ISSN: 2378-315X BBIJ

Biometrics & Biostatistics International Journal
Research Article
Volume 5 Issue 2 - 2017
Ishita Distribution and Its Applications
Rama Shanker* and Kamlesh Kumar Shukla
Department of Statistics, Eritrea Institute of Technology, Eritrea
Received: February 01, 2017 | Published: February 13, 2017
*Corresponding author: Rama Shanker, Department of Statistics, Eritrea Institute of Technology, Eritrea, Email:
Citation: Shanker R, Shukla KK (2017) Ishita Distribution and Its Applications. Biom Biostat Int J 5(2): 00126. DOI: 10.15406/bbij.2017.05.00126

Abstract

In the present paper, a lifetime distribution named, “Ishita distribution” for modeling lifetime data from biomedical science and engineering has been proposed. Statistical properties of the distribution including its shape, moments, skewness, kurtosis, hazard rate function, mean residual life function, stochastic ordering, mean deviations, order statistics, Bonferroni and Lorenz curves, Renyi entropy measure, stress-strength reliability have been discussed. The condition under which Ishita distribution is over-dispersed, equi-dispersed, and under-dispersed are presented along with the conditions under which Akash distribution, introduced by Shanker [1], Lindley distribution, introduced by Lindley [2] and exponential distribution are over-dispersed, equi-dispersed and under-dispersed. Method of maximum likelihood estimation and method of moments have been discussed for estimating the parameter of the proposed distribution. Finally, the goodness of fit of the proposed distribution have been discussed and illustrated with two real lifetime data sets and the fit has been compared with exponential, Lindley and Akash distributions.

Keywords: Akash distribution; Lindley distribution; Moments; Dispersion; Hazard rate function; Mean residual life function; Mean deviations; Order statistics; Stress-strength reliability; Estimation of parameter; Goodness of fit

Introduction

The analyzing and modeling real lifetime data are crucial in many applied sciences including medicine, engineering, insurance and finance, amongst others. The two important one parameter lifetime distributions namely exponential and Lindley [2] are popular for modeling lifetime data from biomedical science and engineering. Recently, Shanker et al. [3] have conducted a comparative and critical study on the modeling of lifetime data from biomedical science and engineering using exponential and Lindley distributions and observed that there are many lifetime data where these two distributions are not suitable due to their shapes, nature of hazard rate functions, and mean residual life, amongst others. While searching a lifetime distribution which gives better fit than exponential and Lindley, Shanker [1] has introduced a lifetime distribution named Akash distribution and showed that Akash distribution gives much better fit than both exponential and Lindley distributions. Shanker et al. [4] have comparative study on the modeling of lifetime data using Akash, Lindley and exponential distribution and observed that there are several situations where these lifetime distributions are not suitable either from theoretical or applied point of view. Therefore, an attempt has been made in this paper to obtain a new lifetime distribution which is flexible than Akash, Lindley and exponential distributions for modeling lifetime data in reliability and in terms of its hazard rate shapes. The new one parameter lifetime distribution is based on a two- component mixture of an exponential distribution having scale parameter θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH4o qCaaa@383B@ and a gamma distribution having shape parameter 3 and scale parameter θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH4o qCaaa@383B@  with their mixing proportion θ 3 θ 3 +2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaO qaaKqzGeGaeqiUde3cdaahaaqabeaajugWaiaaiodaaaaakeaajugi biabeI7aXLqbaoaaCaaaleqabaqcLbmacaaIZaaaaKqzGeGaey4kaS IaaGOmaaaaaaa@421D@ .

Lindley distribution, introduced by Lindley [2] has been defined by the probability density function (p.d.f.) and the cumulative distribution function (c.d.f.) as

f 1 ( x;θ )= θ 2 θ+1 ( 1+x ) e θx ;x>0,θ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGMb qcfa4aaSbaaSqaaKqzGeGaaGymaaWcbeaajuaGdaqadaGcbaqcLbsa caWG4bGaai4oaiabeI7aXbGccaGLOaGaayzkaaqcLbsacqGH9aqpju aGdaWcaaGcbaqcLbsacqaH4oqCjuaGdaahaaWcbeqaaKqzadGaaGOm aaaaaOqaaKqzGeGaeqiUdeNaey4kaSIaaGymaaaajuaGdaqadaGcba qcLbsacaaIXaGaey4kaSIaamiEaaGccaGLOaGaayzkaaqcLbsacaWG Lbqcfa4aaWbaaSqabeaajugWaiabgkHiTiabeI7aXjaadIhaaaqcLb sacaaMc8UaaGPaVlaaykW7caaMc8Uaai4oaiaadIhacqGH+aGpcaaI WaGaaiilaiaaykW7caaMc8UaeqiUdeNaeyOpa4JaaGimaaaa@688B@                                                    (3.1)

F 1 ( x;θ )=1[ 1+ θx θ+1 ] e θx ;x>0,θ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGgb qcfa4aaSbaaSqaaKqzGeGaaGymaaWcbeaajuaGdaqadaGcbaqcLbsa caWG4bGaai4oaiabeI7aXbGccaGLOaGaayzkaaqcLbsacqGH9aqpca aIXaGaeyOeI0scfa4aamWaaOqaaKqzGeGaaGymaiabgUcaRKqbaoaa laaakeaajugibiabeI7aXjaadIhaaOqaaKqzGeGaeqiUdeNaey4kaS IaaGymaaaaaOGaay5waiaaw2faaKqzGeGaamyzaKqbaoaaCaaaleqa baqcLbmacqGHsislcqaH4oqCcaWG4baaaKqzGeGaaGPaVlaaykW7ca GG7aGaamiEaiabg6da+iaaicdacaGGSaGaaGPaVlabeI7aXjabg6da +iaaicdaaaa@6336@                                                     (3.2)

It can be easily verified that the density (3.1) is a two-component mixture of an exponential ( θ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeGaeqiUdehakiaawIcacaGLPaaaaaa@3A66@ distribution and a gamma ( 2,θ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeGaaGOmaiaacYcacqaH4oqCaOGaayjkaiaawMcaaaaa@3BD2@  with their mixing proportion θ θ+1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaO qaaKqzGeGaeqiUdehakeaajugibiabeI7aXjabgUcaRiaaigdaaaaa aa@3CCF@ . Recent years much works have been done on Lindley distribution , its generalization and mixture with other distributions by several authors including Ghitany et al [5], Zakerzadeh and Dolati [6], Mazucheli and Achcar [7], Bakouch et al [8] , Shanker and Mishra [9,10] Shanker et al [11], Shanker and Amanuel [12], Sankaran [13], are some among others.

Akash distribution, introduced by Shanker [1] has been defined by the probability density function (p.d.f.) and the cumulative distribution function (c.d.f.) as

f 2 ( x;θ )= θ 3 θ 2 +2 ( 1+ x 2 ) e θx ;x>0,θ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGMb qcfa4aaSbaaSqaaKqzGeGaaGOmaaWcbeaajuaGdaqadaGcbaqcLbsa caWG4bGaai4oaiabeI7aXbGccaGLOaGaayzkaaqcLbsacqGH9aqpju aGdaWcaaGcbaqcLbsacqaH4oqCjuaGdaahaaWcbeqaaKqzadGaaG4m aaaaaOqaaKqzGeGaeqiUdexcfa4aaWbaaSqabeaajugWaiaaikdaaa qcLbsacqGHRaWkcaaIYaaaaKqbaoaabmaakeaajugibiaaigdacqGH RaWkcaWG4bqcfa4aaWbaaSqabeaajugWaiaaikdaaaaakiaawIcaca GLPaaajugibiaadwgajuaGdaahaaWcbeqaaKqzadGaeyOeI0IaeqiU deNaamiEaaaajugibiaaykW7caaMc8UaaGPaVlaaykW7caGG7aGaam iEaiabg6da+iaaicdacaGGSaGaaGPaVlaaykW7cqaH4oqCcqGH+aGp caaIWaaaaa@6E67@                                                 (3.3)

                                           F 2 ( x,θ )=1[ 1+ θx( θx+2 ) θ 2 +2 ] e θx ;x>0,θ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGgb qcfa4aaSbaaSqaaKqzGeGaaGOmaaWcbeaajuaGdaqadaGcbaqcLbsa caWG4bGaaiilaiabeI7aXbGccaGLOaGaayzkaaqcLbsacqGH9aqpca aIXaGaeyOeI0scfa4aamWaaOqaaKqzGeGaaGymaiabgUcaRKqbaoaa laaakeaajugibiabeI7aXjaadIhajuaGdaqadaGcbaqcLbsacqaH4o qCcaWG4bGaey4kaSIaaGOmaaGccaGLOaGaayzkaaaabaqcLbsacqaH 4oqCjuaGdaahaaWcbeqaaKqzadGaaGOmaaaajugibiabgUcaRiaaik daaaaakiaawUfacaGLDbaajugibiaadwgajuaGdaahaaWcbeqaaKqz adGaeyOeI0IaeqiUdeNaamiEaaaajugibiaaykW7caaMc8Uaai4oai aadIhacqGH+aGpcaaIWaGaaiilaiabeI7aXjabg6da+iaaicdaaaa@6BD3@                                              (3.4)

It can be easily verified that the Akash distribution is a two-component mixture of exponential ( θ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeGaeqiUdehakiaawIcacaGLPaaaaaa@3A66@ distribution and a gamma ( 3,θ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeGaaG4maiaacYcacqaH4oqCaOGaayjkaiaawMcaaaaa@3BD3@ distribution with mixing proportion θ 2 θ 2 +2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaO qaaKqzGeGaeqiUdexcfa4aaWbaaSqabeaajugWaiaaikdaaaaakeaa jugibiabeI7aXLqbaoaaCaaaleqabaqcLbmacaaIYaaaaKqzGeGaey 4kaSIaaGOmaaaaaaa@42A9@ . Shanker [14] has obtained a Poisson mixture of Akash distribution named, “Poisson-Akash distribution (PAD) and discussed its properties, estimation of parameters and applications. Shanker et al. [15] have detailed study on modeling of count data from different fields of knowledge using Poisson-Akash distribution. Shanker and Shukla [16] have obtained weighted Akash distribution and studied its statistical and mathematical properties, estimation of parameters and applications to model lifetime data. Shanker [17] has also obtained a quasi Akash distribution, studied its mathematical and statistical properties, estimation of parameters using both maximum likelihood estimation and method of moments and applications to model lifetime data.

