d.’s sums. By the SLN, we know σ nl ( x 0 ) a.s. E( σ nl ( x 0 ))=0. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabeo 8aZPWaaSbaaKqaafaajugWaiaad6gacqWItecBaSqabaqcLbsacaaI OaGaamiEaOWaaSbaaKqaafaajugOaiaaicdaaSqabaqcLbsacaaIPa GcdaWfGaqaaKqzGeGaeyOKH4kaleqajeaqbaqcLbmacaWGHbGaaGOl aiaadohacaaIUaaaaKqzGeGaamyraiaaiIcacqaHdpWCkmaaBaaaje aqbaqcLbmacaWGUbGaeS4eHWgaleqaaKqzGeGaaGikaiaadIhakmaa BaaajeaqbaqcLbmacaaIWaaaleqaaKqzGeGaaGykaiaaiMcacaaI9a GaaGimaiaai6caaaa@5C65@  Then σ n ( x 0 ) a.s. 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabeo 8aZPWaaSbaaKqaafaajugWaiaad6gaaSqabaqcLbsacaaIOaGaamiE aOWaaSbaaKqaafaajugWaiaaicdaaSqabaqcLbsacaaIPaGcdaWfGa qaaKqzGeGaeyOKH4kaleqajeaqbaqcLbmacaWGHbGaaGOlaiaadoha caaIUaaaaKqzGeGaaGimaaaa@4C29@ . This combined with (22) and (23) yields γ n ( x 0 ) a.s. 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabeo 7aNPWaaSbaaKqaafaajugWaiaad6gaaSqabaqcLbsacaaIOaGaamiE aOWaaSbaaKqaafaajugWaiaaicdaaSqabaqcLbsacaaIPaGcdaWfGa qaaKqzGeGaeyOKH4kaleqajeaqbaqcLboacaWGHbGaaGOlaiaadoha caaIUaaaaKqzGeGaaGimaaaa@4C2D@ . Therefore, by (19) and (20),

B[ a ^ ( x 0 )a( x 0 )] a.s. 0. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaahk eacaaIBbGabCyyayaajaGaaGikaiaadIhakmaaBaaajeaqbaqcLbma caaIWaaaleqaaKqzGeGaaGykaiabgkHiTiaahggacaaIOaGaamiEaO WaaSbaaKqaafaajugOaiaaicdaaSqabaqcLbsacaaIPaGaaGyxaOWa aCbiaeaajugibiabgkziUcWcbeqcbauaaKqzadGaamyyaiaai6caca WGZbGaaGOlaaaajugibiaaicdacaaIUaaaaa@528F@

Proofs of Theorem 3.3 The result follows by the same argument as in part (i) of the proof of Theorem 3.2.

Acknowledgement

This work is partially supported by Natural Science Foundation of Fujian Providence of China (2016J01024) and by Education and Scientific Research Projects for young and middle-aged teachers of Fujian Providence (JAT160383).

Conflict of interest

Author declares that there are no conflicts of interest.

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