eISSN: 2378-315X BBIJ

Biometrics & Biostatistics International Journal
Research Article
Volume 7 Issue 3

Stability within family of Pareto models

Mariam Zahid,1 Shakila Bashir2
1Department of Statistics, Kinnaird College for Women, Pakistan
2Department of Statistics, Forman Christian College, Pakistan
Received: April 16, 2018 | Published: May 09, 2018

Correspondence: Mariam Zahid, Department of Statistics, Kinnaird College for Women, Near Canal bank Jail Road, Lahore, Pakistan, Email

Citation: Zahid M, Bashir S. Stability within family of Pareto models. Biom Biostat Int J. 2018;7(3):177‒186. DOI: 10.15406/bbij.2018.07.00207

Abstract

Transmutation technique is applied to extend the workability and flexibility of weighted Pareto distribution. A weighted probability distribution improves precision for predictability and transmuting the same produces a better model for data fitting. Various statistical properties including moments and quantiles, reliability analysis, mean deviation, order statistics and record values of transmuted weighted Pareto (TWP) distribution are studied. The parameters of the distribution are evaluated using Maximum Likelihood Estimation (MLE). Application study compares different Pareto models to reveal the stability between them. A simulation analysis is also performed.

Keywords: weighted Pareto distribution, moments, reliability analysis, record values, MLE

Introduction

The goal of a probability distribution is to fit maximum data sets with utmost precision and minimum variance so that their behavior can be modeled and predicted. The existing distributions can be made more precise and useful for a wider spectrum of values by applying new techniques such as transmutation.

Pareto distribution was developed by Vilfredo Pareto. This distribution was meant to show the apportionment of income to the population. However, the purpose of the distribution is not restricted to income evaluations only; it uses the precept which requires the data to have a small to large proportion for example, the meteor showers on Earth. Fisher1 introduced the concept of weighted distributions by highlighting the idea of need for surety of occurrence of event. Zelen2 studied weighted distributions and delineated the methodology required to formulate a weighted distribution using cell kinetics. A weighted distribution is found using

f w ( x;θ )= x k .f( x;θ ) μ k ' MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadA gajuaGdaahaaWcbeqcbasaaKqzadGaam4DaaaajuaGdaqadaGcbaqc LbsacaWG4bGaai4oaiabeI7aXbGccaGLOaGaayzkaaqcLbsacqGH9a qpjuaGdaWcaaGcbaqcLbsacaWG4bqcfa4aaWbaaSqabKqaGeaajugW aiaadUgaaaqcLbsacaGGUaGaamOzaKqbaoaabmaakeaajugibiaadI hacaGG7aGaeqiUdehakiaawIcacaGLPaaaaeaajugibiabeY7aTTWa a0baaKqaGeaajugWaiaadUgaaKqaGeaajugWaiaacEcaaaaaaaaa@56F4@     (1)

Where E[ w( x ) ]= μ k ' MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadw eakmaadmaabaqcLbsacaWG3bGcdaqadaqaaKqzGeGaamiEaaGccaGL OaGaayzkaaaacaGLBbGaayzxaaqcLbsacqGH9aqpcqaH8oqBlmaaDa aajeaibaqcLbmacaWGRbaajeaibaqcLbmacaGGNaaaaaaa@460A@ . To derive transmuted weighted Pareto distribution, the weighted form used has k=1. Weighted distribution introduces the surety of occurrence of an event and hence can be considered a vital requisite for all probability distributions.

Transmuting a distribution means that a distribution is elaborated by adding more variables in effort to optimize its adaptability towards data. The Quadratic Rank Transmutation (QRT) technique uses an established formula to derive the new distribution.3 Transmutation, thus, can be carried out using the following relations:

f( x )=g( x )[ ( 1+λ )2λG( x ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamOzaK qbaoaabmaakeaajugibiaadIhaaOGaayjkaiaawMcaaKqzGeGaeyyp a0Jaam4zaKqbaoaabmaakeaajugibiaadIhaaOGaayjkaiaawMcaaK qbaoaadmaakeaajuaGdaqadaGcbaqcLbsacaaIXaGaey4kaSIaeq4U dWgakiaawIcacaGLPaaajugibiabgkHiTiaaikdacqaH7oaBcaWGhb qcfa4aaeWaaOqaaKqzGeGaamiEaaGccaGLOaGaayzkaaaacaGLBbGa ayzxaaaaaa@5258@   (2)

F( x )=( 1+λ )G( x )λ [ G( x ) ] 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamOraK qbaoaabmaakeaajugibiaadIhaaOGaayjkaiaawMcaaKqzGeGaeyyp a0tcfa4aaeWaaOqaaKqzGeGaaGymaiabgUcaRiabeU7aSbGccaGLOa GaayzkaaqcLbsacaWGhbqcfa4aaeWaaOqaaKqzGeGaamiEaaGccaGL OaGaayzkaaqcLbsacqGHsislcqaH7oaBjuaGdaWadaGcbaqcLbsaca WGhbqcfa4aaeWaaOqaaKqzGeGaamiEaaGccaGLOaGaayzkaaaacaGL BbGaayzxaaWcdaahaaqcbasabeaajugWaiaaikdaaaaaaa@54BB@       (3)

Equation (2) and (3) give the formula that can be used to derive the pdf and cdf for the transmuted distribution. g(x) represents the pdf and G(x) is the cdf of the parent distribution and λ[1,1] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeq4UdW MaeyOGIWSaai4waiabgkHiTiaaigdacaGGSaGaaGymaiaac2faaaa@3EDD@ . For λ=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabeU 7aSjabg2da9iaaicdaaaa@3A61@ , the transmuted model changes back to the parent model.

Transmutation technique has been proven very successful in bringing out some useful distributions. Dar et al.4 studied the transmuted weighted exponential distribution. Shahzad et al.5 transmuted the Singh-Maddla distribution. Nasser et al.6 found useful results for transmuted Weibull Logistic distribution. Ashour & Eltehiwy7 studied the transmuted exponentiated Lomax distribution. Aryal & Tsokos8 found that transmuting a distribution helps in advancement of a distribution. Aryal9 found that new generalizations of distributions help in extending the study. Khan et al.10 derived transmuted Kumaraswamy distribution and concluded that for statistical significance of model adequacy the transmuted distribution lead to a better fit than Kumaraswamy distribution. Transmutation studies have gained attention because of their ability to generate new flexible distributions that can help fit data with more precision.

In this paper weighted Pareto distribution is transmuted. Various statistical properties of the new distribution are studied including moments, quantiles, moment generating function (MGF), Bonferroni and Lorenz curves, reliability analysis, order statistics, record values and parameter estimation. Application study compares different Pareto models to see if there is stability between them.

Transmuted weighted pareto distribution

Mir & Ahmad11 derived the weighted Pareto distribution among some other weighted distributions. The pdf and cdf of the weighted Pareto distribution are given below:

f w = α β1 x β ( β1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamOzaK qbaoaaCaaaleqajeaibaqcLbmacaWG3baaaKqzGeGaeyypa0JaeqyS dewcfa4aaWbaaSqabKqaGeaajugWaiabek7aIjabgkHiTiaaigdaaa qcLbsacaWG4bWcdaahaaqcbasabeaajugWaiabgkHiTiabek7aIbaa juaGdaqadaGcbaqcLbsacqaHYoGycqGHsislcaaIXaaakiaawIcaca GLPaaaaaa@4EA6@   (4)

F w =1 α β1 x 1β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamOraK qbaoaaCaaaleqajeaibaqcLbmacaWG3baaaKqzGeGaeyypa0JaaGym aiabgkHiTiabeg7aHTWaaWbaaKqaGeqabaqcLbmacqaHYoGycqGHsi slcaaIXaaaaKqzGeGaamiEaSWaaWbaaKqaGeqabaqcLbmacaaIXaGa eyOeI0IaeqOSdigaaaaa@4A58@     (5)

Here, α is the scale parameter and β is the shape parameter such that α>0 and β>1. These are the main results for the weighted Pareto distribution that will subsequently be transmuted to form a new distribution called the transmuted weighted Pareto distribution (TWP). Equations (2), (4) and (5) are used to find the pdf of the transmuted weighted Pareto distribution:

f TWP ( x;α,β,λ )=( α β1 x β ( β1 ) )[ 1λ+2λ α β1 x 1β ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamOzaK qbaoaaBaaajeaibaqcLbmacaWGubGaam4vaiaadcfaaSqabaqcfa4a aeWaaOqaaKqzGeGaamiEaiaacUdacqaHXoqycaGGSaGaeqOSdiMaai ilaiabeU7aSbGccaGLOaGaayzkaaqcLbsacqGH9aqpjuaGdaqadaGc baqcLbsacqaHXoqylmaaCaaajeaibeqaaKqzadGaeqOSdiMaeyOeI0 IaaGymaaaajugibiaadIhajuaGdaahaaWcbeqcbasaaKqzadGaeyOe I0IaeqOSdigaaKqbaoaabmaakeaajugibiabek7aIjabgkHiTiaaig daaOGaayjkaiaawMcaaaGaayjkaiaawMcaaKqbaoaadmaakeaajugi biaaigdacqGHsislcqaH7oaBcqGHRaWkcaaIYaGaeq4UdWMaeqySde 2cdaahaaqcbasabeaajugWaiabek7aIjabgkHiTiaaigdaaaqcLbsa caWG4bWcdaahaaqcbasabeaajugWaiaaigdacqGHsislcqaHYoGyaa aakiaawUfacaGLDbaaaaa@7445@      (6)

Where, α<x<,α>0 and β>1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibabaaa aaaaaapeGaeqySdeMaeyipaWJaamiEaiabgYda8iabg6HiLkaacYca cqaHXoqycqGH+aGpcaaIWaGaaeiiaiaadggacaWGUbGaamizaiaabc cacqaHYoGycqGH+aGpcaaIXaaaaa@4AB1@ . The graphs of the pdf of transmuted weighted Pareto distribution (TWP) are plotted to show the shape of the distribution (Figure 1).

Figure 1 Graphs of pdf of TWP distribution with different values of α, β and λ.

The graph in Figure 1 (left) uses different values for λ with constant α and β values to show how the new variable impacts the shape of the distribution. Values used for λ are -1, -0.5, 0, 0.5 because λ[ 1,1 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibabaaa aaaaaapeGaeq4UdWMaeyOGIWScdaWadaWdaeaajugib8qacqGHsisl caaIXaGaaiilaiaaigdaaOGaay5waiaaw2faaaaa@4296@ . On the other hand, graph in Figure 1 (right) shows different combinations of all the variables involved to see the changes brought by each variable. Equations (3), (4) and (5) are used to find the cdf for the transmuted weighted Pareto distribution:

F TWP ( x;α,β,λ )=1+ α β1 x 1β ( λ1 )λ α 2( β1 ) x 2( 1β ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamOraS WaaSbaaKqaGeaajugWaiaadsfacaWGxbGaamiuaaqcbasabaqcfa4a aeWaaOqaaKqzGeGaamiEaiaacUdacqaHXoqycaGGSaGaeqOSdiMaai ilaiabeU7aSbGccaGLOaGaayzkaaqcLbsacqGH9aqpcaaIXaGaey4k aSIaeqySde2cdaahaaqcbasabeaajugWaiabek7aIjabgkHiTiaaig daaaqcLbsacaWG4bWcdaahaaqcbasabeaajugWaiaaigdacqGHsisl cqaHYoGyaaqcfa4aaeWaaOqaaKqzGeGaeq4UdWMaeyOeI0IaaGymaa GccaGLOaGaayzkaaqcLbsacqGHsislcqaH7oaBcqaHXoqylmaaCaaa jeaibeqaaKqzadGaaGOmaSWaaeWaaKqaGeaajugWaiabek7aIjabgk HiTiaaigdaaKqaGiaawIcacaGLPaaaaaqcLbsacaWG4bWcdaahaaqc basabeaajugWaiaaikdalmaabmaajeaibaqcLbmacaaIXaGaeyOeI0 IaeqOSdigajeaicaGLOaGaayzkaaaaaaaa@73F1@       (7)

where λ[ 1,1 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibabaaa aaaaaapeGaeq4UdWMaeyOGIWScdaWadaWdaeaajugib8qacqGHsisl caaIXaGaaiilaiaaigdaaOGaay5waiaaw2faaaaa@4296@  and β>1. The cdf is graphically presented in Figure 2.

Figure 2 Graph of cdf of TWP distribution with different α, β and λ values, it shows the cdf with different λ values with constant α and β values (left) and different combinations of α, β and λ values (right).

Moments and other derived measures

This section looks deeper into the distribution by exploring its moments and other properties.

Moment generating function

The moment generating function of the transmuted weighted Pareto distribution is given by:

M x ( t )=E[ e tx ]= 0 e tx f( x )dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamytaK qbaoaaBaaajeaibaqcLbmacaWG4baaleqaaKqbaoaabmaakeaajugi biaadshaaOGaayjkaiaawMcaaKqzGeGaeyypa0JaamyraKqbaoaadm aakeaajugibiaadwgalmaaCaaajeaibeqaaKqzadGaamiDaiaadIha aaaakiaawUfacaGLDbaajugibiabg2da9Kqbaoaapehakeaajugibi aadwgalmaaCaaajeaibeqaaKqzadGaamiDaiaadIhaaaqcLbsacaWG Mbqcfa4aaeWaaOqaaKqzGeGaamiEaaGccaGLOaGaayzkaaqcLbsaca WGKbGaamiEaaqcbasaaKqzadGaaGimaaqcbasaaKqzadGaeyOhIuka jugibiabgUIiYdaaaa@5E77@

= 0 ( 1+tx+ t 2 x 2 2! ++ t n x n n! + ) f( x )dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeyypa0 tcfa4aa8qCaOqaaKqbaoaabmaakeaajugibiaaigdacqGHRaWkcaWG 0bGaamiEaiabgUcaRKqbaoaalaaakeaajugibiaadshalmaaCaaaje aibeqaaKqzadGaaGOmaaaajugibiaadIhalmaaCaaajeaibeqaaKqz adGaaGOmaaaaaOqaaKqzGeGaaGOmaiaacgcaaaGaey4kaSIaeS47IW Kaey4kaSscfa4aaSaaaOqaaKqzGeGaamiDaSWaaWbaaKqaGeqabaqc LbmacaWGUbaaaKqzGeGaamiEaSWaaWbaaKqaGeqabaqcLbmacaWGUb aaaaGcbaqcLbsacaWGUbGaaiyiaaaacqGHRaWkcqWIVlctaOGaayjk aiaawMcaaaqcbasaaKqzadGaaGimaaqcbasaaKqzadGaeyOhIukaju gibiabgUIiYdGaamOzaKqbaoaabmaakeaajugibiaadIhaaOGaayjk aiaawMcaaKqzGeGaamizaiaadIhaaaa@69B0@

= i=0 t i E( X i ) i! MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeyypa0 tcfa4aaabCaOqaaKqbaoaalaaakeaajugibiaadshalmaaCaaajeai beqaaKqzadGaamyAaaaajugibiaadweajuaGdaqadaGcbaqcLbsaca WGybWcdaahaaqcbasabeaajugWaiaadMgaaaaakiaawIcacaGLPaaa aeaajugibiaadMgacaGGHaaaaaqcbasaaKqzadGaamyAaiabg2da9i aaicdaaKqaGeaajugWaiabg6HiLcqcLbsacqGHris5aaaa@4FC8@

= i=0 t i i! ( β1 ) α i [ 2( β1 )iλi ] ( 1β+i )( 22β+i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeyypa0 tcfa4aaabCaOqaaKqbaoaalaaakeaajugibiaadshalmaaCaaajeai beqaaKqzadGaamyAaaaaaOqaaKqzGeGaamyAaiaacgcaaaqcfa4aaS aaaOqaaKqbaoaabmaakeaajugibiabek7aIjabgkHiTiaaigdaaOGa ayjkaiaawMcaaKqzGeGaeqySde2cdaahaaqcbasabeaajugWaiaadM gaaaqcfa4aamWaaOqaaKqzGeGaaGOmaKqbaoaabmaakeaajugibiab ek7aIjabgkHiTiaaigdaaOGaayjkaiaawMcaaKqzGeGaeyOeI0Iaam yAaiabeU7aSjabgkHiTiaadMgaaOGaay5waiaaw2faaaqaaKqbaoaa bmaakeaajugibiaaigdacqGHsislcqaHYoGycqGHRaWkcaWGPbaaki aawIcacaGLPaaajuaGdaqadaGcbaqcLbsacaaIYaGaeyOeI0IaaGOm aiabek7aIjabgUcaRiaadMgaaOGaayjkaiaawMcaaaaaaKqaGeaaju gWaiaadMgacqGH9aqpcaaIWaaajeaibaqcLbmacqGHEisPaKqzGeGa eyyeIuoaaaa@73F9@      (8)

Moments

Moments are defining characteristics of a distribution; the moments for TWP distribution are presented in this section. The rth moment of the TWP distribution, with a random variable X, will be:

m r =E( X r )=( β1 ) ( λ α r r2λ α r r2 α r +2 α r β α r r ) ( rβ+1 )( 22β+r ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaad2 galmaaDaaajeaibaqcLbmacaWGYbaajeaibaaccaqcLbmacqWFYaIO aaqcLbsacqGH9aqpcaWGfbqcfa4aaeWaaOqaaKqzGeGaamiwaKqbao aaCaaaleqajeaibaqcLbmacaWGYbaaaaGccaGLOaGaayzkaaqcLbsa cqGH9aqpjuaGdaqadaGcbaqcLbsacqaHYoGycqGHsislcaaIXaaaki aawIcacaGLPaaajuaGdaWcaaGcbaqcfa4aaeWaaOqaaKqzGeGaeq4U dWMaeqySdewcfa4aaWbaaSqabKqaGeaajugWaiaadkhaaaqcLbsaca WGYbGaeyOeI0IaaGOmaiabeU7aSjabeg7aHTWaaWbaaKqaGeqabaqc LbmacaWGYbaaaKqzGeGaamOCaiabgkHiTiaaikdacqaHXoqyjuaGda ahaaWcbeqcbauaaKqzGdGaamOCaaaajugibiabgUcaRiaaikdacqaH XoqylmaaCaaajeaibeqaaKqzadGaamOCaaaajugibiabek7aIjabgk HiTiabeg7aHTWaaWbaaKqaGeqabaqcLbmacaWGYbaaaKqzGeGaamOC aaGccaGLOaGaayzkaaaabaqcfa4aaeWaaOqaaKqzGeGaamOCaiabgk HiTiabek7aIjabgUcaRiaaigdaaOGaayjkaiaawMcaaKqbaoaabmaa keaajugibiaaikdacqGHsislcaaIYaGaeqOSdiMaey4kaSIaamOCaa GccaGLOaGaayzkaaaaaaaa@881E@       (9)

Result in eq. (9) could be used to derive moments by putting in different values for r.