The new one parameter lifetime distribution has been defined by its probability density function (p.d.f.)

f 3 ( x;θ )= θ 3 θ 3 +2 ( θ+ x 2 ) e θx ;x>0,θ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGMb qcfa4aaSbaaSqaaKqzGeGaaG4maaWcbeaajuaGdaqadaGcbaqcLbsa caWG4bGaai4oaiabeI7aXbGccaGLOaGaayzkaaqcLbsacqGH9aqpju aGdaWcaaGcbaqcLbsacqaH4oqCjuaGdaahaaWcbeqaaKqzadGaaG4m aaaaaOqaaKqzGeGaeqiUdexcfa4aaWbaaSqabeaajugWaiaaiodaaa qcLbsacqGHRaWkcaaIYaaaaKqbaoaabmaakeaajugibiabeI7aXjab gUcaRiaadIhajuaGdaahaaWcbeqaaKqzadGaaGOmaaaaaOGaayjkai aawMcaaKqzGeGaamyzaKqbaoaaCaaaleqabaqcLbmacqGHsislcqaH 4oqCcaWG4baaaKqzGeGaaGPaVlaaykW7caaMc8UaaGPaVlaacUdaca WG4bGaeyOpa4JaaGimaiaacYcacaaMc8UaaGPaVlabeI7aXjabg6da +iaaicdaaaa@6F64@                                                    (3.5)

We would name this probability density function as, “Ishita distribution”. It can be easily verified that the Ishita distribution is a two-component mixture of exponential ( θ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeGaeqiUdehakiaawIcacaGLPaaaaaa@3A66@ distribution and a gamma ( 3,θ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeGaaG4maiaacYcacqaH4oqCaOGaayjkaiaawMcaaaaa@3BD3@ distribution with mixing proportion θ 3 θ 3 +2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaO qaaKqzGeGaeqiUdexcfa4aaWbaaSqabeaajugWaiaaiodaaaaakeaa jugibiabeI7aXLqbaoaaCaaaleqabaqcLbmacaaIZaaaaKqzGeGaey 4kaSIaaGOmaaaaaaa@42AB@ .

The corresponding cumulative distribution function (c.d.f.) of (1.5) can be obtained as

                                        F 3 ( x,θ )=1[ 1+ θx( θx+2 ) θ 3 +2 ] e θx ;x>0,θ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGgb qcfa4aaSbaaSqaaKqzGeGaaG4maaWcbeaajuaGdaqadaGcbaqcLbsa caWG4bGaaiilaiabeI7aXbGccaGLOaGaayzkaaqcLbsacqGH9aqpca aIXaGaeyOeI0scfa4aamWaaOqaaKqzGeGaaGymaiabgUcaRKqbaoaa laaakeaajugibiabeI7aXjaadIhajuaGdaqadaGcbaqcLbsacqaH4o qCcaWG4bGaey4kaSIaaGOmaaGccaGLOaGaayzkaaaabaqcLbsacqaH 4oqCjuaGdaahaaWcbeqaaKqzadGaaG4maaaajugibiabgUcaRiaaik daaaaakiaawUfacaGLDbaajugibiaadwgajuaGdaahaaWcbeqaaKqz adGaeyOeI0IaeqiUdeNaamiEaaaajugibiaaykW7caaMc8Uaai4oai aadIhacqGH+aGpcaaIWaGaaiilaiabeI7aXjabg6da+iaaicdaaaa@6BD5@                                                 (3.6)

The graph of the p.d.f. and the c.d.f. of Ishita distribution for varying values of the parameter θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH4o qCaaa@383B@ are shown in figures 1 and 2.

Figure 1: Graph of the pdf of Ishita distribution for varying values of the parameter θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH4o qCaaa@383B@ .

Figure 2: Graph of the cdf of Ishita distribution for varying values of the parameter θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH4o qCaaa@383B@ .

Statistical Constants

The moment generating function of Ishita distribution (3.5) can be obtained as

M X ( t )= θ 3 θ 3 +2 0 e ( θt )x ( θ+ x 2 ) dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGnb qcfa4aaSbaaSqaaKqzGeGaamiwaaWcbeaajuaGdaqadaGcbaqcLbsa caWG0baakiaawIcacaGLPaaajugibiabg2da9Kqbaoaalaaakeaaju gibiabeI7aXLqbaoaaCaaaleqabaqcLbmacaaIZaaaaaGcbaqcLbsa cqaH4oqCjuaGdaahaaWcbeqaaKqzadGaaG4maaaajugibiabgUcaRi aaikdaaaqcfa4aa8qCaOqaaKqzGeGaamyzaKqbaoaaCaaaleqabaqc LbmacqGHsisllmaabmaabaqcLbmacqaH4oqCcqGHsislcaWG0baali aawIcacaGLPaaajugWaiaaykW7caWG4baaaKqbaoaabmaakeaajugi biabeI7aXjabgUcaRiaadIhajuaGdaahaaWcbeqaaKqzadGaaGOmaa aaaOGaayjkaiaawMcaaKqzGeGaaGPaVdWcbaqcLbmacaaIWaaaleaa jugWaiabg6HiLcqcLbsacqGHRiI8aiaadsgacaWG4baaaa@6EFC@

= θ 3 θ 3 +2 [ θ θt + 2 ( θt ) 3 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGH9a qpjuaGdaWcaaGcbaqcLbsacqaH4oqCjuaGdaahaaWcbeqaaKqzadGa aG4maaaaaOqaaKqzGeGaeqiUdexcfa4aaWbaaSqabeaajugWaiaaio daaaqcLbsacqGHRaWkcaaIYaaaaKqbaoaadmaakeaajuaGdaWcaaGc baqcLbsacqaH4oqCaOqaaKqzGeGaeqiUdeNaeyOeI0IaamiDaaaacq GHRaWkjuaGdaWcaaGcbaqcLbsacaaIYaaakeaajuaGdaqadaGcbaqc LbsacqaH4oqCcqGHsislcaWG0baakiaawIcacaGLPaaajuaGdaahaa WcbeqaaKqzadGaaG4maaaaaaaakiaawUfacaGLDbaaaaa@59D1@

= θ 3 θ 3 +2 [ k=0 ( t θ ) k + 2 θ 3 k=0 ( k+2 k ) ( t θ ) k ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGH9a qpjuaGdaWcaaGcbaqcLbsacqaH4oqCjuaGdaahaaWcbeqaaKqzadGa aG4maaaaaOqaaKqzGeGaeqiUdexcfa4aaWbaaSqabeaajugWaiaaio daaaqcLbsacqGHRaWkcaaIYaaaaKqbaoaadmaakeaajuaGdaaeWbGc baqcfa4aaeWaaOqaaKqbaoaalaaakeaajugibiaadshaaOqaaKqzGe GaeqiUdehaaaGccaGLOaGaayzkaaqcfa4aaWbaaSqabeaajugWaiaa dUgaaaqcLbsacqGHRaWkaSqaaKqzGeGaam4Aaiabg2da9iaaicdaaS qaaKqzGeGaeyOhIukacqGHris5aKqbaoaalaaakeaajugibiaaikda aOqaaKqzGeGaeqiUdexcfa4aaWbaaSqabeaajugWaiaaiodaaaaaaK qbaoaaqahakeaajuaGdaqadaGcbaqcLbsafaqabeGabaaakeaajugi biaadUgacqGHRaWkcaaIYaaakeaajugibiaadUgaaaaakiaawIcaca GLPaaajuaGdaqadaGcbaqcfa4aaSaaaOqaaKqzGeGaamiDaaGcbaqc LbsacqaH4oqCaaaakiaawIcacaGLPaaajuaGdaahaaWcbeqaaKqzad Gaam4AaaaaaSqaaKqzGeGaam4Aaiabg2da9iaaicdaaSqaaKqzGeGa eyOhIukacqGHris5aaGccaGLBbGaayzxaaaaaa@79D8@