m 1 =E( X 1 )=( β1 )[ α( 2βλ3 ) ( 2β )( 32β ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamyBaS Waa0baaKazba2=baqcLbmacaaIXaaajqwaG9FaaGGaaKqzadGae8Nm GikaaKqzGeGaeyypa0JaamyraKqbaoaabmaakeaajugibiaadIfaju aGdaahaaqcbasabKazba2=baqcLbmacaaIXaaaaaGccaGLOaGaayzk aaqcLbsacqGH9aqpjuaGdaqadaGcbaqcLbsacqaHYoGycqGHsislca aIXaaakiaawIcacaGLPaaajuaGdaWadaGcbaqcfa4aaSaaaOqaaKqz GeGaeqySdewcfa4aaeWaaOqaaKqzGeGaaGOmaiabek7aIjabgkHiTi abeU7aSjabgkHiTiaaiodaaOGaayjkaiaawMcaaaqaaKqbaoaabmaa keaajugibiaaikdacqGHsislcqaHYoGyaOGaayjkaiaawMcaaKqbao aabmaakeaajugibiaaiodacqGHsislcaaIYaGaeqOSdigakiaawIca caGLPaaaaaaacaGLBbGaayzxaaaaaa@6D72@     (10)

m 2 =E( X 2 )=( β1 )[ α 2 ( β2λ ) ( 3β )( 2β ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamyBaS Waa0baaKazba2=baqcLbmacaaIYaaajqwaG9FaaGGaaKqzadGae8Nm GikaaKqzGeGaeyypa0JaamyraKqbaoaabmaakeaajugibiaadIfalm aaCaaajqwaG9FabeaajugWaiaaikdaaaaakiaawIcacaGLPaaajugi biabg2da9Kqbaoaabmaakeaajugibiabek7aIjabgkHiTiaaigdaaO GaayjkaiaawMcaaKqbaoaadmaakeaajuaGdaWcaaGcbaqcLbsacqaH XoqyjuaGdaahaaWcbeqcKfay=haajugWaiaaikdaaaqcfa4aaeWaaO qaaKqzGeGaeqOSdiMaeyOeI0IaaGOmaiabgkHiTiabeU7aSbGccaGL OaGaayzkaaaabaqcfa4aaeWaaOqaaKqzGeGaaG4maiabgkHiTiabek 7aIbGccaGLOaGaayzkaaqcfa4aaeWaaOqaaKqzGeGaaGOmaiabgkHi Tiabek7aIbGccaGLOaGaayzkaaaaaaGaay5waiaaw2faaaaa@6FC0@     (11)

m 3 =E( X 3 )=( β1 )[ α 3 ( 2β53λ ) ( 4β )( 52β ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamyBaS Waa0baaKqaGeaajugWaiaaiodaaKqaGeaaiiaajugWaiab=jdiIcaa jugibiabg2da9iaadweajuaGdaqadaGcbaqcLbsacaWGybqcfa4aaW baaSqabKqaGeaajugWaiaaiodaaaaakiaawIcacaGLPaaajugibiab g2da9Kqbaoaabmaakeaajugibiabek7aIjabgkHiTiaaigdaaOGaay jkaiaawMcaaKqbaoaadmaakeaajuaGdaWcaaGcbaqcLbsacqaHXoqy lmaaCaaajeaibeqaaKqzadGaaG4maaaajuaGdaqadaGcbaqcLbsaca aIYaGaeqOSdiMaeyOeI0IaaGynaiabgkHiTiaaiodacqaH7oaBaOGa ayjkaiaawMcaaaqaaKqbaoaabmaakeaajugibiaaisdacqGHsislcq aHYoGyaOGaayjkaiaawMcaaKqbaoaabmaakeaajugibiaaiwdacqGH sislcaaIYaGaeqOSdigakiaawIcacaGLPaaaaaaacaGLBbGaayzxaa aaaa@6B73@    (12)

m 4 =E( X 4 )=( β1 )[ α 4 ( β32λ ) ( 5β )( 3β ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamyBaS Waa0baaKqaGeaajugWaiaaisdaaKqaGeaaiiaajugWaiab=jdiIcaa jugibiabg2da9iaadweajuaGdaqadaGcbaqcLbsacaWGybWcdaahaa qcbasabeaajugWaiaaisdaaaaakiaawIcacaGLPaaajugibiabg2da 9Kqbaoaabmaakeaajugibiabek7aIjabgkHiTiaaigdaaOGaayjkai aawMcaaKqbaoaadmaakeaajuaGdaWcaaGcbaqcLbsacqaHXoqyjuaG daahaaWcbeqcbasaaKqzadGaaGinaaaajuaGdaqadaGcbaqcLbsacq aHYoGycqGHsislcaaIZaGaeyOeI0IaaGOmaiabeU7aSbGccaGLOaGa ayzkaaaabaqcfa4aaeWaaOqaaKqzGeGaaGynaiabgkHiTiabek7aIb GccaGLOaGaayzkaaqcfa4aaeWaaOqaaKqzGeGaaG4maiabgkHiTiab ek7aIbGccaGLOaGaayzkaaaaaaGaay5waiaaw2faaaaa@69FA@   (13)

First moment about origin is the mean of the distribution. Likewise, other higher moments can be used to find the variance, skewness, kurtosis etc.

Variance and coefficient of variation of the TWP distribution are given below:

var(X)=E( X 2 ) [E(X)] 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaciODai aacggacaGGYbGaaiikaiaadIfacaGGPaGaeyypa0JaamyraiaacIca caWGybWcdaahaaqcbasabeaajugWaiaaikdaaaqcLbsacaGGPaGaey OeI0Iaai4waiaadweacaGGOaGaamiwaiaacMcacaGGDbqcfa4aaWba aSqabKqaGeaajugWaiaaikdaaaaaaa@4AD9@

= α 2 ( 2λ β 3 + λ 2 β 3 4 β 3 +16 β 2 5λ β 2 5 λ 2 β 2 +3λβ+7 λ 2 β21β+93 λ 2 ) ( 2β ) 2 ( 32β ) 2 ( 3β ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeyypa0 tcfa4aaSaaaOqaaKqzGeGaeqySdewcfa4aaWbaaSqabKqaGeaajugW aiaaikdaaaqcfa4aaeWaaOqaaKqzGeGaaGOmaiabeU7aSjabek7aIT WaaWbaaKqaGeqabaqcLbmacaaIZaaaaKqzGeGaey4kaSIaeq4UdW2c daahaaqcbasabeaajugWaiaaikdaaaqcLbsacqaHYoGylmaaCaaaje aibeqaaKqzadGaaG4maaaajugibiabgkHiTiaaisdacqaHYoGylmaa CaaajeaibeqaaKqzadGaaG4maaaajugibiabgUcaRiaaigdacaaI2a GaeqOSdiwcfa4aaWbaaSqabKqaGeaajugWaiaaikdaaaqcLbsacqGH sislcaaI1aGaeq4UdWMaeqOSdi2cdaahaaqcbasabeaajugWaiaaik daaaqcLbsacqGHsislcaaI1aGaeq4UdW2cdaahaaqcbasabeaajugW aiaaikdaaaqcLbsacqaHYoGylmaaCaaajeaibeqaaKqzadGaaGOmaa aajugibiabgUcaRiaaiodacqaH7oaBcqaHYoGycqGHRaWkcaaI3aGa eq4UdWwcfa4aaWbaaSqabKqaGeaajugWaiaaikdaaaqcLbsacqaHYo GycqGHsislcaaIYaGaaGymaiabek7aIjabgUcaRiaaiMdacqGHsisl caaIZaGaeq4UdW2cdaahaaqcbasabeaajugWaiaaikdaaaaakiaawI cacaGLPaaaaeaajuaGdaqadaGcbaqcLbsacaaIYaGaeyOeI0IaeqOS digakiaawIcacaGLPaaalmaaCaaajeaibeqaaKqzadGaaGOmaaaaju aGdaqadaGcbaqcLbsacaaIZaGaeyOeI0IaaGOmaiabek7aIbGccaGL OaGaayzkaaWcdaahaaqcbasabeaajugWaiaaikdaaaqcfa4aaeWaaO qaaKqzGeGaaG4maiabgkHiTiabek7aIbGccaGLOaGaayzkaaaaaaaa @A063@   (14)

C V TWP ( X )= Var( X ) E( X ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaam4qai aadAfajuaGdaWgaaqcKfay=haajugWaiaadsfacaWGxbGaamiuaaqc basabaqcfa4aaeWaaOqaaKqzGeGaamiwaaGccaGLOaGaayzkaaqcLb sacqGH9aqpjuaGdaWcaaGcbaqcfa4aaOaaaOqaaKqzGeGaamOvaiaa dggacaWGYbqcfa4aaeWaaOqaaKqzGeGaamiwaaGccaGLOaGaayzkaa aaleqaaaGcbaqcLbsacaWGfbqcfa4aaeWaaOqaaKqzGeGaamiwaaGc caGLOaGaayzkaaaaaaaa@50DA@

= α( 2β )( 32β ) ( 2λ β 3 + λ 2 β 3 4 β 3 +16 β 2 5λ β 2 5 λ 2 β 2 +3λβ+7 λ 2 β21β+93 λ 2 ) ( 2β )( 32β )( β1 )[ α( 2βλ3 ) ] ( 3β ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeyypa0 tcfa4aaSaaaOqaaKqzGeGaeqySdewcfa4aaeWaaOqaaKqzGeGaaGOm aiabgkHiTiabek7aIbGccaGLOaGaayzkaaqcfa4aaeWaaOqaaKqzGe GaaG4maiabgkHiTiaaikdacqaHYoGyaOGaayjkaiaawMcaaKqbaoaa kaaakeaajuaGdaqadaqcLbsaeaqabOqaaKqzGeGaaGOmaiabeU7aSj abek7aITWaaWbaaKqaGeqabaqcLbmacaaIZaaaaKqzGeGaey4kaSIa eq4UdWwcfa4aaWbaaSqabKqaGeaajugWaiaaikdaaaqcLbsacqaHYo GyjuaGdaahaaWcbeqcbasaaKqzadGaaG4maaaajugibiabgkHiTiaa isdacqaHYoGylmaaCaaajeaibeqaaKqzadGaaG4maaaajugibiabgU caRiaaigdacaaI2aGaeqOSdi2cdaahaaqcbasabeaajugWaiaaikda aaqcLbsacqGHsislcaaI1aGaeq4UdWMaeqOSdiwcfa4aaWbaaSqabK qaGeaajugWaiaaikdaaaaakeaajugibiabgkHiTiaaiwdacqaH7oaB lmaaCaaajeaibeqaaKqzadGaaGOmaaaajugibiabek7aITWaaWbaaK qaGeqabaqcLbmacaaIYaaaaKqzGeGaey4kaSIaaG4maiabeU7aSjab ek7aIjabgUcaRiaaiEdacqaH7oaBlmaaCaaajeaibeqaaKqzadGaaG Omaaaajugibiabek7aIjabgkHiTiaaikdacaaIXaGaeqOSdiMaey4k aSIaaGyoaiabgkHiTiaaiodacqaH7oaBjuaGdaahaaWcbeqcbasaaK qzadGaaGOmaaaaaaGccaGLOaGaayzkaaaaleqaaaGcbaqcfa4aaeWa aOqaaKqzGeGaaGOmaiabgkHiTiabek7aIbGccaGLOaGaayzkaaqcfa 4aaeWaaOqaaKqzGeGaaG4maiabgkHiTiaaikdacqaHYoGyaOGaayjk aiaawMcaaKqbaoaabmaakeaajugibiabek7aIjabgkHiTiaaigdaaO GaayjkaiaawMcaaKqbaoaadmaakeaajugibiabeg7aHLqbaoaabmaa keaajugibiaaikdacqaHYoGycqGHsislcqaH7oaBcqGHsislcaaIZa aakiaawIcacaGLPaaaaiaawUfacaGLDbaajuaGdaGcaaGcbaqcfa4a aeWaaOqaaKqzGeGaaG4maiabgkHiTiabek7aIbGccaGLOaGaayzkaa aaleqaaaaaaaa@BD20@    (15)

Skewness and Kurtosis of the TWP distribution reveal information about symmetry and tail of the distribution respectively.

Skewnes s TWP ( X )= m 3 [ Var( X ) ] 3 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaam4uai aadUgacaWGLbGaam4Daiaad6gacaWGLbGaam4CaiaadohalmaaBaaa jeaibaqcLbmacaWGubGaam4vaiaadcfaaKqaGeqaaKqbaoaabmaake aajugibiaadIfaaOGaayjkaiaawMcaaKqzGeGaeyypa0tcfa4aaSaa aOqaaKqzGeGaamyBaSWaaSbaaKqaGeaajugWaiaaiodaaKqaGeqaaa Gcbaqcfa4aamWaaOqaaKqzGeGaamOvaiaadggacaWGYbqcfa4aaeWa aOqaaKqzGeGaamiwaaGccaGLOaGaayzkaaaacaGLBbGaayzxaaWcda ahaaqcbasabeaalmaalaaajeaibaqcLbmacaaIZaaajeaibaqcLbma caaIYaaaaaaaaaaaaa@59FA@   (16)

where,

m 3 = ( β1 ) α 3 ( 2β53λ ) ( 4β )( 52β ) 3 α 3 ( 2βλ3 )( β2λ ) ( β1 ) 2 ( 2β )( 32β )( 3β ) + 2 α 3 ( β1 ) 3 ( 2βλ3 ) ( 2β ) 3 ( 32β ) 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaqcLbsaca WGTbqcfa4aaSbaaSqaaKqzGeGaaG4maaWcbeaajugibiabg2da9Kqb aoaalaaakeaajuaGdaqadaGcbaqcLbsacqaHYoGycqGHsislcaaIXa aakiaawIcacaGLPaaajugibiabeg7aHTWaaWbaaKqaGeqabaqcLbma caaIZaaaaKqbaoaabmaakeaajugibiaaikdacqaHYoGycqGHsislca aI1aGaeyOeI0IaaG4maiabeU7aSbGccaGLOaGaayzkaaaabaqcfa4a aeWaaOqaaKqzGeGaaGinaiabgkHiTiabek7aIbGccaGLOaGaayzkaa qcfa4aaeWaaOqaaKqzGeGaaGynaiabgkHiTiaaikdacqaHYoGyaOGa ayjkaiaawMcaaaaajugibiabgkHiTKqbaoaalaaakeaajugibiaaio dacqaHXoqylmaaCaaajeaibeqaaKqzadGaaG4maaaajuaGdaqadaGc baqcLbsacaaIYaGaeqOSdiMaeyOeI0Iaeq4UdWMaeyOeI0IaaG4maa GccaGLOaGaayzkaaqcfa4aaeWaaOqaaKqzGeGaeqOSdiMaeyOeI0Ia aGOmaiabgkHiTiabeU7aSbGccaGLOaGaayzkaaqcfa4aaeWaaOqaaK qzGeGaeqOSdiMaeyOeI0IaaGymaaGccaGLOaGaayzkaaqcfa4aaWba aSqabKqaGeaajugWaiaaikdaaaaakeaajuaGdaqadaGcbaqcLbsaca aIYaGaeyOeI0IaeqOSdigakiaawIcacaGLPaaajuaGdaqadaGcbaqc LbsacaaIZaGaeyOeI0IaaGOmaiabek7aIbGccaGLOaGaayzkaaqcfa 4aaeWaaOqaaKqzGeGaaG4maiabgkHiTiabek7aIbGccaGLOaGaayzk aaaaaaqaaKqzGeGaey4kaSscfa4aaSaaaOqaaKqzGeGaaGOmaiabeg 7aHTWaaWbaaKqaGeqabaqcLbmacaaIZaaaaKqbaoaabmaakeaajugi biabek7aIjabgkHiTiaaigdaaOGaayjkaiaawMcaaSWaaWbaaKqaGe qabaqcLbmacaaIZaaaaKqbaoaabmaakeaajugibiaaikdacqaHYoGy cqGHsislcqaH7oaBcqGHsislcaaIZaaakiaawIcacaGLPaaaaeaaju aGdaqadaGcbaqcLbsacaaIYaGaeyOeI0IaeqOSdigakiaawIcacaGL PaaajuaGdaahaaWcbeqcbasaaKqzadGaaG4maaaajuaGdaqadaGcba qcLbsacaaIZaGaeyOeI0IaaGOmaiabek7aIbGccaGLOaGaayzkaaWc daahaaqcbasabeaajugWaiaaiodaaaaaaaaaaa@BCDF@

Kurtosi s TWP ( X )= m 4 [ Var( X ) ] 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaam4sai aadwhacaWGYbGaamiDaiaad+gacaWGZbGaamyAaiaadohajuaGdaWg aaqcbasaaKqzadGaamivaiaadEfacaWGqbaaleqaaKqbaoaabmaake aajugibiaadIfaaOGaayjkaiaawMcaaKqzGeGaeyypa0tcfa4aaSaa aOqaaKqzGeGaamyBaKqbaoaaBaaajeaibaqcLbmacaaI0aaaleqaaa Gcbaqcfa4aamWaaOqaaKqzGeGaamOvaiaadggacaWGYbqcfa4aaeWa aOqaaKqzGeGaamiwaaGccaGLOaGaayzkaaaacaGLBbGaayzxaaWcda ahaaqcbasabeaajugWaiaaikdaaaaaaaaa@587A@ (17)

where,

m 4 =( β1 )[ α 4 ( β32λ ) ( 5β )( 3β ) ] 4 α 4 ( β1 ) 2 ( 2βλ3 )( 2β53λ ) ( 2β )( 32β )( 4β )( 52β ) + 6 α 4 ( β1 ) 3 ( 2βλ3 ) 2 ( β2λ ) ( 2β ) 3 ( 32β ) 2 ( 3β ) 3 ( β1 ) 4 α 4 ( 2βλ3 ) 4 ( 2β ) 4 ( 32β ) 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaqcLbsaca WGTbqcfa4aaSbaaKqaGeaajugWaiaaisdaaSqabaqcLbsacqGH9aqp juaGdaqadaGcbaqcLbsacqaHYoGycqGHsislcaaIXaaakiaawIcaca GLPaaajuaGdaWadaGcbaqcfa4aaSaaaOqaaKqzGeGaeqySde2cdaah aaqcbasabeaajugWaiaaisdaaaqcfa4aaeWaaOqaaKqzGeGaeqOSdi MaeyOeI0IaaG4maiabgkHiTiaaikdacqaH7oaBaOGaayjkaiaawMca aaqaaKqbaoaabmaakeaajugibiaaiwdacqGHsislcqaHYoGyaOGaay jkaiaawMcaaKqbaoaabmaakeaajugibiaaiodacqGHsislcqaHYoGy aOGaayjkaiaawMcaaaaaaiaawUfacaGLDbaajugibiabgkHiTKqbao aalaaakeaajugibiaaisdacqaHXoqyjuaGdaahaaWcbeqcbasaaKqz adGaaGinaaaajuaGdaqadaGcbaqcLbsacqaHYoGycqGHsislcaaIXa aakiaawIcacaGLPaaalmaaCaaajeaibeqaaKqzadGaaGOmaaaajuaG daqadaGcbaqcLbsacaaIYaGaeqOSdiMaeyOeI0Iaeq4UdWMaeyOeI0 IaaG4maaGccaGLOaGaayzkaaqcfa4aaeWaaOqaaKqzGeGaaGOmaiab ek7aIjabgkHiTiaaiwdacqGHsislcaaIZaGaeq4UdWgakiaawIcaca GLPaaaaeaajuaGdaqadaGcbaqcLbsacaaIYaGaeyOeI0IaeqOSdiga kiaawIcacaGLPaaajuaGdaqadaGcbaqcLbsacaaIZaGaeyOeI0IaaG Omaiabek7aIbGccaGLOaGaayzkaaqcfa4aaeWaaOqaaKqzGeGaaGin aiabgkHiTiabek7aIbGccaGLOaGaayzkaaqcfa4aaeWaaOqaaKqzGe GaaGynaiabgkHiTiaaikdacqaHYoGyaOGaayjkaiaawMcaaaaaaeaa jugibiabgUcaRKqbaoaalaaakeaajugibiaaiAdacqaHXoqyjuaGda ahaaWcbeqcbasaaKqzadGaaGinaaaajuaGdaqadaGcbaqcLbsacqaH YoGycqGHsislcaaIXaaakiaawIcacaGLPaaalmaaCaaajeaibeqaaK qzadGaaG4maaaajuaGdaqadaGcbaqcLbsacaaIYaGaeqOSdiMaeyOe I0Iaeq4UdWMaeyOeI0IaaG4maaGccaGLOaGaayzkaaqcfa4aaWbaaS qabKqaGeaajugWaiaaikdaaaqcfa4aaeWaaOqaaKqzGeGaeqOSdiMa eyOeI0IaaGOmaiabgkHiTiabeU7aSbGccaGLOaGaayzkaaaabaqcfa 4aaeWaaOqaaKqzGeGaaGOmaiabgkHiTiabek7aIbGccaGLOaGaayzk aaqcfa4aaWbaaSqabKqaGeaajugWaiaaiodaaaqcfa4aaeWaaOqaaK qzGeGaaG4maiabgkHiTiaaikdacqaHYoGyaOGaayjkaiaawMcaaKqb aoaaCaaaleqajeaibaqcLbmacaaIYaaaaKqbaoaabmaakeaajugibi aaiodacqGHsislcqaHYoGyaOGaayjkaiaawMcaaaaajugibiabgkHi TKqbaoaalaaakeaajugibiaaiodajuaGdaqadaGcbaqcLbsacqaHYo GycqGHsislcaaIXaaakiaawIcacaGLPaaalmaaCaaajeaibeqaaKqz adGaaGinaaaajugibiabeg7aHLqbaoaaCaaaleqajeaibaqcLbmaca aI0aaaaKqbaoaabmaakeaajugibiaaikdacqaHYoGycqGHsislcqaH 7oaBcqGHsislcaaIZaaakiaawIcacaGLPaaajuaGdaahaaWcbeqcba saaKqzadGaaGinaaaaaOqaaKqbaoaabmaakeaajugibiaaikdacqGH sislcqaHYoGyaOGaayjkaiaawMcaaKqbaoaaCaaaleqajeaibaqcLb macaaI0aaaaKqbaoaabmaakeaajugibiaaiodacqGHsislcaaIYaGa eqOSdigakiaawIcacaGLPaaajuaGdaahaaWcbeqcbasaaKqzadGaaG inaaaaaaaaaaa@08D7@