= k=0 θ 3 +( k+1 )( k+2 ) θ 3 +2 ( t θ ) k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGH9a qpjuaGdaaeWbGcbaqcfa4aaSaaaOqaaKqzGeGaeqiUdexcfa4aaWba aSqabeaajugWaiaaiodaaaqcLbsacqGHRaWkjuaGdaqadaGcbaqcLb sacaWGRbGaey4kaSIaaGymaaGccaGLOaGaayzkaaqcfa4aaeWaaOqa aKqzGeGaam4AaiabgUcaRiaaikdaaOGaayjkaiaawMcaaaqaaKqzGe GaeqiUdexcfa4aaWbaaSqabeaajugWaiaaiodaaaqcLbsacqGHRaWk caaIYaaaaaWcbaqcLbsacaWGRbGaeyypa0JaaGimaaWcbaqcLbsacq GHEisPaiabggHiLdqcfa4aaeWaaOqaaKqbaoaalaaakeaajugibiaa dshaaOqaaKqzGeGaeqiUdehaaaGccaGLOaGaayzkaaqcfa4aaWbaaS qabeaajugWaiaadUgaaaaaaa@61D0@

The r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCaaaa@36ED@ the moment about origin of Ishita distributon (3.5) is given by

                                               μ r = r![ θ 3 +( r+1 )( r+2 ) ] θ r ( θ 3 +2 ) ;r=1,2,3,... MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH8o qBjuaGdaWgaaWcbaqcLbsacaWGYbaaleqaaKqbaoaaCaaaleqabaqc LbsacWaGGBOmGikaaiabg2da9Kqbaoaalaaakeaajugibiaadkhaca GGHaqcfa4aamWaaOqaaKqzGeGaeqiUdexcfa4aaWbaaSqabeaajugW aiaaiodaaaqcLbsacqGHRaWkjuaGdaqadaGcbaqcLbsacaWGYbGaey 4kaSIaaGymaaGccaGLOaGaayzkaaqcfa4aaeWaaOqaaKqzGeGaamOC aiabgUcaRiaaikdaaOGaayjkaiaawMcaaaGaay5waiaaw2faaaqaaK qzGeGaeqiUdexcfa4aaWbaaSqabeaajugWaiaadkhaaaqcfa4aaeWa aOqaaKqzGeGaeqiUdexcfa4aaWbaaSqabeaajugWaiaaiodaaaqcLb sacqGHRaWkcaaIYaaakiaawIcacaGLPaaaaaqcLbsacaaMc8UaaGPa VlaaykW7caGG7aGaamOCaiabg2da9iaaigdacaGGSaGaaGOmaiaacY cacaaIZaGaaiilaiaac6cacaGGUaGaaiOlaaaa@72C2@                                                                 (2.1)

The first four moments about origin are thus obtained as

μ 1 = θ 3 +6 θ( θ 3 +2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH8o qBjuaGdaWgaaWcbaqcLbsacaaIXaaaleqaaKqbaoaaCaaaleqabaqc LbsacWaGGBOmGikaaiabg2da9KqbaoaalaaakeaajugibiabeI7aXL qbaoaaCaaaleqabaqcLbmacaaIZaaaaKqzGeGaey4kaSIaaGOnaaGc baqcLbsacqaH4oqCjuaGdaqadaGcbaqcLbsacqaH4oqCjuaGdaahaa WcbeqaaKqzadGaaG4maaaajugibiabgUcaRiaaikdaaOGaayjkaiaa wMcaaaaaaaa@52D8@ , μ 2 = 2( θ 3 +12 ) θ 2 ( θ 3 +2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH8o qBjuaGdaWgaaWcbaqcLbsacaaIYaaaleqaaKqbaoaaCaaaleqabaqc LbsacWaGGBOmGikaaiabg2da9Kqbaoaalaaakeaajugibiaaikdaju aGdaqadaGcbaqcLbsacqaH4oqCjuaGdaahaaWcbeqaaKqzadGaaG4m aaaajugibiabgUcaRiaaigdacaaIYaaakiaawIcacaGLPaaaaeaaju gibiabeI7aXLqbaoaaCaaaleqabaqcLbmacaaIYaaaaKqbaoaabmaa keaajugibiabeI7aXLqbaoaaCaaaleqabaqcLbmacaaIZaaaaKqzGe Gaey4kaSIaaGOmaaGccaGLOaGaayzkaaaaaaaa@59A1@

μ 3 = 6( θ 3 +20 ) θ 3 ( θ 3 +2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH8o qBjuaGdaWgaaWcbaqcLbsacaaIZaaaleqaaKqbaoaaCaaaleqabaqc LbsacWaGGBOmGikaaiabg2da9KqbaoaalaaakeaajugibiaaiAdaju aGdaqadaGcbaqcLbsacqaH4oqCjuaGdaahaaWcbeqaaKqzadGaaG4m aaaajugibiabgUcaRiaaikdacaaIWaaakiaawIcacaGLPaaaaeaaju gibiabeI7aXLqbaoaaCaaaleqabaqcLbmacaaIZaaaaKqbaoaabmaa keaajugibiabeI7aXLqbaoaaCaaaleqabaqcLbmacaaIZaaaaKqzGe Gaey4kaSIaaGOmaaGccaGLOaGaayzkaaaaaaaa@59A6@ , μ 4 = 24( θ 3 +30 ) θ 4 ( θ 3 +2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH8o qBjuaGdaWgaaWcbaqcLbsacaaI0aaaleqaaKqbaoaaCaaaleqabaqc LbsacWaGGBOmGikaaiabg2da9Kqbaoaalaaakeaajugibiaaikdaca aI0aqcfa4aaeWaaOqaaKqzGeGaeqiUdexcfa4aaWbaaSqabeaajugW aiaaiodaaaqcLbsacqGHRaWkcaaIZaGaaGimaaGccaGLOaGaayzkaa aabaqcLbsacqaH4oqCjuaGdaahaaWcbeqaaKqzadGaaGinaaaajuaG daqadaGcbaqcLbsacqaH4oqCjuaGdaahaaWcbeqaaKqzadGaaG4maa aajugibiabgUcaRiaaikdaaOGaayjkaiaawMcaaaaaaaa@5A63@

Using relationship between moments about mean and the moments about origin, the moments about mean of Ishita distribution (3.5) can be obtained as

μ 2 = θ 6 +16 θ 3 +12 θ 2 ( θ 3 +2 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaKqbGeaacaaIYaaajuaGbeaacqGH9aqpdaWcaaqaaiabeI7a XnaaCaaabeqcfasaaiaaiAdaaaqcfaOaey4kaSIaaGymaiaaiAdacq aH4oqCdaahaaqcfasabeaacaaIZaaaaKqbakabgUcaRiaaigdacaaI YaaabaGaeqiUde3aaWbaaeqajuaibaGaaGOmaaaajuaGdaqadaqaai abeI7aXnaaCaaabeqcfasaaiaaiodaaaqcfaOaey4kaSIaaGOmaaGa ayjkaiaawMcaamaaCaaajuaibeqaaiaaikdaaaaaaaaa@5118@

μ 3 = 2( θ 9 +30 θ 6 +36 θ 3 +24 ) θ 3 ( θ 3 +2 ) 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaKqbGeaacaaIZaaajuaGbeaacqGH9aqpdaWcaaqaaiaaikda daqadaqaaiabeI7aXnaaCaaabeqcfasaaiaaiMdaaaqcfaOaey4kaS IaaG4maiaaicdacqaH4oqCdaahaaqabKqbGeaacaaI2aaaaKqbakab gUcaRiaaiodacaaI2aGaeqiUde3aaWbaaeqajuaibaGaaG4maaaaju aGcqGHRaWkcaaIYaGaaGinaaGaayjkaiaawMcaaaqaaiabeI7aXnaa Caaabeqcfasaaiaaiodaaaqcfa4aaeWaaeaacqaH4oqCdaahaaqcfa sabeaacaaIZaaaaKqbakabgUcaRiaaikdaaiaawIcacaGLPaaadaah aaqabKqbGeaacaaIZaaaaaaaaaa@5915@

μ 4 = 3( 3 θ 12 +128 θ 9 +408 θ 6 +576 θ 3 +240 ) θ 4 ( θ 3 +2 ) 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaKqbGeaacaaI0aaajuaGbeaacqGH9aqpdaWcaaqaaiaaioda daqadaqaaiaaiodacqaH4oqCdaahaaqabKqbGeaacaaIXaGaaGOmaa aajuaGcqGHRaWkcaaIXaGaaGOmaiaaiIdacqaH4oqCdaahaaqabKqb GeaacaaI5aaaaKqbakabgUcaRiaaisdacaaIWaGaaGioaiabeI7aXn aaCaaabeqcfasaaiaaiAdaaaqcfaOaey4kaSIaaGynaiaaiEdacaaI 2aGaeqiUde3aaWbaaeqajuaibaGaaG4maaaajuaGcqGHRaWkcaaIYa GaaGinaiaaicdaaiaawIcacaGLPaaaaeaacqaH4oqCdaahaaqabKqb GeaacaaI0aaaaKqbaoaabmaabaGaeqiUde3aaWbaaeqajuaibaGaaG 4maaaajuaGcqGHRaWkcaaIYaaacaGLOaGaayzkaaWaaWbaaeqajuai baGaaGinaaaaaaaaaa@633C@

The coefficient of variation ( C.V ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeGaam4qaiaac6cacaWGwbaakiaawIcacaGLPaaaaaa@3B05@ , coefficient of skewness ( β 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqbaoaakaaakeaajugibiabek7aILqbaoaaBaaaleaajugWaiaa igdaaSqabaaabeaaaOGaayjkaiaawMcaaaaa@3DA7@ , coefficient of kurtosis ( β 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacq aHYoGydaWgaaWcbaGaaGOmaaqabaaakiaawIcacaGLPaaaaaa@3A12@ and index of dispersion ( γ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacq aHZoWzaiaawIcacaGLPaaaaaa@3926@  of Ishita distribution (3.5) are thus obtained as

C.V= σ μ 1 = θ 6 +16 θ 3 +12 θ 3 +6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4qai aac6cacaWGwbGaeyypa0ZaaSaaaeaacqaHdpWCaeaacqaH8oqBdaWg aaqaaiaaigdaaeqaamaaCaaabeqaaiadacUHYaIOaaaaaiabg2da9m aalaaabaWaaOaaaeaacqaH4oqCdaahaaqabKqbGeaacaaI2aaaaKqb akabgUcaRiaaigdacaaI2aGaeqiUde3aaWbaaeqajuaibaGaaG4maa aajuaGcqGHRaWkcaaIXaGaaGOmaaqabaaabaGaeqiUde3aaWbaaeqa juaibaGaaG4maaaajuaGcqGHRaWkcaaI2aaaaaaa@52C2@

β 1 = μ 3 μ 2 3/2 = 2( θ 9 +30 θ 6 +36 θ 3 +24 ) ( θ 6 +16 θ 3 +12 ) 3/2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaOaaae aacqaHYoGydaWgaaqaaiaaigdaaeqaaaqabaGaeyypa0ZaaSaaaeaa cqaH8oqBdaWgaaqcfasaaiaaiodaaKqbagqaaaqaaiabeY7aTnaaBa aajuaibaGaaGOmaaqcfayabaWaaWbaaeqajuaibaGaaG4maiaac+ca caaIYaaaaaaajuaGcqGH9aqpdaWcaaqaaiaaikdadaqadaqaaiabeI 7aXnaaCaaabeqcfasaaiaaiMdaaaqcfaOaey4kaSIaaG4maiaaicda cqaH4oqCdaahaaqabKqbGeaacaaI2aaaaKqbakabgUcaRiaaiodaca aI2aGaeqiUde3aaWbaaeqajuaibaGaaG4maaaajuaGcqGHRaWkcaaI YaGaaGinaaGaayjkaiaawMcaaaqaamaabmaabaGaeqiUde3aaWbaae qajuaibaGaaGOnaaaajuaGcqGHRaWkcaaIXaGaaGOnaiabeI7aXnaa CaaabeqcfasaaiaaiodaaaqcfaOaey4kaSIaaGymaiaaikdaaiaawI cacaGLPaaadaahaaqabKqbGeaacaaIZaGaai4laiaaikdaaaaaaaaa @679B@

β 2 = μ 4 μ 2 2 = 3( 3 θ 12 +128 θ 9 +408 θ 6 +576 θ 3 +240 ) ( θ 6 +16 θ 3 +12 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOSdi 2aaSbaaKqbGeaacaaIYaaajuaGbeaacqGH9aqpdaWcaaqaaiabeY7a TnaaBaaajuaibaGaaGinaaqcfayabaaabaGaeqiVd02aaSbaaKqbGe aacaaIYaaajuaGbeaadaahaaqabKqbGeaacaaIYaaaaaaajuaGcqGH 9aqpdaWcaaqaaiaaiodadaqadaqaaiaaiodacqaH4oqCdaahaaqabK qbGeaacaaIXaGaaGOmaaaajuaGcqGHRaWkcaaIXaGaaGOmaiaaiIda cqaH4oqCdaahaaqabKqbGeaacaaI5aaaaKqbakabgUcaRiaaisdaca aIWaGaaGioaiabeI7aXnaaCaaabeqcfasaaiaaiAdaaaqcfaOaey4k aSIaaGynaiaaiEdacaaI2aGaeqiUde3aaWbaaeqajuaibaGaaG4maa aajuaGcqGHRaWkcaaIYaGaaGinaiaaicdaaiaawIcacaGLPaaaaeaa daqadaqaaiabeI7aXnaaCaaabeqcfasaaiaaiAdaaaqcfaOaey4kaS IaaGymaiaaiAdacqaH4oqCdaahaaqabKqbGeaacaaIZaaaaKqbakab gUcaRiaaigdacaaIYaaacaGLOaGaayzkaaWaaWbaaeqajuaibaGaaG Omaaaaaaaaaa@6F8D@

γ= σ 2 μ 1 = θ 6 +16 θ 3 +12 θ( θ 3 +2 )( θ 3 +6 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4SdC Maeyypa0ZaaSaaaeaacqaHdpWCdaahaaqabKqbGeaacaaIYaaaaaqc fayaaiabeY7aTnaaBaaajuaibaGaaGymaaqcfayabaWaaWbaaeqaba Gamai4gkdiIcaaaaGaeyypa0ZaaSaaaeaacqaH4oqCdaahaaqabKqb GeaacaaI2aaaaKqbakabgUcaRiaaigdacaaI2aGaeqiUde3aaWbaae qajuaibaGaaG4maaaajuaGcqGHRaWkcaaIXaGaaGOmaaqaaiabeI7a XnaabmaabaGaeqiUde3aaWbaaeqajuaibaGaaG4maaaajuaGcqGHRa WkcaaIYaaacaGLOaGaayzkaaWaaeWaaeaacqaH4oqCdaahaaqabKqb GeaacaaIZaaaaKqbakabgUcaRiaaiAdaaiaawIcacaGLPaaaaaaaaa@5E11@

The over-dispersion, equi-dispersion, and under-dispersion of Ishita distribution has been presented in (Table 1) along with Akash, Lindley and exponential distributions.

Lifetime Distributions

Over-Dispersion ( μ< σ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacqaH8oqBcqGH8aapcqaHdpWCdaahaaqabKqbGeaacaaIYaaaaaqc faOaayjkaiaawMcaaaaa@3E24@

Equi-Dispersion
( μ= σ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacqaH8oqBcqGH9aqpcqaHdpWCdaahaaqabKqbGeaacaaIYaaaaaqc faOaayjkaiaawMcaaaaa@3E26@

Under-Dispersion
( μ> σ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacqaH8oqBcqGH+aGpcqaHdpWCdaahaaqabKqbGeaacaaIYaaaaaqc faOaayjkaiaawMcaaaaa@3E28@

Ishita

θ<1.535653152 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde NaeyipaWJaaGymaiaac6cacaaI1aGaaG4maiaaiwdacaaI2aGaaGyn aiaaiodacaaIXaGaaGynaiaaikdaaaa@4158@

θ=1.535653152 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde Naeyypa0JaaGymaiaac6cacaaI1aGaaG4maiaaiwdacaaI2aGaaGyn aiaaiodacaaIXaGaaGynaiaaikdaaaa@415A@

θ>1.535653152 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde NaeyOpa4JaaGymaiaac6cacaaI1aGaaG4maiaaiwdacaaI2aGaaGyn aiaaiodacaaIXaGaaGynaiaaikdaaaa@415C@

Akash

θ<1.515400063 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde NaeyipaWJaaGymaiaac6cacaaI1aGaaGymaiaaiwdacaaI0aGaaGim aiaaicdacaaIWaGaaGOnaiaaiodaaaa@414D@

θ=1.515400063 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde Naeyypa0JaaGymaiaac6cacaaI1aGaaGymaiaaiwdacaaI0aGaaGim aiaaicdacaaIWaGaaGOnaiaaiodaaaa@414F@

θ>1.515400063 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde NaeyOpa4JaaGymaiaac6cacaaI1aGaaGymaiaaiwdacaaI0aGaaGim aiaaicdacaaIWaGaaGOnaiaaiodaaaa@4151@

Lindley

θ<1.170086487 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde NaeyipaWJaaGymaiaac6cacaaIXaGaaG4naiaaicdacaaIWaGaaGio aiaaiAdacaaI0aGaaGioaiaaiEdaaaa@415E@

θ=1.170086487 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde Naeyypa0JaaGymaiaac6cacaaIXaGaaG4naiaaicdacaaIWaGaaGio aiaaiAdacaaI0aGaaGioaiaaiEdaaaa@4160@

θ>1.170086487 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde NaeyOpa4JaaGymaiaac6cacaaIXaGaaG4naiaaicdacaaIWaGaaGio aiaaiAdacaaI0aGaaGioaiaaiEdaaaa@4162@

Exponential

θ<1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde NaeyipaWJaaGymaaaa@39F9@

θ=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde Naeyypa0JaaGymaaaa@39FB@

θ>1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde NaeyOpa4JaaGymaaaa@39FD@