Quantiles

The qth quantile of the TWP distribution is found to be

x q = { α 2β [ α β+1 ( 1+λ ) 2 4λq ( 1λ ) α β+1 ] 2λ } 1 1β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamiEaS WaaSbaaKqaGeaajugWaiaadghaaKqaGeqaaKqzGeGaeyypa0tcfa4a aiWaaOqaaKqbaoaalaaakeaajugibiabeg7aHLqbaoaaCaaaleqaje aibaqcLbmacqGHsislcaaIYaGaeqOSdigaaKqbaoaadmaakeaajugi biabeg7aHTWaaWbaaKqaGeqabaqcLbmacqaHYoGycqGHRaWkcaaIXa aaaKqbaoaakaaakeaajuaGdaqadaGcbaqcLbsacaaIXaGaey4kaSIa eq4UdWgakiaawIcacaGLPaaajuaGdaahaaWcbeqcbasaaKqzadGaaG OmaaaajugibiabgkHiTiaaisdacqaH7oaBcaWGXbaaleqaaKqzGeGa eyOeI0scfa4aaeWaaOqaaKqzGeGaaGymaiabgkHiTiabeU7aSbGcca GLOaGaayzkaaqcLbsacqaHXoqyjuaGdaahaaWcbeqcbasaaKqzadGa eqOSdiMaey4kaSIaaGymaaaaaOGaay5waiaaw2faaaqaaKqzGeGaaG OmaiabeU7aSbaaaOGaay5Eaiaaw2haaKqbaoaaCaaaleqajeaibaWc daWcaaqcbasaaKqzadGaaGymaaqcbasaaKqzadGaaGymaiabgkHiTi abek7aIbaaaaaaaa@7870@    (18)

The median of the TWP can thus be found by putting q=0.5

x 0.5 = { α 2β [ α β+1 λ 2 +1 ( 1λ ) α β+1 ] 2λ } 1 1β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamiEaK qbaoaaBaaajeaibaqcLbmacaaIWaGaaiOlaiaaiwdaaSqabaqcLbsa cqGH9aqpjuaGdaGadaGcbaqcfa4aaSaaaOqaaKqzGeGaeqySde2cda ahaaqcbasabeaajugWaiabgkHiTiaaikdacqaHYoGyaaqcfa4aamWa aOqaaKqzGeGaeqySde2cdaahaaqcbasabeaajugWaiabek7aIjabgU caRiaaigdaaaqcfa4aaOaaaOqaaKqzGeGaeq4UdWwcfa4aaWbaaSqa bKqaGeaajugWaiaaikdaaaqcLbsacqGHRaWkcaaIXaaaleqaaKqzGe GaeyOeI0scfa4aaeWaaOqaaKqzGeGaaGymaiabgkHiTiabeU7aSbGc caGLOaGaayzkaaqcLbsacqaHXoqylmaaCaaajeaibeqaaKqzadGaeq OSdiMaey4kaSIaaGymaaaaaOGaay5waiaaw2faaaqaaKqzGeGaaGOm aiabeU7aSbaaaOGaay5Eaiaaw2haaKqbaoaaCaaaleqajeaibaWcda WcaaqcbasaaKqzadGaaGymaaqcbasaaKqzadGaaGymaiabgkHiTiab ek7aIbaaaaaaaa@726D@ (19)

To make skewness and kurtosis yield better results, these measures could be derived using quantiles. The original skewness and kurtosis show infinite measure for heavy-tailed distributions making it less informative. The Bowleys skewness derived as the earliest skewness measure uses average of quartiles minus the median, divided by one half interquartile ranges.

B= Q 3 + Q 1 2 Q 2 Q 3 Q 1 = Q(3/4)+Q(1/4)2Q(2/4) Q(3/4)Q(1/4) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadk eacqGH9aqpkmaalaaabaqcLbsacaWGrbGcdaWgaaqcbauaaKqzadGa aG4maaWcbeaajugibiabgUcaRiaadgfakmaaBaaajeaqbaqcLbkaca aIXaaaleqaaKqzGeGaeyOeI0IaaGOmaiaadgfakmaaBaaajeaqbaqc LbmacaaIYaaaleqaaaGcbaqcLbsacaWGrbGcdaWgaaqcbauaaKqzad GaaG4maaWcbeaajugibiabgkHiTiaadgfakmaaBaaajeaqbaqcLbma caaIXaaaleqaaaaajugibiabg2da9OWaaSaaaeaajugibiaadgfaca GGOaGaaG4maiaac+cacaaI0aGaaiykaiabgUcaRiaadgfacaGGOaGa aGymaiaac+cacaaI0aGaaiykaiabgkHiTiaaikdacaWGrbGaaiikai aaikdacaGGVaGaaGinaiaacMcaaOqaaKqzGeGaamyuaiaacIcacaaI ZaGaai4laiaaisdacaGGPaGaeyOeI0IaamyuaiaacIcacaaIXaGaai 4laiaaisdacaGGPaaaaaaa@6D6E@

For kurtosis, Moors kurtosis uses octiles to make it a better measure.

M= ( E 3 E 1 )+( E 7 E 5 ) E 6 E 2 = Q(3/4)Q(7/8)Q(5/8) Q(6/8)Q(2/8) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaad2 eacqGH9aqpkmaalaaabaqcLbsacaGGOaGaamyraOWaaSbaaKqaafaa jug4aiaaiodaaSqabaqcLbsacqGHsislcaWGfbGcdaWgaaqcbawaaK qzGdGaaGymaaWcbeaajugibiaacMcacqGHRaWkcaGGOaGaamyraOWa aSbaaKqaafaajugWaiaaiEdaaSqabaqcLbsacqGHsislcaWGfbGcda WgaaqcbauaaKqzadGaaGynaaWcbeaajugibiaacMcaaOqaaKqzGeGa amyraOWaaSbaaKqaafaajugWaiaaiAdaaSqabaqcLbsacqGHsislca WGfbGcdaWgaaqcbauaaKqzadGaaGOmaaWcbeaaaaqcLbsacqGH9aqp kmaalaaabaqcLbsacaWGrbGaaiikaiaaiodacaGGVaGaaGinaiaacM cacqGHsislcaWGrbGaaiikaiaaiEdacaGGVaGaaGioaiaacMcacqGH sislcaWGrbGaaiikaiaaiwdacaGGVaGaaGioaiaacMcaaOqaaKqzGe GaamyuaiaacIcacaaI2aGaai4laiaaiIdacaGGPaGaeyOeI0Iaamyu aiaacIcacaaIYaGaai4laiaaiIdacaGGPaaaaaaa@746A@

Bonferroni and Lorenz curves

Bonferroni and Lorenz curves are important for reliability studies. The curves for the function, F(x)=p, are given by:

B( p )= 1 p μ 1 0 q xf( x ) dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamOqaK qbaoaabmaakeaajugibiaadchaaOGaayjkaiaawMcaaKqzGeGaeyyp a0tcfa4aaSaaaOqaaKqzGeGaaGymaaGcbaqcLbsacaWGWbGaeqiVd0 2cdaqhaaqcbasaaKqzadGaaGymaaqcbasaaGGaaKqzadGae8NmGika aaaajuaGdaWdXbGcbaqcLbsacaWG4bGaamOzaKqbaoaabmaakeaaju gibiaadIhaaOGaayjkaiaawMcaaaqcbasaaKqzadGaaGimaaqcbasa aKqzadGaamyCaaqcLbsacqGHRiI8aiaadsgacaWG4baaaa@56C2@ , L( p )=pB( p )= 1 μ 1 0 q xf( x ) dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamitaK qbaoaabmaakeaajugibiaadchaaOGaayjkaiaawMcaaKqzGeGaeyyp a0JaamiCaiaadkeajuaGdaqadaGcbaqcLbsacaWGWbaakiaawIcaca GLPaaajugibiabg2da9KqbaoaalaaakeaajugibiaaigdaaOqaaKqz GeGaeqiVd02cdaqhaaqcbasaaKqzadGaaGymaaqcbasaaGGaaKqzad Gae8NmGikaaaaajuaGdaWdXbGcbaqcLbsacaWG4bGaamOzaKqbaoaa bmaakeaajugibiaadIhaaOGaayjkaiaawMcaaaqcbasaaKqzadGaaG imaaqcbasaaKqzadGaamyCaaqcLbsacqGHRiI8aiaadsgacaWG4baa aa@5CD7@ , where q= F 1 ( p ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamyCai abg2da9iaadAeajuaGdaahaaWcbeqcbasaaKqzadGaeyOeI0IaaGym aaaajuaGdaqadaGcbaqcLbsacaWGWbaakiaawIcacaGLPaaaaaa@40AB@ , F( x )=p, 0p1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibabaaa aaaaaapeGaaeOraOWaaeWaa8aabaqcLbsapeGaaeiEaaGccaGLOaGa ayzkaaqcLbsacqGH9aqpcaqGWbGaaiilaiaabckacaaIWaGaeyizIm QaaeiCaiabgsMiJkaaigdaaaa@475C@

Hence, the Bonferroni and Lorenz curves for TWP respectively are

B( p )= 1 p{ ( β1 )[ α( 2βλ3 ) ( 2β )( 32β ) ] } ( β1 )[ ( α β1 q 2β α ) 2β ( 1λ ) + 2λ( α 2β2 q 32β α ) 32β ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamOqaK qbaoaabmaakeaajugibiaadchaaOGaayjkaiaawMcaaKqzGeGaeyyp a0tcfa4aaSaaaOqaaKqzGeGaaGymaaGcbaqcLbsacaWGWbqcfa4aai WaaOqaaKqbaoaabmaakeaajugibiabek7aIjabgkHiTiaaigdaaOGa ayjkaiaawMcaaKqbaoaadmaakeaajuaGdaWcaaGcbaqcLbsacqaHXo qyjuaGdaqadaGcbaqcLbsacaaIYaGaeqOSdiMaeyOeI0Iaeq4UdWMa eyOeI0IaaG4maaGccaGLOaGaayzkaaaabaqcfa4aaeWaaOqaaKqzGe GaaGOmaiabgkHiTiabek7aIbGccaGLOaGaayzkaaqcfa4aaeWaaOqa aKqzGeGaaG4maiabgkHiTiaaikdacqaHYoGyaOGaayjkaiaawMcaaa aaaiaawUfacaGLDbaaaiaawUhacaGL9baaaaqcfa4aaeWaaOqaaKqz GeGaeqOSdiMaeyOeI0IaaGymaaGccaGLOaGaayzkaaqcfa4aamWaaK qzGeabaeqakeaajuaGdaWcaaGcbaqcfa4aaeWaaOqaaKqzGeGaeqyS dewcfa4aaWbaaSqabKqaGeaajugWaiabek7aIjabgkHiTiaaigdaaa qcLbsacaWGXbqcfa4aaWbaaSqabKqaGeaajugWaiaaikdacqGHsisl cqaHYoGyaaqcLbsacqGHsislcqaHXoqyaOGaayjkaiaawMcaaaqaaK qzGeGaaGOmaiabgkHiTiabek7aIbaajuaGdaqadaGcbaqcLbsacaaI XaGaeyOeI0Iaeq4UdWgakiaawIcacaGLPaaaaeaajugibiabgUcaRK qbaoaalaaakeaajugibiaaikdacqaH7oaBjuaGdaqadaGcbaqcLbsa cqaHXoqyjuaGdaahaaWcbeqcbasaaKqzadGaaGOmaiabek7aIjabgk HiTiaaikdaaaqcLbsacaWGXbqcfa4aaWbaaSqabKqaGeaajugWaiaa iodacqGHsislcaaIYaGaeqOSdigaaKqzGeGaeyOeI0IaeqySdegaki aawIcacaGLPaaaaeaajugibiaaiodacqGHsislcaaIYaGaeqOSdiga aaaakiaawUfacaGLDbaaaaa@AAA8@    (20)

L( p )= 1 ( β1 )[ α( 2βλ3 ) ( 2β )( 32β ) ] ( β1 )[ ( α β1 q 2β α ) 2β ( 1λ ) + ( 2λ( α 2β2 q 32β α ) ) 32β ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamitaK qbaoaabmaakeaajugibiaadchaaOGaayjkaiaawMcaaKqzGeGaeyyp a0tcfa4aaSaaaOqaaKqzGeGaaGymaaGcbaqcfa4aaeWaaOqaaKqzGe GaeqOSdiMaeyOeI0IaaGymaaGccaGLOaGaayzkaaqcfa4aamWaaOqa aKqbaoaalaaakeaajugibiabeg7aHLqbaoaabmaakeaajugibiaaik dacqaHYoGycqGHsislcqaH7oaBcqGHsislcaaIZaaakiaawIcacaGL PaaaaeaajuaGdaqadaGcbaqcLbsacaaIYaGaeyOeI0IaeqOSdigaki aawIcacaGLPaaajuaGdaqadaGcbaqcLbsacaaIZaGaeyOeI0IaaGOm aiabek7aIbGccaGLOaGaayzkaaaaaaGaay5waiaaw2faaaaajuaGda qadaGcbaqcLbsacqaHYoGycqGHsislcaaIXaaakiaawIcacaGLPaaa juaGdaWadaqcLbsaeaqabOqaaKqbaoaalaaakeaajuaGdaqadaGcba qcLbsacqaHXoqyjuaGdaahaaWcbeqcbasaaKqzadGaeqOSdiMaeyOe I0IaaGymaaaajugibiaadghajuaGdaahaaWcbeqcbasaaKqzadGaaG OmaiabgkHiTiabek7aIbaajugibiabgkHiTiabeg7aHbGccaGLOaGa ayzkaaaabaqcLbsacaaIYaGaeyOeI0IaeqOSdigaaKqbaoaabmaake aajugibiaaigdacqGHsislcqaH7oaBaOGaayjkaiaawMcaaaqaaKqz GeGaey4kaSscfa4aaSaaaOqaaKqbaoaabmaakeaajugibiaaikdacq aH7oaBjuaGdaqadaGcbaqcLbsacqaHXoqyjuaGdaahaaWcbeqcbasa aKqzadGaaGOmaiabek7aIjabgkHiTiaaikdaaaqcLbsacaWGXbqcfa 4aaWbaaSqabKqaGeaajugWaiaaiodacqGHsislcaaIYaGaeqOSdiga aKqzGeGaeyOeI0IaeqySdegakiaawIcacaGLPaaaaiaawIcacaGLPa aaaeaajugibiaaiodacqGHsislcaaIYaGaeqOSdigaaaaakiaawUfa caGLDbaaaaa@A886@     (21)

Mean deviation and averages

Mean deviation gives the mean of the absolute deviations from its mean value. Thus, the mean deviation of TWP distribution is calculated as

M D TWP =2μ[ 1+( α β1 μ 1β )( λ1 )λ α 2( β1 ) μ 2( 1β ) ] 2( β1 )[ α β1 μ 2β αλ α β1 μ 2β ( 2β ) + 2λ α 2( β1 ) μ 32β 2λα 32β ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaqcLbsaca WGnbGaamiraKqbaoaaBaaajeaibaqcLbmacaWGubGaam4vaiaadcfa aSqabaqcLbsacqGH9aqpcaaIYaGaeqiVd0wcfa4aamWaaOqaaKqzGe GaaGymaiabgUcaRKqbaoaabmaakeaajugibiabeg7aHLqbaoaaCaaa leqajeaibaqcLbmacqaHYoGycqGHsislcaaIXaaaaKqzGeGaeqiVd0 wcfa4aaWbaaSqabKqaGeaajugWaiaaigdacqGHsislcqaHYoGyaaaa kiaawIcacaGLPaaajuaGdaqadaGcbaqcLbsacqaH7oaBcqGHsislca aIXaaakiaawIcacaGLPaaajugibiabgkHiTiabeU7aSjabeg7aHTWa aWbaaKqaGeqabaqcLbmacaaIYaWcdaqadaqcbasaaKqzadGaeqOSdi MaeyOeI0IaaGymaaqcbaIaayjkaiaawMcaaaaajugibiabeY7aTTWa aWbaaKqaGeqabaqcLbmacaaIYaWcdaqadaqcbasaaKqzadGaaGymai abgkHiTiabek7aIbqcbaIaayjkaiaawMcaaaaaaOGaay5waiaaw2fa aaqaaKqzGeGaeyOeI0IaaGOmaKqbaoaabmaakeaajugibiabek7aIj abgkHiTiaaigdaaOGaayjkaiaawMcaaKqbaoaadmaakeaajuaGdaWc aaGcbaqcLbsacqaHXoqyjuaGdaahaaWcbeqcbasaaKqzadGaeqOSdi MaeyOeI0IaaGymaaaajugibiabeY7aTLqbaoaaCaaaleqajeaibaqc LbmacaaIYaGaeyOeI0IaeqOSdigaaKqzGeGaeyOeI0IaeqySdeMaey OeI0Iaeq4UdWMaeqySdewcfa4aaWbaaSqabKqaGeaajugWaiabek7a IjabgkHiTiaaigdaaaqcLbsacqaH8oqBjuaGdaahaaWcbeqcbasaaK qzadGaaGOmaiabgkHiTiabek7aIbaaaOqaaKqbaoaabmaakeaajugi biaaikdacqGHsislcqaHYoGyaOGaayjkaiaawMcaaaaajugibiabgU caRKqbaoaalaaakeaajugibiaaikdacqaH7oaBcqaHXoqylmaaCaaa jeaibeqaaKqzadGaaGOmaSWaaeWaaKqaGeaajugWaiabek7aIjabgk HiTiaaigdaaKqaGiaawIcacaGLPaaaaaqcLbsacqaH8oqBlmaaCaaa jeaibeqaaKqzadGaaG4maiabgkHiTiaaikdacqaHYoGyaaqcLbsacq GHsislcaaIYaGaeq4UdWMaeqySdegakeaajugibiaaiodacqGHsisl caaIYaGaeqOSdigaaaGccaGLBbGaayzxaaaaaaa@CB9C@     (22)

Harmonic Mean of TWP:

H M TWP = β2 β 2 ( β1 )( α 1 2β α 1 λ α 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamisai aad2eajuaGdaWgaaqcbasaaKqzadGaamivaiaadEfacaWGqbaaleqa aKqzGeGaeyypa0tcfa4aaSaaaOqaaKqzGeGaeqOSdiMaeyOeI0IaaG Omaiabek7aILqbaoaaCaaaleqajeaibaqcLbmacaaIYaaaaaGcbaqc fa4aaeWaaOqaaKqzGeGaeqOSdiMaeyOeI0IaaGymaaGccaGLOaGaay zkaaqcfa4aaeWaaOqaaKqzGeGaeqySdewcfa4aaWbaaSqabKqaGeaa jugWaiabgkHiTiaaigdaaaqcLbsacqGHsislcaaIYaGaeqOSdiMaeq ySdewcfa4aaWbaaSqabKqaGeaajugWaiabgkHiTiaaigdaaaqcLbsa cqGHsislcqaH7oaBcqaHXoqyjuaGdaahaaWcbeqcbasaaKqzadGaey OeI0IaaGymaaaaaOGaayjkaiaawMcaaaaaaaa@6719@     (23)

Geometric mean of TWP:

G M TWP = i=1 n [ ( β1 )( n α 1 n λn α 1 n 1nβ+n 2nλ α 1 n 2n2βn+1 ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaam4rai aad2eajuaGdaWgaaqcbasaaKqzadGaamivaiaadEfacaWGqbaaleqa aKqzGeGaeyypa0tcfa4aaebCaOqaaKqbaoaadmaakeaajuaGdaqada GcbaqcLbsacqaHYoGycqGHsislcaaIXaaakiaawIcacaGLPaaajuaG daqadaGcbaqcfa4aaSaaaOqaaKqzGeGaamOBaiabeg7aHLqbaoaaCa aaleqajeaibaWcdaWcaaqcbasaaKqzadGaaGymaaqcbasaaKqzadGa amOBaaaaaaqcLbsacqaH7oaBcqGHsislcaWGUbGaeqySdewcfa4aaW baaSqabKqaGeaalmaalaaajeaibaqcLbmacaaIXaaajeaibaqcLbma caWGUbaaaaaaaOqaaKqzGeGaaGymaiabgkHiTiaad6gacqaHYoGycq GHRaWkcaWGUbaaaiabgkHiTKqbaoaalaaakeaajugibiaaikdacaWG UbGaeq4UdWMaeqySdewcfa4aaWbaaSqabKqaGeaalmaalaaajeaiba qcLbmacaaIXaaajeaibaqcLbmacaWGUbaaaaaaaOqaaKqzGeGaaGOm aiaad6gacqGHsislcaaIYaGaeqOSdiMaamOBaiabgUcaRiaaigdaaa aakiaawIcacaGLPaaaaiaawUfacaGLDbaaaKqaGeaajugWaiaadMga cqGH9aqpcaaIXaaajeaibaqcLbmacaWGUbaajugibiabg+Givdaaaa@822A@     (24)

Reliability analysis

The reliability analysis involves the evaluation of various processes that assess the quality of life of the data for a time (t).

Reliability function

The reliability function for the TWP distribution tells about the length of life up to a time (t) and thus is an important characteristic to study.

R TWP ( t )= α β1 t 1β ( 1λ+λ α β1 t 1β ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamOuaK qbaoaaBaaajeaibaqcLbmacaWGubGaam4vaiaadcfaaSqabaqcfa4a aeWaaOqaaKqzGeGaamiDaaGccaGLOaGaayzkaaqcLbsacqGH9aqpcq aHXoqyjuaGdaahaaWcbeqcbasaaKqzadGaeqOSdiMaeyOeI0IaaGym aaaajugibiaadshajuaGdaahaaWcbeqcbasaaKqzadGaaGymaiabgk HiTiabek7aIbaajuaGdaqadaGcbaqcLbsacaaIXaGaeyOeI0Iaeq4U dWMaey4kaSIaeq4UdWMaeqySdewcfa4aaWbaaSqabKqaGeaajugWai abek7aIjabgkHiTiaaigdaaaqcLbsacaWG0bqcfa4aaWbaaSqabKqa GeaajugWaiaaigdacqGHsislcqaHYoGyaaaakiaawIcacaGLPaaaaa a@659F@ (25)

Hazard rate

The hazard rate tells about the rate of failure of an item. It predicts the end of its life and can be calculated as follows:

h TWP ( t )= ( β1 )( 1λ+2λ α β1 t 1β ) ( tλt+λ α β1 t 2β ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamiAaK qbaoaaBaaajeaibaqcLbmacaWGubGaam4vaiaadcfaaSqabaqcfa4a aeWaaOqaaKqzGeGaamiDaaGccaGLOaGaayzkaaqcLbsacqGH9aqpju aGdaWcaaGcbaqcfa4aaeWaaOqaaKqzGeGaeqOSdiMaeyOeI0IaaGym aaGccaGLOaGaayzkaaqcfa4aaeWaaOqaaKqzGeGaaGymaiabgkHiTi abeU7aSjabgUcaRiaaikdacqaH7oaBcqaHXoqyjuaGdaahaaWcbeqc basaaKqzadGaeqOSdiMaeyOeI0IaaGymaaaajugibiaadshalmaaCa aajeaibeqaaKqzadGaaGymaiabgkHiTiabek7aIbaaaOGaayjkaiaa wMcaaaqaaKqbaoaabmaakeaajugibiaadshacqGHsislcqaH7oaBca WG0bGaey4kaSIaeq4UdWMaeqySdewcfa4aaWbaaSqabKqaGeaajugW aiabek7aIjabgkHiTiaaigdaaaqcLbsacaWG0bqcfa4aaWbaaSqabK qaGeaajugWaiaaikdacqGHsislcqaHYoGyaaaakiaawIcacaGLPaaa aaaaaa@7672@     (26)

Figure 3 shows the reliability function and hazard rate function graphically. The graph uses different combinations of α, β and λ values to show a decreasing trend of the function (Figure 3).

Figure 3 Reliability and Hazard rate function graphs with different α, β and λ values.

Reversed hazard rate function

Reversed hazard rate comes handy when the time is measured in a reversed manner; therefore, it is tabulated to cover for an occurrence of that sort.

r TWP ( t )= ( β1 ) α β1 t β ( 1λ+2λ α β1 t 1β ) 1+ α β1 t 1β ( λ1 )λ α 2( β1 ) t 2( 1β ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamOCaK qbaoaaBaaajeaibaqcLbmacaWGubGaam4vaiaadcfaaSqabaqcfa4a aeWaaOqaaKqzGeGaamiDaaGccaGLOaGaayzkaaqcLbsacqGH9aqpju aGdaWcaaGcbaqcfa4aaeWaaOqaaKqzGeGaeqOSdiMaeyOeI0IaaGym aaGccaGLOaGaayzkaaqcLbsacqaHXoqyjuaGdaahaaWcbeqcbasaaK qzadGaeqOSdiMaeyOeI0IaaGymaaaajugibiaadshajuaGdaahaaWc beqcbasaaKqzadGaeyOeI0IaeqOSdigaaKqbaoaabmaakeaajugibi aaigdacqGHsislcqaH7oaBcqGHRaWkcaaIYaGaeq4UdWMaeqySdewc fa4aaWbaaSqabKqaGeaajugWaiabek7aIjabgkHiTiaaigdaaaqcLb sacaWG0bqcfa4aaWbaaSqabKqaGeaajugWaiaaigdacqGHsislcqaH YoGyaaaakiaawIcacaGLPaaaaeaajugibiaaigdacqGHRaWkcqaHXo qyjuaGdaahaaWcbeqcbasaaKqzadGaeqOSdiMaeyOeI0IaaGymaaaa jugibiaadshajuaGdaahaaWcbeqcbasaaKqzadGaaGymaiabgkHiTi abek7aIbaajuaGdaqadaGcbaqcLbsacqaH7oaBcqGHsislcaaIXaaa kiaawIcacaGLPaaajugibiabgkHiTiabeU7aSjabeg7aHLqbaoaaCa aaleqajeaibaqcLbmacaaIYaWcdaqadaqcbasaaKqzadGaeqOSdiMa eyOeI0IaaGymaaqcbaIaayjkaiaawMcaaaaajugibiaadshalmaaCa aajeaibeqaaKqzadGaaGOmaSWaaeWaaKqaGeaajugWaiaaigdacqGH sislcqaHYoGyaKqaGiaawIcacaGLPaaaaaaaaaaa@9B40@       (27)

Cumulative hazard rate

Cumulative hazard rate combines all risks that were faced up to a time, t, and this accumulation is referred to as cumulative hazard rate.

CH R TWP ( t )=log[ λ α 2( β1 ) t 2( 1β ) α β1 t 1β ( λ1 ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaam4qai aadIeacaWGsbqcfa4aaSbaaKqaGeaajugWaiaadsfacaWGxbGaamiu aaWcbeaajuaGdaqadaGcbaqcLbsacaWG0baakiaawIcacaGLPaaaju gibiabg2da9iabgkHiTiGacYgacaGGVbGaai4zaKqbaoaadmaakeaa jugibiabeU7aSjabeg7aHTWaaWbaaKqaGeqabaqcLbmacaaIYaWcda qadaqcbasaaKqzadGaeqOSdiMaeyOeI0IaaGymaaqcbaIaayjkaiaa wMcaaaaajugibiaadshajuaGdaahaaWcbeqcbasaaKqzadGaaGOmaS WaaeWaaKqaGeaajugWaiaaigdacqGHsislcqaHYoGyaKqaGiaawIca caGLPaaaaaqcLbsacqGHsislcqaHXoqylmaaCaaajeaibeqaaKqzad GaeqOSdiMaeyOeI0IaaGymaaaajugibiaadshalmaaCaaajeaibeqa aKqzadGaaGymaiabgkHiTiabek7aIbaajuaGdaqadaGcbaqcLbsacq aH7oaBcqGHsislcaaIXaaakiaawIcacaGLPaaaaiaawUfacaGLDbaa aaa@749E@ (28)

Order statistics

It is important to study the range of the probability distribution and to serve the need for range; minimum and maximum pdf’s for the TWP distribution are derived. Most commonly, the pdf for the jth order statistic is used, and is thus derived below

f X ( j ) ( x ( j ) )= n! ( j1 )!( nj )! f X ( x ) [ F X ( x ) ] j1 [ 1 F X ( x ) ] nj MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamOzaS WaaSbaaKqaGeaajugWaiaadIfalmaaBaaajiaibaWcdaqadaqccasa aKqzadGaamOAaaqccaIaayjkaiaawMcaaaqabaaajeaibeaajuaGda qadaGcbaqcLbsacaWG4bWcdaWgaaqcbasaaSWaaeWaaKqaGeaajugW aiaadQgaaKqaGiaawIcacaGLPaaaaeqaaaGccaGLOaGaayzkaaqcLb sacqGH9aqpjuaGdaWcaaGcbaqcLbsacaWGUbGaaiyiaaGcbaqcfa4a aeWaaOqaaKqzGeGaamOAaiabgkHiTiaaigdaaOGaayjkaiaawMcaaK qzGeGaaiyiaKqbaoaabmaakeaajugibiaad6gacqGHsislcaWGQbaa kiaawIcacaGLPaaajugibiaacgcaaaGaamOzaKqbaoaaBaaajeaiba qcLbmacaWGybaaleqaaKqbaoaabmaakeaajugibiaadIhaaOGaayjk aiaawMcaaKqbaoaadmaakeaajugibiaadAeajuaGdaWgaaqcbasaaK qzadGaamiwaaWcbeaajuaGdaqadaGcbaqcLbsacaWG4baakiaawIca caGLPaaaaiaawUfacaGLDbaalmaaCaaajeaibeqaaKqzadGaamOAai abgkHiTiaaigdaaaqcfa4aamWaaOqaaKqzGeGaaGymaiabgkHiTiaa dAeajuaGdaWgaaqcbasaaKqzadGaamiwaaWcbeaajuaGdaqadaGcba qcLbsacaWG4baakiaawIcacaGLPaaaaiaawUfacaGLDbaalmaaCaaa jeaibeqaaKqzadGaamOBaiabgkHiTiaadQgaaaaaaa@7ECD@

= n! ( j1 )!( nj )! [ ( α β1 x ( j ) β ( β1 ) )( 1λ+2λ α β1 x ( j ) 1β ) ] [ 1+ α β1 x ( j ) 1β ( λ1 )λ α 2( β1 ) x ( j ) 2( 1β ) ] j1 [ α β1 x ( j ) 1β ( 1λ )+λ α 2( β1 ) x ( j ) 2( 1β ) ] nj MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaqcLbsacq GH9aqpjuaGdaWcaaGcbaqcLbsacaWGUbGaaiyiaaGcbaqcfa4aaeWa aOqaaKqzGeGaamOAaiabgkHiTiaaigdaaOGaayjkaiaawMcaaKqzGe GaaiyiaKqbaoaabmaakeaajugibiaad6gacqGHsislcaWGQbaakiaa wIcacaGLPaaajugibiaacgcaaaqcfa4aamWaaOqaaKqbaoaabmaake aajugibiabeg7aHLqbaoaaCaaaleqajeaibaqcLbmacqaHYoGycqGH sislcaaIXaaaaKqzGeGaamiEaSWaa0baaKqaGeaalmaabmaajeaiba qcLbmacaWGQbaajeaicaGLOaGaayzkaaaabaqcLbmacqGHsislcqaH YoGyaaqcfa4aaeWaaOqaaKqzGeGaeqOSdiMaeyOeI0IaaGymaaGcca GLOaGaayzkaaaacaGLOaGaayzkaaqcfa4aaeWaaOqaaKqzGeGaaGym aiabgkHiTiabeU7aSjabgUcaRiaaikdacqaH7oaBcqaHXoqyjuaGda ahaaWcbeqcbasaaKqzadGaeqOSdiMaeyOeI0IaaGymaaaajugibiaa dIhalmaaDaaajeaibaWcdaqadaqcbasaaKqzadGaamOAaaqcbaIaay jkaiaawMcaaaqaaKqzadGaaGymaiabgkHiTiabek7aIbaaaOGaayjk aiaawMcaaaGaay5waiaaw2faaaqaaKqbaoaadmaakeaajugibiaaig dacqGHRaWkcqaHXoqyjuaGdaahaaWcbeqcbasaaKqzadGaeqOSdiMa eyOeI0IaaGymaaaajugibiaadIhalmaaDaaajeaibaWcdaqadaqcba saaKqzadGaamOAaaqcbaIaayjkaiaawMcaaaqaaKqzadGaaGymaiab gkHiTiabek7aIbaajuaGdaqadaGcbaqcLbsacqaH7oaBcqGHsislca aIXaaakiaawIcacaGLPaaajugibiabgkHiTiabeU7aSjabeg7aHTWa aWbaaKqaGeqabaqcLbmacaaIYaWcdaqadaqcbasaaKqzadGaeqOSdi MaeyOeI0IaaGymaaqcbaIaayjkaiaawMcaaaaajugibiaadIhalmaa DaaajeaibaWcdaqadaqcbasaaKqzadGaamOAaaqcbaIaayjkaiaawM caaaqaaKqzadGaaGOmaSWaaeWaaKqaGeaajugWaiaaigdacqGHsisl cqaHYoGyaKqaGiaawIcacaGLPaaaaaaakiaawUfacaGLDbaajuaGda ahaaWcbeqcbasaaKqzadGaamOAaiabgkHiTiaaigdaaaqcfa4aamWa aOqaaKqzGeGaeqySdewcfa4aaWbaaSqabKqaGeaajugWaiabek7aIj abgkHiTiaaigdaaaqcLbsacaWG4bWcdaqhaaqcbasaaSWaaeWaaKqa GeaajugWaiaadQgaaKqaGiaawIcacaGLPaaaaeaajugWaiaaigdacq GHsislcqaHYoGyaaqcfa4aaeWaaOqaaKqzGeGaaGymaiabgkHiTiab eU7aSbGccaGLOaGaayzkaaqcLbsacqGHRaWkcqaH7oaBcqaHXoqylm aaCaaajeaibeqaaKqzadGaaGOmaSWaaeWaaKqaGeaajugWaiabek7a IjabgkHiTiaaigdaaKqaGiaawIcacaGLPaaaaaqcLbsacaWG4bWcda qhaaqcbasaaSWaaeWaaKqaGeaajugWaiaadQgaaKqaGiaawIcacaGL PaaaaeaajugWaiaaikdalmaabmaajeaibaqcLbmacaaIXaGaeyOeI0 IaeqOSdigajeaicaGLOaGaayzkaaaaaaGccaGLBbGaayzxaaqcfa4a aWbaaSqabKqaGeaajugWaiaad6gacqGHsislcaWGQbaaaaaaaa@F5C4@        (29)

Order statistics, useful for reliability studies, provide the 1st and nth order pdf for TWP distribution

f ( 1 ) ( x ( 1 ) )=n [ 1F( x ( 1 ) ) ] n1 f( x ( 1 ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamOzaK qbaoaaBaaajeaibaWcdaqadaqcbasaaKqzadGaaGymaaqcbaIaayjk aiaawMcaaaWcbeaajuaGdaqadaGcbaqcLbsacaWG4bqcfa4aaSbaaK qaGeaalmaabmaajeaibaqcLbmacaaIXaaajeaicaGLOaGaayzkaaaa leqaaaGccaGLOaGaayzkaaqcLbsacqGH9aqpcaWGUbqcfa4aamWaaO qaaKqzGeGaaGymaiabgkHiTiaadAeajuaGdaqadaGcbaqcLbsacaWG 4bqcfa4aaSbaaKqaGeaalmaabmaajeaibaqcLbmacaaIXaaajeaica GLOaGaayzkaaaaleqaaaGccaGLOaGaayzkaaaacaGLBbGaayzxaaqc fa4aaWbaaSqabKqaGeaajugWaiaad6gacqGHsislcaaIXaaaaKqzGe GaamOzaKqbaoaabmaakeaajugibiaadIhajuaGdaWgaaqcbasaaSWa aeWaaKqaGeaajugWaiaaigdaaKqaGiaawIcacaGLPaaaaSqabaaaki aawIcacaGLPaaaaaa@639A@

=n [ α β1 x ( 1 ) 1β ( 1λ )+λ α 2( β1 ) x ( 1 ) 2( 1β ) ] n1 [ ( α β1 x ( 1 ) β ( β1 ) )( 1λ+2λ α β1 x ( 1 ) 1β ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeyypa0 JaamOBaKqbaoaadmaakeaajugibiabeg7aHTWaaWbaaKqaGeqabaqc LbmacqaHYoGycqGHsislcaaIXaaaaKqzGeGaamiEaSWaa0baaKqaGe aalmaabmaajeaibaqcLbmacaaIXaaajeaicaGLOaGaayzkaaaabaqc LbmacaaIXaGaeyOeI0IaeqOSdigaaKqbaoaabmaakeaajugibiaaig dacqGHsislcqaH7oaBaOGaayjkaiaawMcaaKqzGeGaey4kaSIaeq4U dWMaeqySde2cdaahaaqcbasabeaajugWaiaaikdalmaabmaajeaiba qcLbmacqaHYoGycqGHsislcaaIXaaajeaicaGLOaGaayzkaaaaaKqz GeGaamiEaSWaa0baaKqaGeaalmaabmaajeaibaqcLbmacaaIXaaaje aicaGLOaGaayzkaaaabaqcLbmacaaIYaWcdaqadaqcbasaaKqzadGa aGymaiabgkHiTiabek7aIbqcbaIaayjkaiaawMcaaaaaaOGaay5wai aaw2faaKqbaoaaCaaaleqajeaibaqcLbmacaWGUbGaeyOeI0IaaGym aaaajuaGdaWadaGcbaqcfa4aaeWaaOqaaKqzGeGaeqySdewcfa4aaW baaSqabKqaGeaajugWaiabek7aIjabgkHiTiaaigdaaaqcLbsacaWG 4bWcdaqhaaqcbasaaSWaaeWaaKqaGeaajugWaiaaigdaaKqaGiaawI cacaGLPaaaaeaajugWaiabgkHiTiabek7aIbaajuaGdaqadaGcbaqc LbsacqaHYoGycqGHsislcaaIXaaakiaawIcacaGLPaaaaiaawIcaca GLPaaajuaGdaqadaGcbaqcLbsacaaIXaGaeyOeI0Iaeq4UdWMaey4k aSIaaGOmaiabeU7aSjabeg7aHTWaaWbaaKqaGeqabaqcLbmacqaHYo GycqGHsislcaaIXaaaaKqzGeGaamiEaSWaa0baaKqaGeaalmaabmaa jeaibaqcLbmacaaIXaaajeaicaGLOaGaayzkaaaabaqcLbmacaaIXa GaeyOeI0IaeqOSdigaaaGccaGLOaGaayzkaaaacaGLBbGaayzxaaaa aa@A7FB@    (30)

f ( n ) ( x ( n ) )=n [ F( x ( n ) ) ] n1 f( x ( n ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamOzaK qbaoaaBaaajeaibaWcdaqadaqcbasaaKqzadGaamOBaaqcbaIaayjk aiaawMcaaaWcbeaajuaGdaqadaGcbaqcLbsacaWG4bWcdaWgaaqcba saaSWaaeWaaKqaGeaajugWaiaad6gaaKqaGiaawIcacaGLPaaaaeqa aaGccaGLOaGaayzkaaqcLbsacqGH9aqpcaWGUbqcfa4aamWaaOqaaK qzGeGaamOraKqbaoaabmaakeaajugibiaadIhalmaaBaaajeaibaWc daqadaqcbasaaKqzadGaamOBaaqcbaIaayjkaiaawMcaaaqabaaaki aawIcacaGLPaaaaiaawUfacaGLDbaalmaaCaaajeaibeqaaKqzadGa amOBaiabgkHiTiaaigdaaaqcLbsacaWGMbqcfa4aaeWaaOqaaKqzGe GaamiEaKqbaoaaBaaajeaibaWcdaqadaqcbasaaKqzadGaamOBaaqc baIaayjkaiaawMcaaaWcbeaaaOGaayjkaiaawMcaaaaa@6128@