Table 1: Over-dispersion, equi-dispersion and under-dispersion of Ishita, Akash, Lindley and exponential distributions for the parameter θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde haaa@383A@

Hazard Rate Function And Mean Residual Life Function

Let X MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiwaa aa@3761@ be a continuous random variable with p.d.f. f( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aabmaabaGaamiEaaGaayjkaiaawMcaaaaa@39F5@ and c.d.f. F( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOram aabmaabaGaamiEaaGaayjkaiaawMcaaaaa@39D5@ . The hazard rate function (also known as the failure rate function) h( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiAam aabmaabaGaamiEaaGaayjkaiaawMcaaaaa@39F7@ and the mean residual life function m( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyBam aabmaabaGaamiEaaGaayjkaiaawMcaaaaa@39FC@ of X MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiwaa aa@3761@ are respectively defined as

h( x )= lim Δx0 P( X<x+Δx|X>x ) Δx = f( x ) 1F( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiAam aabmaabaGaamiEaaGaayjkaiaawMcaaiabg2da9maaxababaGaciiB aiaacMgacaGGTbaabaGaeyiLdqKaamiEaiabgkziUkaaicdaaeqaam aalaaabaGaamiuamaabmaabaWaaqGaaeaacaWGybGaeyipaWJaamiE aiabgUcaRiabgs5aejaadIhaaiaawIa7aiaadIfacqGH+aGpcaWG4b aacaGLOaGaayzkaaaabaGaeyiLdqKaamiEaaaacqGH9aqpdaWcaaqa aiaadAgadaqadaqaaiaadIhaaiaawIcacaGLPaaaaeaacaaIXaGaey OeI0IaamOramaabmaabaGaamiEaaGaayjkaiaawMcaaaaaaaa@5BF4@                             (5.1)

and

 m( x )=E[ Xx|X>x ]= 1 1F( x ) x [ 1F( t ) ] dt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaaeiiai aad2gadaqadaqaaiaadIhaaiaawIcacaGLPaaacqGH9aqpcaWGfbWa amWaaeaadaabcaqaaiaadIfacqGHsislcaWG4baacaGLiWoacaWGyb GaeyOpa4JaamiEaaGaay5waiaaw2faaiaaysW7cqGH9aqpcaaMe8+a aSaaaeaacaaIXaaabaGaaGymaiabgkHiTiaadAeadaqadaqaaiaadI haaiaawIcacaGLPaaaaaWaa8qmaeaadaWadaqaaiaaigdacqGHsisl caWGgbWaaeWaaeaacaWG0baacaGLOaGaayzkaaaacaGLBbGaayzxaa aajuaibaGaamiEaaqaaiabg6HiLcqcfaOaey4kIipacaaMe8UaaGPa VlaadsgacaWG0baaaa@60A0@             (5.2)

The corresponding hazard rate function, h( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiAam aabmaabaGaamiEaaGaayjkaiaawMcaaaaa@39F7@ and the mean residual life function, m( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyBam aabmaabaGaamiEaaGaayjkaiaawMcaaaaa@39FC@ of the Ishita distribution (1.5) are thus obtained as

h( x )= θ 3 ( θ+ x 2 ) θx( θx+2 )+( θ 3 +2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiAam aabmaabaGaamiEaaGaayjkaiaawMcaaiabg2da9maalaaabaGaeqiU de3aaWbaaeqajuaibaGaaG4maaaajuaGdaqadaqaaiabeI7aXjabgU caRiaadIhadaahaaqabKqbGeaacaaIYaaaaaqcfaOaayjkaiaawMca aaqaaiabeI7aXjaadIhadaqadaqaaiabeI7aXjaadIhacqGHRaWkca aIYaaacaGLOaGaayzkaaGaey4kaSYaaeWaaeaacqaH4oqCdaahaaqa bKqbGeaacaaIZaaaaKqbakabgUcaRiaaikdaaiaawIcacaGLPaaaaa aaaa@54FD@                                                                                        (5.3)

and             m( x )= θ 3 +2 [ θx( θx+2 )+( θ 3 +2 ) ] e θx x [ θt( θt+2 )+( θ 3 +2 ) θ 3 +2 ] e θt dt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyBam aabmaabaGaamiEaaGaayjkaiaawMcaaiabg2da9maalaaabaGaeqiU de3aaWbaaeqajuaibaGaaG4maaaajuaGcqGHRaWkcaaIYaaabaWaam WaaeaacqaH4oqCcaWG4bWaaeWaaeaacqaH4oqCcaWG4bGaey4kaSIa aGOmaaGaayjkaiaawMcaaiabgUcaRmaabmaabaGaeqiUde3aaWbaae qajuaibaGaaG4maaaajuaGcqGHRaWkcaaIYaaacaGLOaGaayzkaaaa caGLBbGaayzxaaGaamyzamaaCaaabeqcfasaaiabgkHiTiabeI7aXj aadIhaaaaaaKqbaoaapehabaWaamWaaeaadaWcaaqaaiabeI7aXjaa dshadaqadaqaaiabeI7aXjaadshacqGHRaWkcaaIYaaacaGLOaGaay zkaaGaey4kaSYaaeWaaeaacqaH4oqCdaahaaqabKqbGeaacaaIZaaa aKqbakabgUcaRiaaikdaaiaawIcacaGLPaaaaeaacqaH4oqCdaahaa qabKqbGeaacaaIZaaaaKqbakabgUcaRiaaikdaaaaacaGLBbGaayzx aaGaamyzamaaCaaabeqcfasaaiabgkHiTiabeI7aXjaadshaaaqcfa OaamizaiaadshaaKqbGeaacaWG4baabaGaeyOhIukajuaGcqGHRiI8 aaaa@7AD3@  

= θ 2 x 2 +4θx+( θ 3 +6 ) θ[ θx( θx+2 )+( θ 3 +2 ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyypa0 ZaaSaaaeaacqaH4oqCdaahaaqabKqbGeaacaaIYaaaaKqbakaadIha daahaaqabKqbGeaacaaIYaaaaKqbakabgUcaRiaaisdacqaH4oqCca WG4bGaey4kaSYaaeWaaeaacqaH4oqCdaahaaqabKqbGeaacaaIZaaa aKqbakabgUcaRiaaiAdaaiaawIcacaGLPaaaaeaacqaH4oqCdaWada qaaiabeI7aXjaadIhadaqadaqaaiabeI7aXjaadIhacqGHRaWkcaaI YaaacaGLOaGaayzkaaGaey4kaSYaaeWaaeaacqaH4oqCdaahaaqabK qbGeaacaaIZaaaaKqbakabgUcaRiaaikdaaiaawIcacaGLPaaaaiaa wUfacaGLDbaaaaaaaa@5CC1@  (5.4)

It can be easily verified that h( 0 )= θ 3 θ 3 +2 =f( 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiAam aabmaabaGaaGimaaGaayjkaiaawMcaaiabg2da9maalaaabaGaeqiU de3aaWbaaeqajuaibaGaaG4maaaaaKqbagaacqaH4oqCdaahaaqabK qbGeaacaaIZaaaaKqbakabgUcaRiaaikdaaaGaeyypa0JaamOzamaa bmaabaGaaGimaaGaayjkaiaawMcaaaaa@473E@ and m( 0 )= θ 3 +6 θ( θ 3 +2 ) = μ 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyBam aabmaabaGaaGimaaGaayjkaiaawMcaaiabg2da9maalaaabaGaeqiU de3aaWbaaeqajuaibaGaaG4maaaajuaGcqGHRaWkcaaI2aaabaGaeq iUde3aaeWaaeaacqaH4oqCdaahaaqabKqbGeaacaaIZaaaaKqbakab gUcaRiaaikdaaiaawIcacaGLPaaaaaGaeyypa0JaeqiVd02cdaWgaa qcfayaaKqzadGaaGymaaqcfayabaWcdaahaaqcfayabeaajugWaiad acUHYaIOaaaaaa@52AE@ .It is also obvious from the graphs of h( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiAam aabmaabaGaamiEaaGaayjkaiaawMcaaaaa@39F7@  and m( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyBam aabmaabaGaamiEaaGaayjkaiaawMcaaaaa@39FC@ that h( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiAam aabmaabaGaamiEaaGaayjkaiaawMcaaaaa@39F7@  is an increasing function of ( xandθ1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacaWG4bGaaGPaVlaaykW7caqGHbGaaeOBaiaabsgacaaMc8UaaGPa VlabeI7aXjabgsMiJkaaigdaaiaawIcacaGLPaaaaaa@4618@  and ( x1andθ>1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacaWG4bGaeyyzImRaaGymaiaaykW7caaMc8Uaaeyyaiaab6gacaqG KbGaaGPaVlaaykW7cqaH4oqCcqGH+aGpcaaIXaaacaGLOaGaayzkaa aaaa@47EC@  and decreasing function of ( 0<x<1andθ>1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacaaIWaGaeyipaWJaamiEaiabgYda8iaaigdacaaMc8UaaGPaVlaa bggacaqGUbGaaeizaiaaykW7caaMc8UaeqiUdeNaeyOpa4JaaGymaa GaayjkaiaawMcaaaaa@48E8@ , whereas m( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyBam aabmaabaGaamiEaaGaayjkaiaawMcaaaaa@39FC@ is a decreasing function of x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEaa aa@3781@ , and θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde haaa@383A@ . The graph of the hazard rate function and mean residual life function of Ishita distribution (5.5) are shown in (Figures 3&4).