=n [ 1+ α β1 x ( n ) 1β ( λ1 )λ α 2( β1 ) x ( n ) 2( 1β ) ] n1 ×[ ( α β1 x ( n ) β ( β1 ) )( 1λ+2λ α β1 x ( n ) 1β ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaqcLbsacq GH9aqpcaWGUbqcfa4aamWaaOqaaKqzGeGaaGymaiabgUcaRiabeg7a HTWaaWbaaKqaGeqabaqcLbmacqaHYoGycqGHsislcaaIXaaaaKqzGe GaamiEaSWaa0baaKqaGeaalmaabmaajeaibaqcLbmacaWGUbaajeai caGLOaGaayzkaaaabaqcLbmacaaIXaGaeyOeI0IaeqOSdigaaKqbao aabmaakeaajugibiabeU7aSjabgkHiTiaaigdaaOGaayjkaiaawMca aKqzGeGaeyOeI0Iaeq4UdWMaeqySde2cdaahaaqcbasabeaajugWai aaikdalmaabmaajeaibaqcLbmacqaHYoGycqGHsislcaaIXaaajeai caGLOaGaayzkaaaaaKqzGeGaamiEaSWaa0baaKqaGeaalmaabmaaje aibaqcLbmacaWGUbaajeaicaGLOaGaayzkaaaajqwaG9FaaKqzadGa aGOmaSWaaeWaaKazba2=baqcLbmacaaIXaGaeyOeI0IaeqOSdigajq waG9VaayjkaiaawMcaaaaaaOGaay5waiaaw2faaKqbaoaaCaaaleqa jeaibaqcLbmacaWGUbGaeyOeI0IaaGymaaaaaOqaaKqzGeGaey41aq Bcfa4aamWaaOqaaKqbaoaabmaakeaajugibiabeg7aHLqbaoaaCaaa leqajeaibaqcLbmacqaHYoGycqGHsislcaaIXaaaaKqzGeGaamiEaS Waa0baaKqaGeaalmaabmaajeaibaqcLbmacaWGUbaajeaicaGLOaGa ayzkaaaabaqcLbmacqGHsislcqaHYoGyaaqcfa4aaeWaaOqaaKqzGe GaeqOSdiMaeyOeI0IaaGymaaGccaGLOaGaayzkaaaacaGLOaGaayzk aaqcfa4aaeWaaOqaaKqzGeGaaGymaiabgkHiTiabeU7aSjabgUcaRi aaikdacqaH7oaBcqaHXoqyjuaGdaahaaWcbeqcbasaaKqzadGaeqOS diMaeyOeI0IaaGymaaaajugibiaadIhalmaaDaaabaWaaeWaaeaaju gWaiaad6gaaSGaayjkaiaawMcaaaqcbasaaKqzadGaaGymaiabgkHi Tiabek7aIbaaaOGaayjkaiaawMcaaaGaay5waiaaw2faaaaaaa@B287@    (31)

The joint pdf for X(i) and X(j) is also found for the TWP distribution

f X ( i ) , X ( j ) ( u,v )= n! ( i1 )!( j1i )!( nj )! f X ( u ) f X ( v ) [ F X ( u ) ] i1 × [ F X ( v ) F X ( u ) ] j1i [ 1 F X ( v ) ] nj MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaqcLbsaca WGMbqcfa4aaSbaaKqaGeaajugWaiaadIfajuaGdaWgaaqccasaaKqb aoaabmaajiaibaqcLbmacaWGPbaajiaicaGLOaGaayzkaaaabeaaju gWaiaacYcacaWGybqcfa4aaSbaaKGaGeaajuaGdaqadaqccasaaKqz adGaamOAaaqccaIaayjkaiaawMcaaaqabaaaleqaaKqbaoaabmaake aajugibiaadwhacaGGSaGaamODaaGccaGLOaGaayzkaaqcLbsacqGH 9aqpjuaGdaWcaaGcbaqcLbsacaWGUbGaaiyiaaGcbaqcfa4aaeWaaO qaaKqzGeGaamyAaiabgkHiTiaaigdaaOGaayjkaiaawMcaaKqzGeGa aiyiaKqbaoaabmaakeaajugibiaadQgacqGHsislcaaIXaGaeyOeI0 IaamyAaaGccaGLOaGaayzkaaqcLbsacaGGHaqcfa4aaeWaaOqaaKqz GeGaamOBaiabgkHiTiaadQgaaOGaayjkaiaawMcaaKqzGeGaaiyiaa aacaWGMbqcfa4aaSbaaKqaGeaajugWaiaadIfaaSqabaqcfa4aaeWa aOqaaKqzGeGaamyDaaGccaGLOaGaayzkaaqcLbsacaWGMbqcfa4aaS baaKqaGeaajugWaiaadIfaaKqaGeqaaKqbaoaabmaakeaajugibiaa dAhaaOGaayjkaiaawMcaaKqbaoaadmaakeaajugibiaadAeajuaGda WgaaqcbasaaKqzadGaamiwaaWcbeaajuaGdaqadaGcbaqcLbsacaWG 1baakiaawIcacaGLPaaaaiaawUfacaGLDbaajuaGdaahaaqcbasabe aajugWaiaadMgacqGHsislcaaIXaaaaaGcbaqcLbsacqGHxdaTjuaG daWadaGcbaqcLbsacaWGgbqcfa4aaSbaaKqaGeaajugWaiaadIfaaS qabaqcfa4aaeWaaOqaaKqzGeGaamODaaGccaGLOaGaayzkaaqcLbsa cqGHsislcaWGgbqcfa4aaSbaaKqaGeaajugWaiaadIfaaSqabaqcfa 4aaeWaaOqaaKqzGeGaamyDaaGccaGLOaGaayzkaaaacaGLBbGaayzx aaqcfa4aaWbaaSqabKqaGeaajugWaiaadQgacqGHsislcaaIXaGaey OeI0IaamyAaaaajuaGdaWadaGcbaqcLbsacaaIXaGaeyOeI0IaamOr aKqbaoaaBaaajeaibaqcLbmacaWGybaaleqaaKqbaoaabmaakeaaju gibiaadAhaaOGaayjkaiaawMcaaaGaay5waiaaw2faaKqbaoaaCaaa leqajeaibaqcLbmacaWGUbGaeyOeI0IaamOAaaaaaaaa@B3F4@

= n! ( i1 )!( j1i )!( nj )! [ ( α β1 u β ( β1 ) )( 1λ+2λ α β1 u 1β ) ] ×[ ( α β1 v β ( β1 ) )( 1λ+2λ α β1 v 1β ) ] [ 1+ α β1 u 1β ( λ1 )λ α 2( β1 ) u 2( 1β ) ] i1 × [ α β1 v 1β ( λ1 )λ α 2( β1 ) v 2( 1β ) + α β1 u 1β ( 1λ )+λ α 2( β1 ) u 2( 1β ) ] j1i × [ α β1 v 1β ( 1λ )+λ α 2( β1 ) v 2( 1β ) ] nj MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaqcLbsacq GH9aqpjuaGdaWcaaGcbaqcLbsacaWGUbGaaiyiaaGcbaqcfa4aaeWa aOqaaKqzGeGaamyAaiabgkHiTiaaigdaaOGaayjkaiaawMcaaKqzGe GaaiyiaKqbaoaabmaakeaajugibiaadQgacqGHsislcaaIXaGaeyOe I0IaamyAaaGccaGLOaGaayzkaaqcLbsacaGGHaqcfa4aaeWaaOqaaK qzGeGaamOBaiabgkHiTiaadQgaaOGaayjkaiaawMcaaKqzGeGaaiyi aaaajuaGdaWadaGcbaqcfa4aaeWaaOqaaKqzGeGaeqySdewcfa4aaW baaSqabKqaGeaajugWaiabek7aIjabgkHiTiaaigdaaaqcLbsacaWG 1bqcfa4aaWbaaSqabKqaGeaajugWaiabgkHiTiabek7aIbaajuaGda qadaGcbaqcLbsacqaHYoGycqGHsislcaaIXaaakiaawIcacaGLPaaa aiaawIcacaGLPaaajuaGdaqadaGcbaqcLbsacaaIXaGaeyOeI0Iaeq 4UdWMaey4kaSIaaGOmaiabeU7aSjabeg7aHTWaaWbaaKqaGeqabaqc LbmacqaHYoGycqGHsislcaaIXaaaaKqzGeGaamyDaKqbaoaaCaaale qajeaibaqcLbmacaaIXaGaeyOeI0IaeqOSdigaaaGccaGLOaGaayzk aaaacaGLBbGaayzxaaaabaqcLbsacqGHxdaTjuaGdaWadaGcbaqcfa 4aaeWaaOqaaKqzGeGaeqySdewcfa4aaWbaaSqabKqaGeaajugWaiab ek7aIjabgkHiTiaaigdaaaqcLbsacaWG2bqcfa4aaWbaaSqabKqaGe aajugWaiabgkHiTiabek7aIbaajuaGdaqadaGcbaqcLbsacqaHYoGy cqGHsislcaaIXaaakiaawIcacaGLPaaaaiaawIcacaGLPaaajuaGda qadaGcbaqcLbsacaaIXaGaeyOeI0Iaeq4UdWMaey4kaSIaaGOmaiab eU7aSjabeg7aHLqbaoaaCaaaleqajeaibaqcLbmacqaHYoGycqGHsi slcaaIXaaaaKqzGeGaamODaKqbaoaaCaaaleqajeaibaqcLbmacaaI XaGaeyOeI0IaeqOSdigaaaGccaGLOaGaayzkaaaacaGLBbGaayzxaa qcfa4aamWaaOqaaKqzGeGaaGymaiabgUcaRiabeg7aHTWaaWbaaKqa GeqabaqcLbmacqaHYoGycqGHsislcaaIXaaaaKqzGeGaamyDaKqbao aaCaaaleqajeaibaqcLbmacaaIXaGaeyOeI0IaeqOSdigaaKqbaoaa bmaakeaajugibiabeU7aSjabgkHiTiaaigdaaOGaayjkaiaawMcaaK qzGeGaeyOeI0Iaeq4UdWMaeqySdegddaahaaqcKfaG=hqabaqcLbka caaIYaaddaqadaqcKfaG=haajugOaiabek7aIjabgkHiTiaaigdaaK azba4=caGLOaGaayzkaaaaaKqzGeGaamyDaKqbaoaaCaaaleqajeai baqcLbmacaaIYaWcdaqadaqcbasaaKqzadGaaGymaiabgkHiTiabek 7aIbqcbaIaayjkaiaawMcaaaaaaOGaay5waiaaw2faaKqbaoaaCaaa leqajeaibaqcLbmacaWGPbGaeyOeI0IaaGymaaaaaOqaaKqzGeGaey 41aqBcfa4aamWaaOqaaKqzGeGaeqySdewcfa4aaWbaaSqabKqaGeaa jugWaiabek7aIjabgkHiTiaaigdaaaqcLbsacaWG2bqcfa4aaWbaaS qabKqaGeaajugWaiaaigdacqGHsislcqaHYoGyaaqcfa4aaeWaaOqa aKqzGeGaeq4UdWMaeyOeI0IaaGymaaGccaGLOaGaayzkaaqcLbsacq GHsislcqaH7oaBcqaHXoqyjuaGdaahaaWcbeqcbasaaKqzadGaaGOm aSWaaeWaaKqaGeaajugWaiabek7aIjabgkHiTiaaigdaaKqaGiaawI cacaGLPaaaaaqcLbsacaWG2bWcdaahaaqcbasabeaajugWaiaaikda lmaabmaajeaibaqcLbmacaaIXaGaeyOeI0IaeqOSdigajeaicaGLOa GaayzkaaaaaKqzGeGaey4kaSIaeqySdewcfa4aaWbaaSqabKazba4= baqcLbkacqaHYoGycqGHsislcaaIXaaaaKqzGeGaamyDaKqbaoaaCa aaleqajeaibaqcLbmacaaIXaGaeyOeI0IaeqOSdigaaKqbaoaabmaa keaajugibiaaigdacqGHsislcqaH7oaBaOGaayjkaiaawMcaaKqzGe Gaey4kaSIaeq4UdWMaeqySdewcfa4aaWbaaSqabKqaGeaajugWaiaa ikdalmaabmaajeaibaqcLbmacqaHYoGycqGHsislcaaIXaaajeaica GLOaGaayzkaaaaaKqzGeGaamyDaKqbaoaaCaaaleqajeaibaqcLbma caaIYaWcdaqadaqcbasaaKqzadGaaGymaiabgkHiTiabek7aIbqcba IaayjkaiaawMcaaaaaaOGaay5waiaaw2faaKqbaoaaCaaaleqajeai baqcLbmacaWGQbGaeyOeI0IaaGymaiabgkHiTiaadMgaaaaakeaaju gibiabgEna0Mqbaoaadmaakeaajugibiabeg7aHLqbaoaaCaaaleqa jeaibaqcLbmacqaHYoGycqGHsislcaaIXaaaaKqzGeGaamODaKqbao aaCaaaleqajeaibaqcLbmacaaIXaGaeyOeI0IaeqOSdigaaKqbaoaa bmaakeaajugibiaaigdacqGHsislcqaH7oaBaOGaayjkaiaawMcaaK qzGeGaey4kaSIaeq4UdWMaeqySdewcfa4aaWbaaSqabKqaGeaajugW aiaaikdalmaabmaajeaibaqcLbmacqaHYoGycqGHsislcaaIXaaaje aicaGLOaGaayzkaaaaaKqzGeGaamODaKqbaoaaCaaaleqajeaibaqc LbmacaaIYaWcdaqadaqcbasaaKqzadGaaGymaiabgkHiTiabek7aIb qcbaIaayjkaiaawMcaaaaaaOGaay5waiaaw2faaKqbaoaaCaaaleqa jeaibaqcLbmacaWGUbGaeyOeI0IaamOAaaaaaaaa@8AF2@      (32)

Random number generation and parameter estimation

Random number generation

The inversion method is used to generate random numbers for the TWP distribution

1+ α β1 x 1β ( λ1 )λ α 2( β1 ) x 2( 1β ) =u MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaaGymai abgUcaRiabeg7aHLqbaoaaCaaaleqajeaibaqcLbmacqaHYoGycqGH sislcaaIXaaaaKqzGeGaamiEaSWaaWbaaKqaGeqabaqcLbmacaaIXa GaeyOeI0IaeqOSdigaaKqbaoaabmaakeaajugibiabeU7aSjabgkHi TiaaigdaaOGaayjkaiaawMcaaKqzGeGaeyOeI0Iaeq4UdWMaeqySde 2cdaahaaqcbasabeaajugWaiaaikdalmaabmaajeaibaqcLbmacqaH YoGycqGHsislcaaIXaaajeaicaGLOaGaayzkaaaaaKqzGeGaamiEaS WaaWbaaKqaGeqabaqcLbmacaaIYaWcdaqadaqcbasaaKqzadGaaGym aiabgkHiTiabek7aIbqcbaIaayjkaiaawMcaaaaajugibiabg2da9i aadwhaaaa@65AC@

Here, u~U( 0,1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamyDai aac6hacaWGvbqcfa4aaeWaaOqaaKqzGeGaaGimaiaacYcacaaIXaaa kiaawIcacaGLPaaaaaa@3E2F@ . After calculation, result for x is

x= { α 2β [ α β+1 λ 2 4λu+2λ+1 ( 1λ ) α β+1 ] 2λ } 1 1β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamiEai abg2da9KqbaoaacmaakeaajuaGdaWcaaGcbaqcLbsacqaHXoqyjuaG daahaaWcbeqcbasaaKqzadGaeyOeI0IaaGOmaiabek7aIbaajuaGda WadaGcbaqcLbsacqaHXoqyjuaGdaahaaWcbeqcbasaaKqzadGaeqOS diMaey4kaSIaaGymaaaajuaGdaGcaaGcbaqcLbsacqaH7oaBlmaaCa aajeaibeqaaKqzadGaaGOmaaaajugibiabgkHiTiaaisdacqaH7oaB caWG1bGaey4kaSIaaGOmaiabeU7aSjabgUcaRiaaigdaaSqabaqcLb sacqGHsisljuaGdaqadaGcbaqcLbsacaaIXaGaeyOeI0Iaeq4UdWga kiaawIcacaGLPaaajugibiabeg7aHLqbaoaaCaaaleqajeaibaqcLb macqaHYoGycqGHRaWkcaaIXaaaaaGccaGLBbGaayzxaaaabaqcLbsa caaIYaGaeq4UdWgaaaGccaGL7bGaayzFaaWcdaahaaqcbasabeaalm aalaaajeaibaqcLbmacaaIXaaajeaibaqcLbmacaaIXaGaeyOeI0Ia eqOSdigaaaaaaaa@75DA@       (33)

Equation (33) can be used to get random numbers when the parameters α, β and λ are known.

Method of moments

One of the techniques for parameter estimation is to use method of moments. This process uses moments of the distribution to estimate parameters. Since there are three parameters, there will be three equations:

β ^ = 5 α ^ n+ α ^ λ ^ n7 i=1 n x i + α ^ 2 n 2 +2 α ^ 2 λ ^ n 2 + α ^ 2 λ ^ 2 n 2 +2 α ^ i=1 n x i n 6 α ^ i=1 n x i λ ^ n+ i=1 n x i 2 2( 2 α ^ n2 i=1 n x i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGafqOSdi MbaKaacqGH9aqpjuaGdaWcaaGcbaqcLbsacaaI1aGafqySdeMbaKaa caWGUbGaey4kaSIafqySdeMbaKaacuaH7oaBgaqcaiaad6gacqGHsi slcaaI3aqcfa4aaabCaOqaaKqzGeGaamiEaSWaaSbaaKqaGeaajugW aiaadMgaaKqaGeqaaaqaaKqzadGaamyAaiabg2da9iaaigdaaKqaGe aajugWaiaad6gaaKqzGeGaeyyeIuoacqGHRaWkjuaGdaGcaaqcLbsa eaqabOqaaKqzGeGafqySdeMbaKaajuaGdaahaaWcbeqcbasaaKqzad GaaGOmaaaajugibiaad6gajuaGdaahaaWcbeqcbasaaKqzadGaaGOm aaaajugibiabgUcaRiaaikdacuaHXoqygaqcaKqbaoaaCaaaleqaje aibaqcLbmacaaIYaaaaKqzGeGafq4UdWMbaKaacaWGUbqcfa4aaWba aSqabKqaGeaajugWaiaaikdaaaqcLbsacqGHRaWkcuaHXoqygaqcaK qbaoaaCaaaleqajeaibaqcLbmacaaIYaaaaKqzGeGafq4UdWMbaKaa juaGdaahaaWcbeqcbasaaKqzadGaaGOmaaaajugibiaad6gajuaGda ahaaWcbeqcbasaaKqzadGaaGOmaaaajugibiabgUcaRiaaikdacuaH XoqygaqcaKqbaoaaqahakeaajugibiaadIhalmaaBaaajeaqbaqcLb oacaWGPbaajeaqbeaaaKqaGeaajugWaiaadMgacqGH9aqpcaaIXaaa jeaibaqcLbmacaWGUbaajugibiabggHiLdGaamOBaaGcbaqcLbsacq GHsislcaaI2aGafqySdeMbaKaajuaGdaaeWbGcbaqcLbsacaWG4bqc fa4aaSbaaKqaGeaajugWaiaadMgaaSqabaaajeaibaqcLbmacaWGPb Gaeyypa0JaaGymaaqcbasaaKqzadGaamOBaaqcLbsacqGHris5aiqb eU7aSzaajaGaamOBaiabgUcaRKqbaoaaqahakeaajugibiaadIhaju aGdaqhaaqcbasaaKqzadGaamyAaaqcbasaaKqzadGaaGOmaaaaaKqa GeaajugWaiaadMgacqGH9aqpcaaIXaaajeaibaqcLbmacaWGUbaaju gibiabggHiLdaaaSqabaaakeaajugibiaaikdajuaGdaqadaGcbaqc LbsacaaIYaGafqySdeMbaKaacaWGUbGaeyOeI0IaaGOmaKqbaoaaqa hakeaajugibiaadIhajuaGdaWgaaqcbauaaKqzGdGaamyAaaqcbaua baaajeaibaqcLbmacaWGPbGaeyypa0JaaGymaaqcbasaaKqzadGaam OBaaqcLbsacqGHris5aaGccaGLOaGaayzkaaaaaaaa@CABF@     (34)

λ ^ = i=1 n x i 2 β ^ 2 5 i=1 n x i 2 β ^ +6 i=1 n x i 2 β ^ 2 α ^ 2 n+3 β ^ α ^ 2 n2 α ^ 2 n α ^ 2 n( 1 β ^ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGafq4UdW MbaKaacqGH9aqpjuaGdaWcaaGcbaqcfa4aaabCaOqaaKqzGeGaamiE aKqbaoaaDaaaleaajugibiaadMgaaSqaaKqzGeGaaGOmaaaaaKqaGe aajugWaiaadMgacqGH9aqpcaaIXaaajeaibaqcLbmacaWGUbaajugi biabggHiLdGafqOSdiMbaKaajuaGdaahaaWcbeqaaKqzGeGaaGOmaa aacqGHsislcaaI1aqcfa4aaabCaOqaaKqzGeGaamiEaSWaa0baaKqa GeaajugWaiaadMgaaKqaGeaajugWaiaaikdaaaaajeaibaqcLbmaca WGPbGaeyypa0JaaGymaaqcbasaaKqzadGaamOBaaqcLbsacqGHris5 aiqbek7aIzaajaGaey4kaSIaaGOnaKqbaoaaqahakeaajugibiaadI hajuaGdaqhaaqcbauaaKqzGdGaamyAaaqcbasaaKqzadGaaGOmaaaa aKqaGeaajugWaiaadMgacqGH9aqpcaaIXaaajeaibaqcLbmacaWGUb aajugibiabggHiLdGaeyOeI0IafqOSdiMbaKaajuaGdaahaaWcbeqc basaaKqzadGaaGOmaaaajugibiqbeg7aHzaajaqcfa4aaWbaaKqaGe qabaqcLbmacaaIYaaaaKqzGeGaamOBaiabgUcaRiaaiodacuaHYoGy gaqcaiqbeg7aHzaajaqcfa4aaWbaaKqaGeqabaqcLbmacaaIYaaaaK qzGeGaamOBaiabgkHiTiaaikdacuaHXoqygaqcaKqbaoaaCaaajeai beqaaKqzadGaaGOmaaaajugibiaad6gaaOqaaKqzGeGafqySdeMbaK aajuaGdaahaaWcbeqcbasaaKqzadGaaGOmaaaajugibiaad6gajuaG daqadaGcbaqcLbsacaaIXaGaeyOeI0IafqOSdiMbaKaaaOGaayjkai aawMcaaaaaaaa@9B0E@     (35)