Figure 3: Graph of hazard rate function of Ishita distribution for varying values of parameter θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH4o qCaaa@383B@ .

Figure 4: Graph of mean residual life function of Ishita distribution for varying values of parameter θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH4o qCaaa@383B@ .

Stochastic Orderings

Stochastic ordering of positive continuous random variables is an important tool for judging their comparative behavior. A random variable X MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiwaa aa@3761@ is said to be smaller than a random variable Y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamywaa aa@3762@ in the

  1. Stochastic order ( X st Y ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacaWGybGaeyizIm6aaSbaaKqbGeaacaWGZbGaamiDaaqcfayabaGa amywaaGaayjkaiaawMcaaaaa@3E4B@ if F X ( x ) F Y ( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOram aaBaaajuaibaGaamiwaaqcfayabaWaaeWaaeaacaWG4baacaGLOaGa ayzkaaGaeyyzImRaamOramaaBaaajuaibaGaamywaaqcfayabaWaae WaaeaacaWG4baacaGLOaGaayzkaaaaaa@4261@ for all x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEaa aa@3781@
  2. Hazard rate order ( X hr Y ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacaWGybGaeyizIm6aaSbaaKqbGeaacaWGObGaamOCaaqcfayabaGa amywaaGaayjkaiaawMcaaaaa@3E3E@ if h X ( x ) h Y ( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiAam aaBaaajuaibaGaamiwaaqcfayabaWaaeWaaeaacaWG4baacaGLOaGa ayzkaaGaeyyzImRaamiAamaaBaaajuaibaGaamywaaqcfayabaWaae WaaeaacaWG4baacaGLOaGaayzkaaaaaa@42A5@  for all x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEaa aa@3781@
  3. Mean residual life order ( X mrl Y ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacaWGybGaeyizIm6aaSbaaKqbGeaacaWGTbGaamOCaiaadYgaaKqb agqaaiaadMfaaiaawIcacaGLPaaaaaa@3F34@ if m X ( x ) m Y ( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyBam aaBaaajuaibaGaamiwaaqcfayabaWaaeWaaeaacaWG4baacaGLOaGa ayzkaaGaeyizImQaamyBamaaBaaajuaibaGaamywaaqcfayabaWaae WaaeaacaWG4baacaGLOaGaayzkaaaaaa@429E@ for all x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEaa aa@3781@
  4. Likelihood ratio order ( X lr Y ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacaWGybGaeyizIm6aaSbaaKqbGeaacaWGSbGaamOCaaqcfayabaGa amywaaGaayjkaiaawMcaaaaa@3E42@ if f X ( x ) f Y ( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacaWGMbWaaSbaaKqbGeaacaWGybaajuaGbeaadaqadaqaaiaadIha aiaawIcacaGLPaaaaeaacaWGMbWaaSbaaKqbGeaacaWGzbaajuaGbe aadaqadaqaaiaadIhaaiaawIcacaGLPaaaaaaaaa@40EB@  decreases in x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEaa aa@3781@ .

The following results due to Shaked and Shanthikumar [18] are well known for establishing stochastic ordering of distributions

X lr YX hr YX mrl Y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiwai abgsMiJoaaBaaajuaibaGaamiBaiaadkhaaKqbagqaaiaadMfacqGH shI3caWGybGaeyizIm6aaSbaaKqbGeaacaWGObGaamOCaaqcfayaba GaamywaiabgkDiElaadIfacqGHKjYOdaWgaaqcfasaaiaad2gacaWG YbGaamiBaaqcfayabaGaamywaaaa@4ECB@                                                                 (6.1)

X st Y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaCbeae aacqGHthY3aeaacaWGybGaeyizIm6aaSbaaKqbGeaacaWGZbGaamiD aaqcfayabaGaamywaaqabaaaaa@3F4F@

The Ishita distribution is ordered with respect to the strongest ‘likelihood ratio’ ordering as shown in the following theorem:

Theorem

Let X MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiwaa aa@3761@ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaGae8hpI4 haaa@3761@  Ishita distributon ( θ 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacqaH4oqCdaWgaaqaaiaaigdaaeqaaaGaayjkaiaawMcaaaaa@3A9F@  and Y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamywaa aa@3762@ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaGae8hpI4 haaa@3761@  Ishita distribution ( θ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacqaH4oqCdaWgaaqcfasaaiaaikdaaKqbagqaaaGaayjkaiaawMca aaaa@3B5C@ . If θ 1 > θ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde 3aaSbaaKqbGeaacaaIXaaajuaGbeaacqGH+aGpcqaH4oqCdaWgaaqc fasaaiaaikdaaKqbagqaaaaa@3E29@ , then X lr Y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiwai abgsMiJoaaBaaajuaibaGaamiBaiaadkhaaKqbagqaaiaadMfaaaa@3CB9@ and hence X hr Y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiwai abgsMiJoaaBaaajuaibaGaamiAaiaadkhaaKqbagqaaiaadMfaaaa@3CB5@ , X mrl Y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiwai abgsMiJoaaBaaajuaibaGaamyBaiaadkhacaWGSbaajuaGbeaacaWG zbaaaa@3DAB@ and X st Y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiwai abgsMiJoaaBaaajuaibaGaam4CaiaadshaaKqbagqaaiaadMfaaaa@3CC2@ .

Proof

We have

f X ( x,θ ) f Y ( x,θ ) = θ 1 3 ( θ 2 3 +2 ) θ 2 3 ( θ 1 3 +2 ) ( θ 1 + x 2 θ 2 + x 2 ) e ( θ 1 θ 2 )x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSqaae aacaWGMbWaaSbaaKqbGeaacaWGybaajuaGbeaadaqadaqaaiaadIha caGGSaGaeqiUdehacaGLOaGaayzkaaaabaGaamOzamaaBaaajuaiba GaamywaaqcfayabaWaaeWaaeaacaWG4bGaaiilaiabeI7aXbGaayjk aiaawMcaaaaacqGH9aqpdaWcaaqaaiabeI7aXnaaBaaajuaibaGaaG ymaaqcfayabaWaaWbaaeqajuaibaGaaG4maaaajuaGdaqadaqaaiab eI7aXnaaBaaajuaibaGaaGOmaaqcfayabaWaaWbaaeqajuaibaGaaG 4maaaajuaGcqGHRaWkcaaIYaaacaGLOaGaayzkaaaabaGaeqiUde3a aSbaaKqbGeaacaaIYaaajuaGbeaadaahaaqabKqbGeaacaaIZaaaaK qbaoaabmaabaGaeqiUde3aaSbaaKqbGeaacaaIXaaajuaGbeaadaah aaqabKqbGeaacaaIZaaaaKqbakabgUcaRiaaikdaaiaawIcacaGLPa aaaaWaaeWaaeaadaWcaaqaaiabeI7aXnaaBaaajuaibaGaaGymaaqc fayabaGaey4kaSIaamiEamaaCaaabeqcfasaaiaaikdaaaaajuaGba GaeqiUde3aaSbaaKqbGeaacaaIYaaajuaGbeaacqGHRaWkcaWG4bWa aWbaaeqajuaibaGaaGOmaaaaaaaajuaGcaGLOaGaayzkaaGaamyzam aaCaaabeqcfasaaiabgkHiTKqbaoaabmaajuaibaGaeqiUdexcfa4a aSbaaKqbGeaacaaIXaaabeaacqGHsislcqaH4oqCjuaGdaWgaaqcfa saaiaaikdaaeqaaaGaayjkaiaawMcaaiaadIhaaaaaaa@7CDD@   ; x>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaai4oai aabccacaWG4bGaeyOpa4JaaGimaaaa@3AA5@  

Now

ln f X ( x,θ ) f Y ( x,θ ) =ln[ θ 1 3 ( θ 2 3 +2 ) θ 2 3 ( θ 1 3 +2 ) ]+ln( θ 1 + x 2 θ 2 + x 2 )( θ 1 θ 2 )x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaciiBai aac6gadaWcbaqaaiaadAgadaWgaaqcfasaaiaadIfaaKqbagqaamaa bmaabaGaamiEaiaacYcacqaH4oqCaiaawIcacaGLPaaaaeaacaWGMb WaaSbaaKqbGeaacaWGzbaajuaGbeaadaqadaqaaiaadIhacaGGSaGa eqiUdehacaGLOaGaayzkaaaaaiabg2da9iGacYgacaGGUbWaamWaae aadaWcaaqaaiabeI7aXnaaBaaajuaibaGaaGymaaqcfayabaWaaWba aeqajuaibaGaaG4maaaajuaGdaqadaqaaiabeI7aXnaaBaaajuaiba GaaGOmaaqcfayabaWaaWbaaeqajuaibaGaaG4maaaajuaGcqGHRaWk caaIYaaacaGLOaGaayzkaaaabaGaeqiUde3aaSbaaKqbGeaacaaIYa aajuaGbeaadaahaaqabKqbGeaacaaIZaaaaKqbaoaabmaabaGaeqiU de3aaSbaaKqbGeaacaaIXaaajuaGbeaadaahaaqabKqbGeaacaaIZa aaaKqbakabgUcaRiaaikdaaiaawIcacaGLPaaaaaaacaGLBbGaayzx aaGaey4kaSIaciiBaiaac6gadaqadaqaamaalaaabaGaeqiUde3aaS baaKqbGeaacaaIXaaajuaGbeaacqGHRaWkcaWG4bWaaWbaaeqajuai baGaaGOmaaaaaKqbagaacqaH4oqCdaWgaaqcfasaaiaaikdaaKqbag qaaiabgUcaRiaadIhadaahaaqabKqbGeaacaaIYaaaaaaaaKqbakaa wIcacaGLPaaacqGHsisldaqadaqaaiabeI7aXnaaBaaajuaibaGaaG ymaaqcfayabaGaeyOeI0IaeqiUde3aaSbaaKqbGeaacaaIYaaajuaG beaaaiaawIcacaGLPaaacaWG4baaaa@8367@

This gives                         d dx ln f X ( x,θ ) f Y ( x,θ ) = 2( θ 1 θ 2 ) ( θ 1 + x 2 )( θ 2 + x 2 ) ( θ 1 θ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacaWGKbaabaGaamizaiaadIhaaaGaciiBaiaac6gadaWcbaqaaiaa dAgadaWgaaqcfasaaiaadIfaaKqbagqaamaabmaabaGaamiEaiaacY cacqaH4oqCaiaawIcacaGLPaaaaeaacaWGMbWaaSbaaKqbGeaacaWG zbaajuaGbeaadaqadaqaaiaadIhacaGGSaGaeqiUdehacaGLOaGaay zkaaaaaiabg2da9iabgkHiTmaalaaabaGaaGOmamaabmaabaGaeqiU de3aaSbaaKqbGeaacaaIXaaajuaGbeaacqGHsislcqaH4oqCdaWgaa qcfasaaiaaikdaaKqbagqaaaGaayjkaiaawMcaaaqaamaabmaabaGa eqiUde3aaSbaaKqbGeaacaaIXaaajuaGbeaacqGHRaWkcaWG4bWaaW baaeqajuaibaGaaGOmaaaaaKqbakaawIcacaGLPaaadaqadaqaaiab eI7aXnaaBaaajuaibaGaaGOmaaqcfayabaGaey4kaSIaamiEamaaCa aabeqcfasaaiaaikdaaaaajuaGcaGLOaGaayzkaaaaaiabgkHiTmaa bmaabaGaeqiUde3aaSbaaKqbGeaacaaIXaaajuaGbeaacqGHsislcq aH4oqCdaWgaaqcfasaaiaaikdaaKqbagqaaaGaayjkaiaawMcaaaaa @70EE@  

Thus for θ 1 > θ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde 3aaSbaaKqbGeaacaaIXaaajuaGbeaacqGH+aGpcqaH4oqCdaWgaaqc fasaaiaaikdaaKqbagqaaaaa@3E29@ , d dx ln f X ( x,θ ) f Y ( x,θ ) <0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacaWGKbaabaGaamizaiaadIhaaaGaciiBaiaac6gadaWcbaqaaiaa dAgadaWgaaqcfasaaiaadIfaaKqbagqaamaabmaabaGaamiEaiaacY cacqaH4oqCaiaawIcacaGLPaaaaeaacaWGMbWaaSbaaKqbGeaacaWG zbaajuaGbeaadaqadaqaaiaadIhacaGGSaGaeqiUdehacaGLOaGaay zkaaaaaiabgYda8iaaicdaaaa@4C39@ . This means that X lr Y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiwai abgsMiJoaaBaaajuaibaGaamiBaiaadkhaaKqbagqaaiaadMfaaaa@3CB9@ and hence X hr Y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiwai abgsMiJoaaBaaajuaibaGaamiAaiaadkhaaKqbagqaaiaadMfaaaa@3CB5@ , X mrl Y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiwai abgsMiJoaaBaaajuaibaGaamyBaiaadkhacaWGSbaajuaGbeaacaWG zbaaaa@3DAB@ and X st Y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiwai abgsMiJoaaBaaajuaibaGaam4CaiaadshaaKqbagqaaiaadMfaaaa@3CC2@ .

Mean Deviations

Generally the amount of scatter in a population is measured to some extent by the totality of deviations usually from their mean and median. These are known as the mean deviation about the mean and the mean deviation about the median defined as

δ 1 ( X )= 0 | xμ | f( x )dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiTdq 2aaSbaaKqbGeaacaaIXaaajuaGbeaadaqadaqaaiaadIfaaiaawIca caGLPaaacqGH9aqpdaWdXbqaamaaemaabaGaamiEaiabgkHiTiabeY 7aTbGaay5bSlaawIa7aaqcfasaaiaaicdaaeaacqGHEisPaKqbakab gUIiYdGaaGPaVlaadAgadaqadaqaaiaadIhaaiaawIcacaGLPaaaca WGKbGaamiEaaaa@5009@  and δ 2 ( X )= 0 | xM | f( x )dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiTdq 2aaSbaaKqbGeaacaaIYaaajuaGbeaadaqadaqaaiaadIfaaiaawIca caGLPaaacqGH9aqpdaWdXbqaamaaemaabaGaamiEaiabgkHiTiaad2 eaaiaawEa7caGLiWoaaKqbGeaacaaIWaaabaGaeyOhIukajuaGcqGH RiI8aiaaykW7caWGMbWaaeWaaeaacaWG4baacaGLOaGaayzkaaGaam izaiaadIhaaaa@4F26@ , respectively, where μ=E( X ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 Maeyypa0JaamyramaabmaabaGaamiwaaGaayjkaiaawMcaaaaa@3C70@  and M=Median ( X ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamytai abg2da9iaab2eacaqGLbGaaeizaiaabMgacaqGHbGaaeOBaiaabcca daqadaqaaiaadIfaaiaawIcacaGLPaaaaaa@40C5@ . The measures δ 1 ( X ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiTdq 2aaSbaaeaacaaIXaaabeaadaqadaqaaiaadIfaaiaawIcacaGLPaaa aaa@3B6B@  and δ 2 ( X ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiTdq 2aaSbaaeaacaaIYaaabeaadaqadaqaaiaadIfaaiaawIcacaGLPaaa aaa@3B6C@ can be calculated using the following relationships

δ 1 ( X )= 0 μ ( μx ) f( x )dx+ μ ( xμ ) f( x )dxa MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiTdq 2aaSbaaKqbGeaacaaIXaaajuaGbeaadaqadaqaaiaadIfaaiaawIca caGLPaaacqGH9aqpdaWdXbqaamaabmaabaGaeqiVd0MaeyOeI0Iaam iEaaGaayjkaiaawMcaaaqcfasaaiaaicdaaeaacqaH8oqBaKqbakab gUIiYdGaamOzamaabmaabaGaamiEaaGaayjkaiaawMcaaiaadsgaca WG4bGaey4kaSYaa8qCaeaadaqadaqaaiaadIhacqGHsislcqaH8oqB aiaawIcacaGLPaaaaKqbGeaacqaH8oqBaeaacqGHEisPaKqbakabgU IiYdGaamOzamaabmaabaGaamiEaaGaayjkaiaawMcaaiaadsgacaWG 4bGaamyyaaaa@5FA6@

=μF( μ ) 0 μ xf( x )dx μ[ 1F( μ ) ]+ μ x f( x )dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyypa0 JaeqiVd0MaamOramaabmaabaGaeqiVd0gacaGLOaGaayzkaaGaeyOe I0Yaa8qCaeaacaWG4bGaaGPaVlaadAgadaqadaqaaiaadIhaaiaawI cacaGLPaaacaWGKbGaamiEaaqcfasaaiaaicdaaeaacqaH8oqBaKqb akabgUIiYdGaeyOeI0IaeqiVd02aamWaaeaacaaIXaGaeyOeI0Iaam OramaabmaabaGaeqiVd0gacaGLOaGaayzkaaaacaGLBbGaayzxaaGa ey4kaSYaa8qCaeaacaWG4bGaaGPaVdqcfasaaiabeY7aTbqaaiabg6 HiLcqcfaOaey4kIipacaWGMbWaaeWaaeaacaWG4baacaGLOaGaayzk aaGaamizaiaadIhaaaa@64CF@

=2μF( μ )2μ+2 μ x f( x )dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyypa0 JaaGOmaiabeY7aTjaadAeadaqadaqaaiabeY7aTbGaayjkaiaawMca aiabgkHiTiaaikdacqaH8oqBcqGHRaWkcaaIYaWaa8qCaeaacaWG4b GaaGPaVdqcfasaaiabeY7aTbqaaiabg6HiLcqcfaOaey4kIipacaWG MbWaaeWaaeaacaWG4baacaGLOaGaayzkaaGaamizaiaadIhaaaa@5116@