α ^ = 20 i=1 n x i 3 13 i=1 n x i 3 β ^ +2 i=1 n x i 3 β ^ 2 n( 5+2 β ^ 2 +3 λ ^ 7 β ^ 3 β ^ λ ^ ) 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGafqySde MbaKaacqGH9aqpjuaGdaGcbaGcbaqcfa4aaSaaaOqaaKqzGeGaaGOm aiaaicdajuaGdaaeWbGcbaqcLbsacaWG4bWcdaqhaaqcbasaaKqzad GaamyAaaqcbasaaKqzadGaaG4maaaajugibiabgkHiTiaaigdacaaI ZaaaleaajugibiaadMgacqGH9aqpcaaIXaaajqwaG9FaaKqzadGaam OBaaqcLbsacqGHris5aKqbaoaaqahakeaajugibiaadIhalmaaDaaa jeaibaqcLbmacaWGPbaajeaibaqcLbmacaaIZaaaaaqcbasaaKqzad GaamyAaiabg2da9iaaigdaaKqaGeaajugWaiaad6gaaKqzGeGaeyye IuoacuaHYoGygaqcaiabgUcaRiaaikdajuaGdaaeWbGcbaqcLbsaca WG4bWcdaqhaaqcbasaaKqzadGaamyAaaqcbasaaKqzadGaaG4maaaa aKqaafaajug4aiaadMgacqGH9aqpcaaIXaaajeaibaqcLbmacaWGUb aajugibiabggHiLdGafqOSdiMbaKaalmaaCaaajeaqbeqaaKqzGdGa aGOmaaaaaOqaaKqzGeGaamOBaKqbaoaabmaakeaajugibiaaiwdacq GHRaWkcaaIYaGafqOSdiMbaKaajuaGdaahaaWcbeqcbauaaKqzGdGa aGOmaaaajugibiabgUcaRiaaiodacuaH7oaBgaqcaiabgkHiTiaaiE dacuaHYoGygaqcaiabgkHiTiaaiodacuaHYoGygaqcaiqbeU7aSzaa jaaakiaawIcacaGLPaaaaaaaleaajugibiaaiodaaaaaaa@906D@              (36)

Equations (34), (35) and (36) express the parameters; they can be further solved simultaneously to get cleaner expressions for the parameters.

 Maximum likelihood estimation

Widely used technique for evaluating the parameters of the distribution is that of Maximum Likelihood Estimation technique. If x 1 , x 2 ,, x n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadI hakmaaBaaaleaajugibiaaigdaaSqabaqcLbsacaGGSaGaamiEaOWa aSbaaSqaaKqzGeGaaGOmaaWcbeaajugibiaacYcacqWIMaYscaGGSa GaamiEaOWaaSbaaSqaaKqzGeGaamOBaaWcbeaaaaa@42EE@  is a random sample of size n from the TWP distribution then its log-likelihood function will be:

L= i=1 n f TWP ( x i ;α,β,λ )= ( β1 ) n ( α n( β1 ) i=1 n x i β ) i=1 n [ ( 1λ+2λ α β1 x 1β ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamitai abg2da9KqbaoaarahakeaajugibiaadAgajuaGdaWgaaWcbaqcLbsa caWGubGaam4vaiaadcfaaSqabaqcfa4aaeWaaOqaaKqzGeGaamiEaS WaaSbaaKqaGeaajugWaiaadMgaaKqaGeqaaKqzGeGaai4oaiabeg7a HjaacYcacqaHYoGycaGGSaGaeq4UdWgakiaawIcacaGLPaaajugibi abg2da9Kqbaoaabmaakeaajugibiabek7aIjabgkHiTiaaigdaaOGa ayjkaiaawMcaaSWaaWbaaKqaGeqabaqcLbmacaWGUbaaaKqbaoaabm aakeaajugibiabeg7aHTWaaWbaaKqaGeqabaqcLbmacaWGUbWcdaqa daqcbasaaKqzadGaeqOSdiMaeyOeI0IaaGymaaqcbaIaayjkaiaawM caaaaajuaGdaqeWbGcbaqcLbsacaWG4bqcfa4aa0baaKqaGeaajugW aiaadMgaaKqaGeaajugWaiabgkHiTiabek7aIbaaaKqaGeaajugWai aadMgacqGH9aqpcaaIXaaajeaqbaqcLboacaWGUbaajugibiabg+Gi vdaakiaawIcacaGLPaaajuaGdaqeWbGcbaqcfa4aamWaaOqaaKqbao aabmaakeaajugibiaaigdacqGHsislcqaH7oaBcqGHRaWkcaaIYaGa eq4UdWMaeqySdewcfa4aaWbaaSqabKqaGeaajugWaiabek7aIjabgk HiTiaaigdaaaqcLbsacaWG4bqcfa4aaWbaaKqaGeqabaqcLbmacaaI XaGaeyOeI0IaeqOSdigaaaGccaGLOaGaayzkaaaacaGLBbGaayzxaa aajeaibaqcLbmacaWGPbGaeyypa0JaaGymaaqcbasaaKqzadGaamOB aaqcLbsacqGHpis1aaqcbasaaKqzadGaamyAaiabg2da9iaaigdaaK qaGeaajugWaiaad6gaaKqzGeGaey4dIunaaaa@A206@

Using LL=ln L, log-likelihood function is derived

LL=nln( β1 )+n( β1 )lnαβ i=1 n ln x i + i=1 n ln [ ( 1λ+2λ α β1 x 1β ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamitai aadYeacqGH9aqpcaWGUbGaciiBaiaac6gajuaGdaqadaGcbaqcLbsa cqaHYoGycqGHsislcaaIXaaakiaawIcacaGLPaaajugibiabgUcaRi aad6gajuaGdaqadaGcbaqcLbsacqaHYoGycqGHsislcaaIXaaakiaa wIcacaGLPaaajugibiGacYgacaGGUbGaeqySdeMaeyOeI0IaeqOSdi wcfa4aaabCaOqaaKqzGeGaciiBaiaac6gacaWG4bqcfa4aaSbaaSqa aKqzGeGaamyAaaWcbeaajugibiabgUcaRaqcbasaaKqzadGaamyAai abg2da9iaaigdaaKqaafaajug4aiaad6gaaKqzGeGaeyyeIuoajuaG daaeWbGcbaqcLbsaciGGSbGaaiOBaaqcbasaaKqzadGaamyAaiabg2 da9iaaigdaaKqaGeaajugWaiaad6gaaKqzGeGaeyyeIuoajuaGdaWa daGcbaqcfa4aaeWaaOqaaKqzGeGaaGymaiabgkHiTiabeU7aSjabgU caRiaaikdacqaH7oaBcqaHXoqylmaaCaaajeaibeqaaKqzadGaeqOS diMaeyOeI0IaaGymaaaajugibiaadIhalmaaCaaajeaqbeqaaKqzGd GaaGymaiabgkHiTiabek7aIbaaaOGaayjkaiaawMcaaaGaay5waiaa w2faaaaa@8718@      (37)

To estimate the parameters, equation (37) is differentiated with respect to β and λ and put equal to zero so as to get the respective parameters.

LL β = n β1 +nlnα i=1 n ln x i + i=1 n 2 α β1 lnαλ x i 1β 2 α β1 λ x i 1β ln( x i ) ( 2 α β1 λ x i 1β λ+1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGdaWcaa GcbaqcLbsacqGHciITcaWGmbGaamitaaGcbaqcLbsacqGHciITcqaH YoGyaaGaeyypa0tcfa4aaSaaaOqaaKqzGeGaamOBaaGcbaqcLbsacq aHYoGycqGHsislcaaIXaaaaiabgUcaRiaad6gaciGGSbGaaiOBaiab eg7aHjabgkHiTKqbaoaaqahakeaajugibiGacYgacaGGUbGaamiEaS WaaSbaaKqaGeaajugWaiaadMgaaKqaGeqaaaqaaKqzadGaamyAaiab g2da9iaaigdaaKqaGeaajugWaiaad6gaaKqzGeGaeyyeIuoacqGHRa WkjuaGdaaeWbGcbaqcfa4aaSaaaOqaaKqzGeGaaGOmaiabeg7aHLqb aoaaCaaajeaibeqaaKqzadGaeqOSdiMaeyOeI0IaaGymaaaajugibi GacYgacaGGUbGaeqySdeMaeq4UdWMaamiEaKqbaoaaDaaajeaibaqc LbmacaWGPbaajeaibaqcLbmacaaIXaGaeyOeI0IaeqOSdigaaKqzGe GaeyOeI0IaaGOmaiabeg7aHLqbaoaaCaaajeaibeqaaKqzadGaeqOS diMaeyOeI0IaaGymaaaajugibiabeU7aSjaadIhajuaGdaqhaaqcba saaKqzadGaamyAaaqcbasaaKqzadGaaGymaiabgkHiTiabek7aIbaa jugibiGacYgacaGGUbqcfa4aaeWaaOqaaKqzGeGaamiEaKqbaoaaBa aajeaibaqcLbmacaWGPbaajeaibeaaaOGaayjkaiaawMcaaaqaaKqb aoaabmaakeaajugibiaaikdacqaHXoqyjuaGdaahaaqcbasabeaaju gWaiabek7aIjabgkHiTiaaigdaaaqcLbsacqaH7oaBcaWG4bqcfa4a a0baaKqaGeaajugWaiaadMgaaKqaGeaajugWaiaaigdacqGHsislcq aHYoGyaaqcLbsacqGHsislcqaH7oaBcqGHRaWkcaaIXaaakiaawIca caGLPaaaaaaajeaibaqcLbmacaWGPbGaeyypa0JaaGymaaqcbasaaK qzadGaamOBaaqcLbsacqGHris5aaaa@B42D@    (38)

LL λ = i=1 n 2 α β1 x i 1β 1 ( 1λ+2λ α β1 x i 1β ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGdaWcaa GcbaqcLbsacqGHciITcaWGmbGaamitaaGcbaqcLbsacqGHciITcqaH 7oaBaaGaeyypa0tcfa4aaabCaOqaaKqbaoaalaaakeaajugibiaaik dacqaHXoqylmaaCaaajeaibeqaaKqzadGaeqOSdiMaeyOeI0IaaGym aaaajugibiaadIhalmaaDaaajeaibaqcLbmacaWGPbaajeaibaqcLb macaaIXaGaeyOeI0IaeqOSdigaaKqzGeGaeyOeI0IaaGymaaGcbaqc fa4aaeWaaOqaaKqzGeGaaGymaiabgkHiTiabeU7aSjabgUcaRiaaik dacqaH7oaBcqaHXoqyjuaGdaahaaWcbeqaaKqzadGaeqOSdiMaeyOe I0scLbsacaaIXaaaaiaadIhalmaaDaaajeaibaqcLbmacaWGPbaaje aibaqcLbmacaaIXaGaeyOeI0IaeqOSdigaaaGccaGLOaGaayzkaaaa aaqcbasaaKqzadGaamyAaiabg2da9iaaigdaaKqaafaajug4aiaad6 gaaKqzGeGaeyyeIuoaaaa@7553@     (39)

For TWP distribution xα MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWG4bGaey yzImRaeqySdegaaa@3AA0@ , α is the lower limit for this distribution so the maximum likelihood estimate of α will be the first statistic value i.e., x ( 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadI hakmaaBaaajeaibaWcdaqadaqcbasaaKqzadGaaGymaaqcbaIaayjk aiaawMcaaaWcbeaaaaa@3BFB@ . The log-likelihood function is numerically maximised by using the R software.

Simulation

Simulation can help in understanding the data sets for the particular distribution. Inverse CDF technique is used to simulate the data set in R. The values used for parameters are α=1, β=6 and λ=-0.2. A data set of size 100 is thus simulated for the TWP distribution (Table 1).

Data generated for α=1, β=3 and λ=-0.2

2.171301

2.469904

3.627952

4.122543

2.290153

1.961311

2.02465

2.788962

1.912685

2.237079

1.993091

2.611573

2.926888

2.658368

2.705542

2.050194

2.19487

2.333488

2.010353

2.462991

4.240425

2.054078

2.150241

2.836084

3.139823

3.276202

3.642714

1.914709

2.589984

1.980117

2.079828

2.081706

1.925856

1.989657

1.95235

3.188533

3.776208

2.161587

3.04785

2.287527

1.908391

1.909436

2.133601

1.945302

2.128175

2.208188

2.512261

2.831646

1.927698

2.205562

2.107547

2.102916

2.061781

2.129319

1.963237

2.142851

1.952253

2.75405

2.385384

2.056045

2.490634

1.987443

2.111349

2.256456

3.406278

2.197887

1.921594

3.08898

2.598769

2.323328

2.2008

2.32725

3.405701

2.567741

2.850449

1.931747

1.956453

2.343114

2.214165

2.446729

2.566559

1.917651

2.910597

2.54337

2.346076

1.959292

1.939467

2.359965

2.056514

2.491718

3.358692

2.762751

2.22607

2.009319

3.075688

2.161866

1.906318

2.630087

2.90743

3.515726

Table 1 Results from the simulation study of the TWP distribution

The data given above is used to estimate parameters for the distribution. Estimated values are given in Table 2.

Model

Estimates

Standard Error

Transmuted Weighted Pareto

β ^ =5.695927 λ ^ =0.109770 α ^ =min(x)=1.906318 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaqcLbsacu aHYoGygaqcaiabg2da9iaabwdacaqGUaGaaeOnaiaabMdacaqG1aGa aeyoaiaabkdacaqG3aaakeaajugibiqbeU7aSzaajaGaeyypa0Jaey OeI0Iaaeimaiaab6cacaqGXaGaaeimaiaabMdacaqG3aGaae4naiaa bcdaaOqaaKqzGeGafqySdeMbaKaacqGH9aqpciGGTbGaaiyAaiaac6 gacaGGOaGaamiEaiaacMcacqGH9aqpcaqGXaGaaeOlaiaabMdacaqG WaGaaeOnaiaabodacaqGXaGaaeioaaaaaa@5806@

0.681109
0.241365

Table 2 Estimated values of parameters for TWP distribution

The variance covariance matrix for the TWP distribution with the above data will be: | 0.4639094 -0.12210893 -0.12210893 0.05825721 | MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaaemaabaqcLb safaqabeGacaaakeaajugibiaabcdacaqGUaGaaeinaiaabAdacaqG ZaGaaeyoaiaabcdacaqG5aGaaeinaaGcbaqcLbsacaqGTaGaaeimai aab6cacaqGXaGaaeOmaiaabkdacaqGXaGaaeimaiaabIdacaqG5aGa ae4maaGcbaqcLbsacaqGTaGaaeimaiaab6cacaqGXaGaaeOmaiaabk dacaqGXaGaaeimaiaabIdacaqG5aGaae4maaGcbaqcLbsacaqGWaGa aeOlaiaabcdacaqG1aGaaeioaiaabkdacaqG1aGaae4naiaabkdaca qGXaaaaaGccaGLhWUaayjcSdaaaa@590C@

This shows that the variance of MLE of β and λ are Var ( β ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaacI caiiaacuWFYoGygaWeaiaacMcaaaa@3C18@ = 0.4639094 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaabcdacaqGUa GaaeinaiaabAdacaqGZaGaaeyoaiaabcdacaqG5aGaaeinaaaa@3C37@  and Var ( λ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaacI cacuaH7oaBgaWeaiaacMcaaaa@3C26@ = 0.05825721 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaabcdacaqGUa GaaeimaiaabwdacaqG4aGaaeOmaiaabwdacaqG3aGaaeOmaiaabgda aaa@3CE5@  respectively.

Record values

Record values show in a systematic way the arrangement of the random variable. Bashir & Ahmad12 characterise a weighted Pareto distribution based on its upper record values. Therefore, as it is an important area to study the record values for TWP distribution are also studied to reveal information about sequence of random variables.

f n ( x )= [ R( x ) ] n1 Γn f( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamOzaS WaaSbaaKqaGeaajugWaiaad6gaaKqaGeqaaKqbaoaabmaakeaajugi biaadIhaaOGaayjkaiaawMcaaKqzGeGaeyypa0tcfa4aaSaaaOqaaK qbaoaadmaakeaajugibiaadkfajuaGdaqadaGcbaqcLbsacaWG4baa kiaawIcacaGLPaaaaiaawUfacaGLDbaajuaGdaahaaWcbeqcbasaaK qzadGaamOBaiabgkHiTiaaigdaaaaakeaajugibiabfo5ahjaad6ga aaGaamOzaKqbaoaabmaakeaajugibiaadIhaaOGaayjkaiaawMcaaa aa@53E5@

Where, R( x )=ln[ 1F( x ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadk fakmaabmaabaqcLbsacaWG4baakiaawIcacaGLPaaajugibiabg2da 9iabgkHiTiGacYgacaGGUbGcdaWadaqaaKqzGeGaaGymaiabgkHiTi aadAeakmaabmaabaqcLbsacaWG4baakiaawIcacaGLPaaaaiaawUfa caGLDbaaaaa@475A@ and f(x) is the pdf of the TWP distribution.

= ( β1 )( α β1 x u β )( 1λ+2λ α β1 x u 1β ) Γn × [ ln( α β1 x u 1β λ α β1 x u 1β +λ α 2( β1 ) x u 2( 1β ) ) ] n1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaqcLbsacq GH9aqpjuaGdaWcaaGcbaqcfa4aaeWaaOqaaKqzGeGaeqOSdiMaeyOe I0IaaGymaaGccaGLOaGaayzkaaqcfa4aaeWaaOqaaKqzGeGaeqySde wcfa4aaWbaaSqabKqaGeaajugWaiabek7aIjabgkHiTiaaigdaaaqc LbsacaWG4bWcdaqhaaqcbasaaKqzadGaamyDaaqcbasaaKqzadGaey OeI0IaeqOSdigaaaGccaGLOaGaayzkaaqcfa4aaeWaaOqaaKqzGeGa aGymaiabgkHiTiabeU7aSjabgUcaRiaaikdacqaH7oaBcqaHXoqyju aGdaahaaWcbeqcKfaG=haajugOaiabek7aIjabgkHiTiaaigdaaaqc LbsacaWG4baddaqhaaqcKfaG=haajugOaiaadwhaaKazba4=baqcLb kacaaIXaGaeyOeI0IaeqOSdigaaaGccaGLOaGaayzkaaaabaqcLbsa cqqHtoWrcaWGUbaaaaGcbaqcLbsacqGHxdaTjuaGdaWadaGcbaqcLb sacqGHsislciGGSbGaaiOBaKqbaoaabmaakeaajugibiabeg7aHTWa aWbaaKqaGeqabaqcLbmacqaHYoGycqGHsislcaaIXaaaaKqzGeGaam iEaKqbaoaaDaaajeaibaqcLbmacaWG1baajeaibaqcLbmacaaIXaGa eyOeI0IaeqOSdigaaKqzGeGaeyOeI0Iaeq4UdWMaeqySde2cdaahaa qcbasabeaajugWaiabek7aIjabgkHiTiaaigdaaaqcLbsacaWG4bWc daqhaaqcbasaaKqzadGaamyDaaqcbasaaKqzadGaaGymaiabgkHiTi abek7aIbaajugibiabgUcaRiabeU7aSjabeg7aHLqbaoaaCaaaleqa jeaibaqcLbmacaaIYaWcdaqadaqcbasaaKqzadGaeqOSdiMaeyOeI0 IaaGymaaqcbaIaayjkaiaawMcaaaaajugibiaadIhalmaaDaaajeai baqcLbmacaWG1baajeaibaqcLbmacaaIYaWcdaqadaqcbasaaKqzad GaaGymaiabgkHiTiabek7aIbqcbaIaayjkaiaawMcaaaaaaOGaayjk aiaawMcaaaGaay5waiaaw2faaKqbaoaaCaaaleqajeaibaqcLbmaca WGUbGaeyOeI0IaaGymaaaaaaaa@BD12@         (40)

Solving the above equation for mean and variance may lead to more complex expressions. Therefore, numerical values of α, β and λ (estimated parameters from the simulation study) are used to find the mean and variance of upper record values (Table 3).

Parameters

n

Mean

Variance

α = 1.9,

2

0.718

2.095

β = 5.7,

3

0.914

3.723

λ = -0.1

4

1.162

6.593

5

1.477

11.652

Table 3 Mean and Variance of the upper record values of TWP distribution

The joint pdf of X U ( i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamiwaS WaaSbaaKqaGeaajugWaiaadwfalmaaBaaajiaibaWcdaqadaqccasa aKqzadGaamyAaaqccaIaayjkaiaawMcaaaqabaaajeaibeaaaaa@3E1C@ and X U ( j ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamiwaO WaaSbaaKqaGeaajugWaiaadwfalmaaBaaajiaibaWcdaqadaqccasa aKqzadGaamOAaaqccaIaayjkaiaawMcaaaqabaaaleqaaaaa@3DFD@ is

f i,j ( x,y )= [ R( x ) ] i1 Γi r( x ) [ R( y )R( x ) ] ji1 Γ( ji ) f( y ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamOzaS WaaSbaaKqaGeaajugWaiaadMgacaGGSaGaamOAaaqcbasabaqcfa4a aeWaaOqaaKqzGeGaamiEaiaacYcacaWG5baakiaawIcacaGLPaaaju gibiabg2da9KqbaoaalaaakeaajuaGdaWadaGcbaqcLbsacaWGsbqc fa4aaeWaaOqaaKqzGeGaamiEaaGccaGLOaGaayzkaaaacaGLBbGaay zxaaWcdaahaaqcbasabeaajugWaiaadMgacqGHsislcaaIXaaaaaGc baqcLbsacqqHtoWrcaWGPbaaaiaadkhajuaGdaqadaGcbaqcLbsaca WG4baakiaawIcacaGLPaaajuaGdaWcaaGcbaqcfa4aamWaaOqaaKqz GeGaamOuaKqbaoaabmaakeaajugibiaadMhaaOGaayjkaiaawMcaaK qzGeGaeyOeI0IaamOuaKqbaoaabmaakeaajugibiaadIhaaOGaayjk aiaawMcaaaGaay5waiaaw2faaKqbaoaaCaaaleqajeaibaqcLbmaca WGQbGaeyOeI0IaamyAaiabgkHiTiaaigdaaaaakeaajugibiabfo5a hLqbaoaabmaakeaajugibiaadQgacqGHsislcaWGPbaakiaawIcaca GLPaaaaaqcLbsacaWGMbqcfa4aaeWaaOqaaKqzGeGaamyEaaGccaGL OaGaayzkaaaaaa@7837@

= [ ln( α β1 x 1β λ α β1 x 1β +λ α 2( β1 ) x 2( 1β ) ) ] i1 Γi ×[ ( β1 ) α β1 x β ( 1λ+2λ α β1 x 1β ) ( xλx+λ α β1 x 2β ) ] × [ ln( α β1 y 1β λ α β1 y 1β +λ α 2( β1 ) y 2( 1β ) ) +ln( α β1 x 1β λ α β1 x 1β +λ α 2( β1 ) x 2( 1β ) ) ] ji1 Γ( ji ) ×( β1 )( α β1 y β )[ 1λ+2λ α β1 y 1β ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaqcLbsacq GH9aqpjuaGdaWcaaGcbaqcfa4aamWaaOqaaKqzGeGaeyOeI0IaciiB aiaac6gajuaGdaqadaGcbaqcLbsacqaHXoqyjuaGdaahaaWcbeqcba saaKqzadGaeqOSdiMaeyOeI0IaaGymaaaajugibiaadIhalmaaCaaa jeaibeqaaKqzadGaaGymaiabgkHiTiabek7aIbaajugibiabgkHiTi abeU7aSjabeg7aHTWaaWbaaKqaGeqabaqcLbmacqaHYoGycqGHsisl caaIXaaaaKqzGeGaamiEaSWaaWbaaKqaGeqabaqcLbmacaaIXaGaey OeI0IaeqOSdigaaKqzGeGaey4kaSIaeq4UdWMaeqySdewcfa4aaWba aSqabKqaGeaajugWaiaaikdalmaabmaajeaibaqcLbmacqaHYoGycq GHsislcaaIXaaajeaicaGLOaGaayzkaaaaaKqzGeGaamiEaSWaaWba aKqaGeqabaqcLbmacaaIYaWcdaqadaqcbasaaKqzadGaaGymaiabgk HiTiabek7aIbqcbaIaayjkaiaawMcaaaaaaOGaayjkaiaawMcaaaGa ay5waiaaw2faaSWaaWbaaKqaGeqabaqcLbmacaWGPbGaeyOeI0IaaG ymaaaaaOqaaKqzGeGaeu4KdCKaamyAaaaaaOqaaKqzGeGaey41aqBc fa4aamWaaOqaaKqbaoaalaaakeaajuaGdaqadaGcbaqcLbsacqaHYo GycqGHsislcaaIXaaakiaawIcacaGLPaaajugibiabeg7aHLqbaoaa CaaaleqajeaibaqcLbmacqaHYoGycqGHsislcaaIXaaaaKqzGeGaam iEaSWaaWbaaKqaGeqabaqcLbmacqGHsislcqaHYoGyaaqcfa4aaeWa aOqaaKqzGeGaaGymaiabgkHiTiabeU7aSjabgUcaRiaaikdacqaH7o aBcqaHXoqylmaaCaaajeaibeqaaKqzadGaeqOSdiMaeyOeI0IaaGym aaaajugibiaadIhalmaaCaaajeaibeqaaKqzadGaaGymaiabgkHiTi abek7aIbaaaOGaayjkaiaawMcaaaqaaKqbaoaabmaakeaajugibiaa dIhacqGHsislcqaH7oaBcaWG4bGaey4kaSIaeq4UdWMaeqySde2cda ahaaqcbasabeaajugWaiabek7aIjabgkHiTiaaigdaaaqcLbsacaWG 4bWcdaahaaqcbasabeaajugWaiaaikdacqGHsislcqaHYoGyaaaaki aawIcacaGLPaaaaaaacaGLBbGaayzxaaaabaqcLbsacqGHxdaTjuaG daWcaaGcbaqcfa4aamWaaKqzGeabaeqakeaajugibiabgkHiTiGacY gacaGGUbqcfa4aaeWaaOqaaKqzGeGaeqySde2cdaahaaqcbasabeaa jugWaiabek7aIjabgkHiTiaaigdaaaqcLbsacaWG5bWcdaahaaqcba sabeaajugWaiaaigdacqGHsislcqaHYoGyaaqcLbsacqGHsislcqaH 7oaBcqaHXoqyjuaGdaahaaWcbeqcbasaaKqzadGaeqOSdiMaeyOeI0 IaaGymaaaajugibiaadMhalmaaCaaajeaibeqaaKqzadGaaGymaiab gkHiTiabek7aIbaajugibiabgUcaRiabeU7aSjabeg7aHTWaaWbaaK qaGeqabaqcLbmacaaIYaWcdaqadaqcbasaaKqzadGaeqOSdiMaeyOe I0IaaGymaaqcbaIaayjkaiaawMcaaaaajugibiaadMhalmaaCaaaje aibeqaaKqzadGaaGOmaSWaaeWaaKqaGeaajugWaiaaigdacqGHsisl cqaHYoGyaKqaGiaawIcacaGLPaaaaaaakiaawIcacaGLPaaaaeaaju gibiabgUcaRiGacYgacaGGUbqcfa4aaeWaaOqaaKqzGeGaeqySdewc fa4aaWbaaSqabKqaGeaajugWaiabek7aIjabgkHiTiaaigdaaaqcLb sacaWG4bWcdaahaaqcbasabeaajugWaiaaigdacqGHsislcqaHYoGy aaqcLbsacqGHsislcqaH7oaBcqaHXoqymmaaCaaajqwaa+Fabeaaju gOaiabek7aIjabgkHiTiaaigdaaaqcLbsacaWG4bWcdaahaaqcbasa beaajugWaiaaigdacqGHsislcqaHYoGyaaqcLbsacqGHRaWkcqaH7o aBcqaHXoqylmaaCaaajeaibeqaaKqzadGaaGOmaSWaaeWaaKqaGeaa jugWaiabek7aIjabgkHiTiaaigdaaKqaGiaawIcacaGLPaaaaaqcLb sacaWG4bWcdaahaaqcbasabeaajugWaiaaikdalmaabmaajeaibaqc LbmacaaIXaGaeyOeI0IaeqOSdigajeaicaGLOaGaayzkaaaaaaGcca GLOaGaayzkaaaaaiaawUfacaGLDbaajuaGdaahaaWcbeqcbasaaKqz adGaamOAaiabgkHiTiaadMgacqGHsislcaaIXaaaaaGcbaqcLbsacq qHtoWrjuaGdaqadaGcbaqcLbsacaWGQbGaeyOeI0IaamyAaaGccaGL OaGaayzkaaaaaaqaaKqzGeGaey41aqBcfa4aaeWaaOqaaKqzGeGaeq OSdiMaeyOeI0IaaGymaaGccaGLOaGaayzkaaqcfa4aaeWaaOqaaKqz GeGaeqySdewcfa4aaWbaaSqabKqaGeaajugWaiabek7aIjabgkHiTi aaigdaaaqcLbsacaWG5bqcfa4aaWbaaSqabKqaGeaajugWaiabgkHi Tiabek7aIbaaaOGaayjkaiaawMcaaKqbaoaadmaakeaajugibiaaig dacqGHsislcqaH7oaBcqGHRaWkcaaIYaGaeq4UdWMaeqySdewcfa4a aWbaaSqabKazba4=baqcLbkacqaHYoGycqGHsislcaaIXaaaaKqzGe GaamyEaKqbaoaaCaaaleqajeaibaqcLbmacaaIXaGaeyOeI0IaeqOS digaaaGccaGLBbGaayzxaaaaaaa@811D@ (41)

where r(x) is the hazard rate function. The conditional pdf of X U ( j ) X U ( i ) = x i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaalaaabaqcLb sacaWGybWcdaWgaaqcbasaaKqzadGaamyvaSWaaSbaaKGaGeaalmaa bmaajiaibaqcLbmacaWGQbaajiaicaGLOaGaayzkaaaabeaaaKqaGe qaaaGcbaqcLbsacaWGybGcdaWgaaqcbasaaKqzadGaamyvaSWaaSba aKGaGeaalmaabmaajiaibaqcLbmacaWGPbaajiaicaGLOaGaayzkaa aabeaaaSqabaaaaKqzGeGaeyypa0JaamiEaSWaaSbaaKqaGeaajugW aiaadMgaaKqaGeqaaaaa@4B96@ is

f( X U ( j ) = y i X U ( i ) = x i )= [ R( y )R( x ) ] ji1 ( ji1 )! f( y ) 1F( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamOzaK qbaoaabmaakeaajuaGdaWcaaGcbaqcLbsacaWGybWcdaWgaaqcbasa aKqzadGaamyvaSWaaSbaaKGaGeaalmaabmaajiaibaqcLbmacaWGQb aajiaicaGLOaGaayzkaaaabeaaaKqaGeqaaKqzGeGaeyypa0JaamyE aSWaaSbaaKqaGeaajugWaiaadMgaaKqaGeqaaaGcbaqcLbsacaWGyb qcfa4aaSbaaKqaGeaajugWaiaadwfalmaaBaaajiaibaWcdaqadaqc casaaKqzadGaamyAaaqccaIaayjkaiaawMcaaaqabaaaleqaaKqzGe Gaeyypa0JaamiEaSWaaSbaaKqaGeaajugWaiaadMgaaKqaGeqaaaaa aOGaayjkaiaawMcaaKqzGeGaeyypa0tcfa4aaSaaaOqaaKqbaoaadm aakeaajugibiaadkfajuaGdaqadaGcbaqcLbsacaWG5baakiaawIca caGLPaaajugibiabgkHiTiaadkfajuaGdaqadaGcbaqcLbsacaWG4b aakiaawIcacaGLPaaaaiaawUfacaGLDbaajuaGdaahaaWcbeqcbasa aKqzadGaamOAaiabgkHiTiaadMgacqGHsislcaaIXaaaaaGcbaqcfa 4aaeWaaOqaaKqzGeGaamOAaiabgkHiTiaadMgacqGHsislcaaIXaaa kiaawIcacaGLPaaajugibiaacgcaaaqcfa4aaSaaaOqaaKqzGeGaam OzaKqbaoaabmaakeaajugibiaadMhaaOGaayjkaiaawMcaaaqaaKqz GeGaaGymaiabgkHiTiaadAeajuaGdaqadaGcbaqcLbsacaWG4baaki aawIcacaGLPaaaaaaaaa@8117@

= [ ln( α β1 y 1β λ α β1 y 1β +λ α 2( β1 ) y 2( 1β ) ) +ln( α β1 x 1β λ α β1 x 1β +λ α 2( β1 ) x 2( 1β ) ) ] ji1 ( ji1 )! × ( β1 )( α β1 y β )[ 1λ+2λ α β1 y 1β ] α β1 x 1β ( 1λ+λ α β1 x 1β ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaqcLbsacq GH9aqpjuaGdaWcaaGcbaqcfa4aamWaaKqzGeabaeqakeaajugibiab gkHiTiGacYgacaGGUbqcfa4aaeWaaOqaaKqzGeGaeqySdewcfa4aaW baaSqabKqaGeaajugWaiabek7aIjabgkHiTiaaigdaaaqcLbsacaWG 5bWcdaahaaqcbasabeaajugWaiaaigdacqGHsislcqaHYoGyaaqcLb sacqGHsislcqaH7oaBcqaHXoqylmaaCaaajeaibeqaaKqzadGaeqOS diMaeyOeI0IaaGymaaaajugibiaadMhalmaaCaaajeaibeqaaKqzad GaaGymaiabgkHiTiabek7aIbaajugibiabgUcaRiabeU7aSjabeg7a HLqbaoaaCaaaleqajeaibaqcLbmacaaIYaWcdaqadaqcbasaaKqzad GaeqOSdiMaeyOeI0IaaGymaaqcbaIaayjkaiaawMcaaaaajugibiaa dMhalmaaCaaajeaibeqaaKqzadGaaGOmaSWaaeWaaKqaGeaajugWai aaigdacqGHsislcqaHYoGyaKqaGiaawIcacaGLPaaaaaaakiaawIca caGLPaaaaeaajugibiabgUcaRiGacYgacaGGUbqcfa4aaeWaaOqaaK qzGeGaeqySde2cdaahaaqcbasabeaajugWaiabek7aIjabgkHiTiaa igdaaaqcLbsacaWG4bWcdaahaaqcbasabeaajugWaiaaigdacqGHsi slcqaHYoGyaaqcLbsacqGHsislcqaH7oaBcqaHXoqylmaaCaaajeai beqaaKqzadGaeqOSdiMaeyOeI0IaaGymaaaajugibiaadIhajuaGda ahaaWcbeqcbasaaKqzadGaaGymaiabgkHiTiabek7aIbaajugibiab gUcaRiabeU7aSjabeg7aHTWaaWbaaKqaGeqabaqcLbmacaaIYaWcda qadaqcbasaaKqzadGaeqOSdiMaeyOeI0IaaGymaaqcbaIaayjkaiaa wMcaaaaajugibiaadIhalmaaCaaajeaibeqaaKqzadGaaGOmaSWaae WaaKqaGeaajugWaiaaigdacqGHsislcqaHYoGyaKqaGiaawIcacaGL PaaaaaaakiaawIcacaGLPaaaaaGaay5waiaaw2faaSWaaWbaaKqaGe qabaqcLbmacaWGQbGaeyOeI0IaamyAaiabgkHiTiaaigdaaaaakeaa juaGdaqadaGcbaqcLbsacaWGQbGaeyOeI0IaamyAaiabgkHiTiaaig daaOGaayjkaiaawMcaaKqzGeGaaiyiaaaaaOqaaKqzGeGaey41aqBc fa4aaSaaaOqaaKqbaoaabmaakeaajugibiabek7aIjabgkHiTiaaig daaOGaayjkaiaawMcaaKqbaoaabmaakeaajugibiabeg7aHLqbaoaa CaaaleqajeaibaqcLbmacqaHYoGycqGHsislcaaIXaaaaKqzGeGaam yEaSWaaWbaaKqaGeqabaqcLbmacqGHsislcqaHYoGyaaaakiaawIca caGLPaaajuaGdaWadaGcbaqcLbsacaaIXaGaeyOeI0Iaeq4UdWMaey 4kaSIaaGOmaiabeU7aSjabeg7aHTWaaWbaaKqaGeqabaqcLbmacqaH YoGycqGHsislcaaIXaaaaKqzGeGaamyEaSWaaWbaaKqaGeqabaqcLb macaaIXaGaeyOeI0IaeqOSdigaaaGccaGLBbGaayzxaaaabaqcLbsa cqaHXoqylmaaCaaajeaibeqaaKqzadGaeqOSdiMaeyOeI0IaaGymaa aajugibiaadIhalmaaCaaajeaibeqaaKqzadGaaGymaiabgkHiTiab ek7aIbaajuaGdaqadaGcbaqcLbsacaaIXaGaeyOeI0Iaeq4UdWMaey 4kaSIaeq4UdWMaeqySde2cdaahaaqcbasabeaajugWaiabek7aIjab gkHiTiaaigdaaaqcLbsacaWG4bWcdaahaaqcbasabeaajugWaiaaig dacqGHsislcqaHYoGyaaaakiaawIcacaGLPaaaaaaaaaa@1234@  (42)

For j=i+1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamOAai abg2da9iaadMgacqGHRaWkcaaIXaaaaa@3ADA@

f( y i+1 X U( i ) = x i )= f( y i+1 ) 1F( x i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamOzaK qbaoaabmaakeaajuaGdaWcaaGcbaqcLbsacaWG5bqcfa4aaSbaaKqa GeaajugWaiaadMgacqGHRaWkcaaIXaaaleqaaaGcbaqcLbsacaWGyb WcdaWgaaqcbasaaKqzadGaamyvaSWaaeWaaKqaGeaajugWaiaadMga aKqaGiaawIcacaGLPaaaaeqaaaaajugibiabg2da9iaadIhalmaaBa aajeaibaqcLbmacaWGPbaajeaibeaaaOGaayjkaiaawMcaaKqzGeGa eyypa0tcfa4aaSaaaOqaaKqzGeGaamOzaKqbaoaabmaakeaajugibi aadMhajuaGdaWgaaqcbasaaKqzadGaamyAaiabgUcaRiaaigdaaSqa baaakiaawIcacaGLPaaaaeaajugibiaaigdacqGHsislcaWGgbqcfa 4aaeWaaOqaaKqzGeGaamiEaSWaaSbaaKqaGeaajugWaiaadMgaaKqa GeqaaaGccaGLOaGaayzkaaaaaaaa@62C9@

= ( β1 )( α β1 y i+1 β )( 1λ+2λ α β1 y i+1 1β ) α β1 x i 1β ( 1λ+λ α β1 x i 1β ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeyypa0 tcfa4aaSaaaOqaaKqbaoaabmaakeaajugibiabek7aIjabgkHiTiaa igdaaOGaayjkaiaawMcaaKqbaoaabmaakeaajugibiabeg7aHTWaaW baaKqaGeqabaqcLbmacqaHYoGycqGHsislcaaIXaaaaKqzGeGaamyE aSWaa0baaKqaGeaajugWaiaadMgacqGHRaWkcaaIXaaajeaibaqcLb macqGHsislcqaHYoGyaaaakiaawIcacaGLPaaajuaGdaqadaGcbaqc LbsacaaIXaGaeyOeI0Iaeq4UdWMaey4kaSIaaGOmaiabeU7aSjabeg 7aHTWaaWbaaKqaGeqabaqcLbmacqaHYoGycqGHsislcaaIXaaaaKqz GeGaamyEaSWaa0baaKqaGeaajugWaiaadMgacqGHRaWkcaaIXaaaje aibaqcLbmacaaIXaGaeyOeI0IaeqOSdigaaaGccaGLOaGaayzkaaaa baqcLbsacqaHXoqylmaaCaaajeaibeqaaKqzadGaeqOSdiMaeyOeI0 IaaGymaaaajugibiaadIhalmaaDaaajeaibaWcdaahaaqccasabeaa jugWaiaadMgaaaaajeaibaWcdaahaaqccasabeaajugWaiaaigdacq GHsislcqaHYoGyaaaaaKqbaoaabmaakeaajugibiaaigdacqGHsisl cqaH7oaBcqGHRaWkcqaH7oaBcqaHXoqyjuaGdaahaaWcbeqcbasaaK qzadGaeqOSdiMaeyOeI0IaaGymaaaajugibiaadIhalmaaDaaajeai baWcdaahaaqccasabeaajugWaiaadMgaaaaajeaibaWcdaahaaqcca sabeaajugWaiaaigdacqGHsislcqaHYoGyaaaaaaGccaGLOaGaayzk aaaaaaaa@942B@ (43)

Application

Two real life examples are used to get results for the TWP distribution. The data of remission times, in months, of people with Bladder cancer as recorded by Lee & Wang13 is used for this application. The data is given in Table 4. Since weighted distributions can be length-biased and area-biased, comparison is conducted for transmuted versions of both of these types of weighted distributions. Transmuted length-biased Pareto (TLbP) is the one studied throughout the length of this study (referred to as TWP) and transmuted area biased Pareto (TAbP) is derived using k=2 in equation 1 and then transmuted in the same manner as TWP. Henceforth, the parameters are evaluated for TWP, TAbP, Transmuted Pareto (TP), Weighted Pareto (WP) and Pareto (P) distributions. The results for the estimates are given in (Table 5).

Remission times of Bladder Cancer patients

0.08

2.09

3.48

4.87

6.94

8.66

13.11

23.63

0.2

2.23

3.52

4.98

6.97

9.02

13.29

0.4

2.26

3.57

5.06

7.09

9.22

13.8

25.74

0.5

2.46

3.64

5.09

7.26

9.47

14.24

25.82

0.51

2.54

3.7

5.17

7.28

9.74

14.76

26.31

0.81

2.62

3.82

5.32

7.32

10.06

14.77

32.15

2.64

3.88

5.32

7.39

10.34

14.83

34.26

0.9

2.69

4.18

5.34

7.59

10.66

15.96

36.66

1.05

2.69

4.23

5.41

7.62

10.75

16.62

43.01

1.19

2.75

4.26

5.41

7.63

17.12

46.12

1.26

2.83

4.33

7.66

11.25

17.14

79.05

1.35

2.87

5.62

7.87

11.64

17.36

1.4

3.02

4.34

5.71

7.93

11.79

18.1

1.46

4.4

5.85

8.26

11.98

19.13

1.76

3.25

4.5

6.25

8.37

12.02

2.02

3.31

4.51

6.54

8.53

12.03

20.28

2.02

3.36

6.76

12.07

21.73

2.07

3.36

6.93

8.65

12.63

22.69

5.49

Table 4 Data of remission of Bladder Cancer patients as recorded by Lee & Wang13

Model

Estimates

-2log lik

AIC

AICC

BIC

TWP

β ^ =1.3366013 λ ^ =0.9754199 α ^ =min(x)=0.08 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaqcLbsacu aHYoGygaqcaiabg2da9iaaigdacaGGUaGaaG4maiaaiodacaaI2aGa aGOnaiaaicdacaaIXaGaaG4maaGcbaqcLbsacuaH7oaBgaqcaiabg2 da9iabgkHiTiaaicdacaGGUaGaaGyoaiaaiEdacaaI1aGaaGinaiaa igdacaaI5aGaaGyoaaGcbaqcLbsacuaHXoqygaqcaiabg2da9iGac2 gacaGGPbGaaiOBaiaacIcacaWG4bGaaiykaiabg2da9iaaicdacaGG UaGaaGimaiaaiIdaaaaa@5714@

1005.038

1011.038

1011.231

1014.742

TAbP

β ^ =2.336608 λ ^ =0.9754183 α ^ =min(x)=0.08 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaqcLbsacu aHYoGygaqcaiabg2da9iaaikdacaGGUaGaaG4maiaaiodacaaI2aGa aGOnaiaaicdacaaI4aaakeaajugibiqbeU7aSzaajaGaeyypa0Jaey OeI0IaaGimaiaac6cacaaI5aGaaG4naiaaiwdacaaI0aGaaGymaiaa iIdacaaIZaaakeaajugibiqbeg7aHzaajaGaeyypa0JaciyBaiaacM gacaGGUbGaaiikaiaadIhacaGGPaGaeyypa0JaaGimaiaac6cacaaI WaGaaGioaaaaaa@5658@

1005.038

1011.038

1011.231

1014.742

TP

β ^ =0.3365833 λ ^ =0.9754229 α ^ =min(x)=0.08 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaqcLbsacu aHYoGygaqcaiabg2da9iaaicdacaGGUaGaaG4maiaaiodacaaI2aGa aGynaiaaiIdacaaIZaGaaG4maaGcbaqcLbsacuaH7oaBgaqcaiabg2 da9iabgkHiTiaaicdacaGGUaGaaGyoaiaaiEdacaaI1aGaaGinaiaa ikdacaaIYaGaaGyoaaGcbaqcLbsacuaHXoqygaqcaiabg2da9iGac2 gacaGGPbGaaiOBaiaacIcacaWG4bGaaiykaiabg2da9iaaicdacaGG UaGaaGimaiaaiIdaaaaa@5716@

1005.038

1011.038

1011.231

1014.742

WP

β ^ =1.23369 α ^ =min(x)=0.08 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaqcLbsacu aHYoGygaqcaiabg2da9iaaigdacaGGUaGaaGOmaiaaiodacaaIZaGa aGOnaiaaiMdaaOqaaKqzGeGafqySdeMbaKaacqGH9aqpciGGTbGaai yAaiaac6gacaGGOaGaamiEaiaacMcacqGH9aqpcaaIWaGaaiOlaiaa icdacaaI4aaaaaa@4AA2@

1077.046

1081.046

1081.142

1081.898

P

β ^ =0.23369 α ^ =min(x)=0.08 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaqcLbsacu aHYoGygaqcaiabg2da9iaaicdacaGGUaGaaGOmaiaaiodacaaIZaGa aGOnaiaaiMdaaOqaaKqzGeGafqySdeMbaKaacqGH9aqpciGGTbGaai yAaiaac6gacaGGOaGaamiEaiaacMcacqGH9aqpcaaIWaGaaiOlaiaa icdacaaI4aaaaaa@4AA1@

1077.046

1081.046

1081.142

1081.898

Table 5 Estimated value of parameters for different distributions

Table 5 shows a difference in estimates of all the distributions but -2 log likelihood, AIC, AICc and BIC are the same i.e., 1005.038, 1011.038, 1011.231 and 1014.742 respectively for TWP, TAbP and TP distributions. Also for WP and P distributions the -2 log liklihood, AIC, AICc and BIC are same i.e., 1077.046, 1081.046, 1081.142 and 1081.898 respectively. This may mean that TWP, TAbP and TP distributions respond to data in a similar way and WP and P distributions respond in a similar manner with TWP, TabP and TP being better than the others because of the lower -2 log likelihood, AIC, AICc and BIC values. This behavior, thus, exhibits the stability between family of Pareto models. It should also be borne in mind that these values are correct to three decimal places and may show differences if higher points are taken into consideration.

The TWP distribution takes the surety of occurrence into account by incorporating the weighted aspect into it and is further transmuted by adding a variable to make it more flexible. TWP, thus, presents a comprehensive model that accounts for a transmuted version of a weighted distribution.

The variance-covariance matrix of the MLE of the TWP distribution is as following. Variances of MLE of β and λ.

Var ( β ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaacI caiiaacuWFYoGygaWeaiaacMcaaaa@3C18@ = 0.0004906221 and Var ( λ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaacI cacuaH7oaBgaWeaiaacMcaaaa@3C26@ = 0.0005991575.

| 0.0004906221 -0.0000358219 -0.0000358219 0.0005991575 | MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaaemaabaqcLb safaqabeGacaaakeaajugibiaabcdacaqGUaGaaeimaiaabcdacaqG WaGaaeinaiaabMdacaqGWaGaaeOnaiaabkdacaqGYaGaaeymaaGcba qcLbsacaqGTaGaaeimaiaab6cacaqGWaGaaeimaiaabcdacaqGWaGa ae4maiaabwdacaqG4aGaaeOmaiaabgdacaqG5aaakeaajugibiaab2 cacaqGWaGaaeOlaiaabcdacaqGWaGaaeimaiaabcdacaqGZaGaaeyn aiaabIdacaqGYaGaaeymaiaabMdaaOqaaKqzGeGaaeimaiaab6caca qGWaGaaeimaiaabcdacaqG1aGaaeyoaiaabMdacaqGXaGaaeynaiaa bEdacaqG1aaaaaGccaGLhWUaayjcSdaaaa@5F5B@

The confidence intervals are

| 2.5% 97.5% β 1.2951272 1.3820321 λ -0.9981309 -0.8927093 | MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaaemaabaqcLb safaqabeWadaaakeaaaeaajugibiaaikdacaGGUaGaaGynaiaacwca aOqaaKqzGeGaaGyoaiaaiEdacaGGUaGaaGynaiaacwcaaOqaaKqzGe GaeqOSdigakeaajugibiaabgdacaqGUaGaaeOmaiaabMdacaqG1aGa aeymaiaabkdacaqG3aGaaeOmaaGcbaqcLbsacaqGXaGaaeOlaiaabo dacaqG4aGaaeOmaiaabcdacaqGZaGaaeOmaiaabgdaaOqaaKqzGeGa eq4UdWgakeaajugibiaab2cacaqGWaGaaeOlaiaabMdacaqG5aGaae ioaiaabgdacaqGZaGaaeimaiaabMdaaOqaaKqzGeGaaeylaiaabcda caqGUaGaaeioaiaabMdacaqGYaGaae4naiaabcdacaqG5aGaae4maa aaaOGaay5bSlaawIa7aaaa@6330@

The second data used is that of the survival times of patients who got better after chemotherapy treatment as reported by Bekker et al.14 The data is given below Table 6, the data is used to get estimated parameters for different Pareto distributions.

Survival times after chemotherapy treatment

0.047

0.115

0.121

0.132

0.164

0.197

0.203

0.26

0.282

0.296

0.334

0.395

0.458

0.466

0.501

0.507

0.529

0.534

0.54

0.641

0.644

0.696

0.841

0.863

1.099

1.219

1.271

1.326

1.447

1.485

1.553

1.581

1.589

2.178

2.343

2.416

2.444

2.825

2.83

3.578

3.658

3.743

3.978

4.003

4.033

Table 6 Data of survival times (years) from chemotherapy treatment

Table 7 shows that TWP, TabP and TP have lower -2 log likelihood, AIC, AICc and BIC values as compared to WP and P distribution which means they are better amongst others. The similarity in results is indicated in this application also which again draws attention to the stability between Pareto models.

Model

Estimates

-2log lik

AIC

AICC

BIC

TWP

β ^ =1.5024309 λ ^ =0.9209614 α ^ =min(x)=0.047 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaqcLbsacu aHYoGygaqcaiabg2da9iaaigdacaGGUaGaaGynaiaaicdacaaIYaGa aGinaiaaiodacaaIWaGaaGyoaaGcbaqcLbsacuaH7oaBgaqcaiabg2 da9iabgkHiTiaaicdacaGGUaGaaGyoaiaaikdacaaIWaGaaGyoaiaa iAdacaaIXaGaaGinaaGcbaqcLbsacuaHXoqygaqcaiabg2da9iGac2 gacaGGPbGaaiOBaiaacIcacaWG4bGaaiykaiabg2da9iaaicdacaGG UaGaaGimaiaaisdacaaI3aaaaaa@57C5@

145.484

151.484

152.069

153.097

TAbP

β ^ =2.5024359 λ ^ =0.9209386 α ^ =min(x)=0.047 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaqcLbsacu aHYoGygaqcaiabg2da9iaaikdacaGGUaGaaGynaiaaicdacaaIYaGa aGinaiaaiodacaaI1aGaaGyoaaGcbaqcLbsacuaH7oaBgaqcaiabg2 da9iabgkHiTiaaicdacaGGUaGaaGyoaiaaikdacaaIWaGaaGyoaiaa iodacaaI4aGaaGOnaaGcbaqcLbsacuaHXoqygaqcaiabg2da9iGac2 gacaGGPbGaaiOBaiaacIcacaWG4bGaaiykaiabg2da9iaaicdacaGG UaGaaGimaiaaisdacaaI3aaaaaa@57D1@

145.484

151.484

152.069

153.097

TP

β ^ =0.5024354 λ ^ =0.9208684 α ^ =min(x)=0.047 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaqcLbsacu aHYoGygaqcaiabg2da9iaaicdacaGGUaGaaGynaiaaicdacaaIYaGa aGinaiaaiodacaaI1aGaaGinaaGcbaqcLbsacuaH7oaBgaqcaiabg2 da9iabgkHiTiaaicdacaGGUaGaaGyoaiaaikdacaaIWaGaaGioaiaa iAdacaaI4aGaaGinaaGcbaqcLbsacuaHXoqygaqcaiabg2da9iGac2 gacaGGPbGaaiOBaiaacIcacaWG4bGaaiykaiabg2da9iaaicdacaGG UaGaaGimaiaaisdacaaI3aaaaaa@57CA@

145.484

151.484

152.069

153.097

WP

β ^ =1.353038 α ^ =min(x)=0.047 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaqcLbsacu aHYoGygaqcaiabg2da9iaaigdacaGGUaGaaG4maiaaiwdacaaIZaGa aGimaiaaiodacaaI4aaakeaajugibiqbeg7aHzaajaGaeyypa0Jaci yBaiaacMgacaGGUbGaaiikaiaadIhacaGGPaGaeyypa0JaaGimaiaa c6cacaaIWaGaaGinaiaaiEdaaaaa@4C18@

163.461

169.461

170.046

167.268

P

β ^ =0.3530415 α ^ =min(x)=0.047 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaqcLbsacu aHYoGygaqcaiabg2da9iaaicdacaGGUaGaaG4maiaaiwdacaaIZaGa aGimaiaaisdacaaIXaGaaGynaaGcbaqcLbsacuaHXoqygaqcaiabg2 da9iGac2gacaGGPbGaaiOBaiaacIcacaWG4bGaaiykaiabg2da9iaa icdacaGGUaGaaGimaiaaisdacaaI3aaaaaa@4CD0@

163.461

169.461

170.046

167.268

Table 7 Estimated value of parameters for different distributions

The variance-covariance matrix of the MLE of the TWP distribution is as following. Variances of MLE of β and λ.

Var ( β ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaacI caiiaacuWFYoGygaWeaiaacMcaaaa@3C18@ = 0.0032758564 and Var ( λ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaacI cacuaH7oaBgaWeaiaacMcaaaa@3C26@ = 0.0061652295.

| 0.0032758564 -0.0006416901 -0.0006416901 0.0061652295 | MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaaemaabaqcLb safaqabeGacaaakeaajugibiaabcdacaqGUaGaaeimaiaabcdacaqG ZaGaaeOmaiaabEdacaqG1aGaaeioaiaabwdacaqG2aGaaeinaaGcba qcLbsacaqGTaGaaeimaiaab6cacaqGWaGaaeimaiaabcdacaqG2aGa aeinaiaabgdacaqG2aGaaeyoaiaabcdacaqGXaaakeaajugibiaab2 cacaqGWaGaaeOlaiaabcdacaqGWaGaaeimaiaabAdacaqG0aGaaeym aiaabAdacaqG5aGaaeimaiaabgdaaOqaaKqzGeGaaeimaiaab6caca qGWaGaaeimaiaabAdacaqGXaGaaeOnaiaabwdacaqGYaGaaeOmaiaa bMdacaqG1aaaaaGccaGLhWUaayjcSdaaaa@5F64@

The confidence intervals are

| 2.5% 97.5% β 1.3980604 1.6232607 λ -0.9940705 -0.6582288 | MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaaemaabaqcLb safaqabeWadaaakeaaaeaajugibiaaikdacaGGUaGaaGynaiaacwca aOqaaKqzGeGaaGyoaiaaiEdacaGGUaGaaGynaiaacwcaaOqaaKqzGe GaeqOSdigakeaajugibiaabgdacaqGUaGaae4maiaabMdacaqG4aGa aeimaiaabAdacaqGWaGaaeinaaGcbaqcLbsacaqGXaGaaeOlaiaabA dacaqGYaGaae4maiaabkdacaqG2aGaaeimaiaabEdaaOqaaKqzGeGa eq4UdWgakeaajugibiaab2cacaqGWaGaaeOlaiaabMdacaqG5aGaae inaiaabcdacaqG3aGaaeimaiaabwdaaOqaaKqzGeGaaeylaiaabcda caqGUaGaaeOnaiaabwdacaqG4aGaaeOmaiaabkdacaqG4aGaaeioaa aaaOGaay5bSlaawIa7aaaa@6335@

Conclusion

Weighted Pareto distribution is used in this study to derive a new distribution. The parent distribution is extended using a QRT map in order to transmute it. The resulting transmuted weighted Pareto distribution is then studied. Statistical properties of the distribution are elaborated. Moments, quantiles, MGF, reliability measures, order statistics and record values are derived. Two applications are used to study the parameters of TWP distribution. The comparison of TWP distribution with TAbP, TP, WP and P distributions reveals it to be a better model than Pareto and Weighted Pareto distributions but shows stability when compared with transmuted area-biased Pareto and transmuted Pareto distributions. It can be concluded that since TWP deals with the weighted Pareto distribution, TWP can be considered theoretically more advanced than others in the family of Pareto models.

Acknowledgement

None.

Conflict of interest

Authors declare that there is no conflict of interest.

References

  1. Fisher RA. The Effects of Methods of Ascertainment upon the Estimation of Frequencies. Ann. Eugenics. 1934;(6):13–25.
  2. Zelen M. Problems in Cell Kinetics and the Early Detection of Disease, in Reliability and Biometry. SIAM Philadelphia. 1974;6(5):701–706.
  3. Shaw W, Buckley I. The Alchemy of Probability Distributions: Beyond Gram–Charlier Expansions and a Skew–Kurtotic– Normal Distribution. Research Report; 2007.
  4. Dar AA, Ahmed A, Reshi JA. Transmuted Weighted Exponential Distribution and Its Application. Journal of Statistics Applications & Probability. 2017;6(1):219–232.
  5. Shahzad MN, Merovci F, Asghar Z. Transmuted Singh–Maddala Distribution: A New Flexible and Upside–Down Bathtub Shaped Hazard Function Distribution. Revista Colombiana de Estadísti. 2017;40(1):1–27.
  6. Nassar MM, Radwan SS, Elmasry A. Transmuted Weibull Logistic Distribution. International Journal of Innovative Research & Development. 2017;6(4):122–131.
  7. Ashour SK, Eltehiwy MA. Transmuted Lomax Distribution. American Journal of Applied Mathematics and Statistics. 2013;1(6):121−127.
  8. Aryal GR, Tsokos CP. Transmuted Weibull distribution: A Generalization of the Weibull Probability Distribution. European Journal of Pure and Applied Mathematics. 2011;4(2):89−102.
  9. Aryal GR. Transmuted Log–Logistic Distribution. Journal of Statistics Applications & Probability. 2013;2(1):11−20.
  10. Khan MS, King R, Hudson IL. Transmuted Kumaraswamy Distribution. Statistics in Trnsition. 2016;17(2):183–210.
  11. Mir KA, Ahmad M. Size–Biased Distributions and Their Applications. Pakistan Journal of Statistic. 2009;25:283–294.
  12. Bashir S, Ahmad M. Record Values from Size–Biased Pareto Distribution and a Characterization. International Journal of Engineering Research and General Science. 2004;2(4):101–109.
  13. Lee ET, Wang JW. Statistical Methods for Survival Data Analysis. 3rd ed. New York: Wiley; 2003.
  14. Bekker A, Roux J, Mostert P. A generalization of the compound Rayleigh distribution: using a Bayesian methods on cancer survival times. Commun Stat Theory Methods. 2007;29(7):1419–1433.
©2018 Zahid et al. This is an open access article distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and build upon your work non-commercially.
© 2019 MedCrave Publishing, All rights reserved. No part of this content may be reproduced or transmitted in any form or by any means as per the standard guidelines of fair use.
Creative Commons License Open Access by MedCrave Publishing is licensed under a Creative Commons Attribution 4.0 International License.
Based on a work at https://medcraveonline.com
Best viewed in Mozilla Firefox | Google Chrome | Above IE 7.0 version | Opera |Privacy Policy