=2μF( μ )2 0 μ x f( x )dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyypa0 JaaGOmaiabeY7aTjaadAeadaqadaqaaiabeY7aTbGaayjkaiaawMca aiabgkHiTiaaikdadaWdXbqaaiaadIhacaaMc8oajuaibaGaaGimaa qaaiabeY7aTbqcfaOaey4kIipacaWGMbWaaeWaaeaacaWG4baacaGL OaGaayzkaaGaamizaiaadIhaaaa@4D0B@                                                                     (7.1)

and

δ 2 ( X )= 0 M ( Mx ) f( x )dx+ M ( xM ) f( x )dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiTdq 2aaSbaaKqbGeaacaaIYaaajuaGbeaadaqadaqaaiaadIfaaiaawIca caGLPaaacqGH9aqpdaWdXbqaamaabmaabaGaamytaiabgkHiTiaadI haaiaawIcacaGLPaaaaKqbGeaacaaIWaaabaGaamytaaqcfaOaey4k IipacaWGMbWaaeWaaeaacaWG4baacaGLOaGaayzkaaGaamizaiaadI hacqGHRaWkdaWdXbqaamaabmaabaGaamiEaiabgkHiTiaad2eaaiaa wIcacaGLPaaaaKqbGeaacaWGnbaabaGaeyOhIukajuaGcqGHRiI8ai aadAgadaqadaqaaiaadIhaaiaawIcacaGLPaaacaWGKbGaamiEaaaa @5B31@

=MF( M ) 0 M xf( x )dx M[ 1F( M ) ]+ M x f( x )dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyypa0 JaamytaiaaykW7caWGgbWaaeWaaeaacaWGnbaacaGLOaGaayzkaaGa eyOeI0Yaa8qCaeaacaWG4bGaaGPaVlaadAgadaqadaqaaiaadIhaai aawIcacaGLPaaacaWGKbGaamiEaaqcfasaaiaaicdaaeaacaWGnbaa juaGcqGHRiI8aiabgkHiTiaad2eadaWadaqaaiaaigdacqGHsislca WGgbWaaeWaaeaacaWGnbaacaGLOaGaayzkaaaacaGLBbGaayzxaaGa ey4kaSYaa8qCaeaacaWG4bGaaGPaVdqcfasaaiaad2eaaeaacqGHEi sPaKqbakabgUIiYdGaamOzamaabmaabaGaamiEaaGaayjkaiaawMca aiaadsgacaWG4baaaa@6102@

=μ+2 M x f( x )dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyypa0 JaeyOeI0IaeqiVd0Maey4kaSIaaGOmamaapehabaGaamiEaiaaykW7 aKqbGeaacaWGnbaabaGaeyOhIukajuaGcqGHRiI8aiaadAgadaqada qaaiaadIhaaiaawIcacaGLPaaacaWGKbGaamiEaaaa@48FA@

=μ2 0 M x f( x )dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyypa0 JaeqiVd0MaeyOeI0IaaGOmamaapehabaGaamiEaiaaykW7aKqbGeaa caaIWaaabaGaamytaaqcfaOaey4kIipacaWGMbWaaeWaaeaacaWG4b aacaGLOaGaayzkaaGaamizaiaadIhaaaa@4761@                                                                         (7.2)

Using p.d.f. (1.5) and expression for the mean of Ishita distribution, we get

0 μ x f 3 ( x,θ )dx=μ { θ 4 μ+ θ 2 ( μ 3 +1 )+3 θ 2 μ 2 +6( θμ+1 ) } e θμ θ( θ 3 +2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aa8qCae aacaWG4bGaaGPaVdqcfasaaiaaicdaaeaacqaH8oqBaKqbakabgUIi YdGaamOzamaaBaaajuaibaGaaG4maaqcfayabaWaaeWaaeaacaWG4b GaaiilaiabeI7aXbGaayjkaiaawMcaaiaadsgacaWG4bGaeyypa0Ja eqiVd0MaeyOeI0YaaSaaaeaadaGadaqaaiabeI7aXnaaCaaabeqcfa saaiaaisdaaaqcfaOaeqiVd0Maey4kaSIaeqiUde3aaWbaaeqajuai baGaaGOmaaaajuaGdaqadaqaaiabeY7aTnaaCaaabeqcfasaaiaaio daaaqcfaOaey4kaSIaaGymaaGaayjkaiaawMcaaiabgUcaRiaaioda cqaH4oqCdaahaaqabKqbGeaacaaIYaaaaKqbakabeY7aTnaaCaaabe qcfasaaiaaikdaaaqcfaOaey4kaSIaaGOnaiaaykW7daqadaqaaiab eI7aXjaaykW7cqaH8oqBcqGHRaWkcaaIXaaacaGLOaGaayzkaaaaca GL7bGaayzFaaGaamyzamaaCaaabeqcfasaaiabgkHiTiabeI7aXjaa ykW7cqaH8oqBaaaajuaGbaGaeqiUde3aaeWaaeaacqaH4oqCdaahaa qabKqbGeaacaaIZaaaaKqbakabgUcaRiaaikdaaiaawIcacaGLPaaa aaaaaa@80DE@                             (7.3)  

       0 M x f 3 ( x,θ )dx=μ { θ 4 M+ θ 3 ( M 3 +1 )+3 θ 2 M 2 +6( θM+1 ) } e θM θ( θ 3 +2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aa8qCae aacaWG4bGaaGPaVdqcfasaaiaaicdaaeaacaWGnbaajuaGcqGHRiI8 aiaadAgadaWgaaqcfasaaiaaiodaaKqbagqaamaabmaabaGaamiEai aacYcacqaH4oqCaiaawIcacaGLPaaacaWGKbGaamiEaiabg2da9iab eY7aTjabgkHiTmaalaaabaWaaiWaaeaacqaH4oqCdaahaaqabKqbGe aacaaI0aaaaKqbakaad2eacqGHRaWkcqaH4oqCdaahaaqabKqbGeaa caaIZaaaaKqbaoaabmaabaGaamytamaaCaaabeqcfasaaiaaiodaaa qcfaOaey4kaSIaaGymaaGaayjkaiaawMcaaiabgUcaRiaaiodacaaM c8UaeqiUde3aaWbaaeqajuaibaGaaGOmaaaajuaGcaWGnbWaaWbaae qajuaibaGaaGOmaaaajuaGcqGHRaWkcaaI2aGaaGPaVpaabmaabaGa eqiUdeNaamytaiabgUcaRiaaigdaaiaawIcacaGLPaaaaiaawUhaca GL9baacaWGLbWaaWbaaeqajuaibaGaeyOeI0IaeqiUdeNaaGPaVlaa d2eaaaaajuaGbaGaeqiUde3aaeWaaeaacqaH4oqCdaahaaqabKqbGe aacaaIZaaaaKqbakabgUcaRiaaikdaaiaawIcacaGLPaaaaaaaaa@7B87@                           (7.4)

Using expressions from (5.1), (5.2), (5.3), and (5.4), the mean deviation about mean, δ 1 ( X ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiTdq 2aaSbaaeaacaaIXaaabeaadaqadaqaaiaadIfaaiaawIcacaGLPaaa aaa@3B6B@  and the mean deviation about median, δ 2 ( X ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiTdq 2aaSbaaeaacaaIYaaabeaadaqadaqaaiaadIfaaiaawIcacaGLPaaa aaa@3B6C@  of Ishita distribution are obtained as

                           δ 1 ( X )= 2{ θ 2 μ 2 +4θμ+( θ 3 +6 ) } e θμ θ( θ 3 +2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiTdq 2aaSbaaKqbGeaacaaIXaaajuaGbeaadaqadaqaaiaadIfaaiaawIca caGLPaaacqGH9aqpdaWcaaqaaiaaikdadaGadaqaaiabeI7aXnaaCa aabeqcfasaaiaaikdaaaqcfaOaeqiVd02aaWbaaeqajuaibaGaaGOm aaaajuaGcqGHRaWkcaaI0aGaeqiUdeNaaGPaVlabeY7aTjabgUcaRm aabmaabaGaeqiUde3aaWbaaeqajuaibaGaaG4maaaajuaGcqGHRaWk caaI2aaacaGLOaGaayzkaaaacaGL7bGaayzFaaGaamyzamaaCaaabe qcfasaaiabgkHiTiabeI7aXjaaykW7cqaH8oqBaaaajuaGbaGaeqiU de3aaeWaaeaacqaH4oqCdaahaaqabKqbGeaacaaIZaaaaKqbakabgU caRiaaikdaaiaawIcacaGLPaaaaaaaaa@6499@                                                          (7.5)

δ 2 ( X )= 2{ θ 4 M+ θ 3 ( M 3 +1 )+3 θ 2 M 2 +6( θM+1 ) } e θM θ( θ 3 +2 ) μ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiTdq 2aaSbaaKqbGeaacaaIYaaajuaGbeaadaqadaqaaiaadIfaaiaawIca caGLPaaacqGH9aqpdaWcaaqaaiaaikdadaGadaqaaiabeI7aXnaaCa aabeqcfasaaiaaisdaaaqcfaOaamytaiabgUcaRiabeI7aXnaaCaaa beqcfasaaiaaiodaaaqcfa4aaeWaaeaacaWGnbWaaWbaaeqajuaiba GaaG4maaaajuaGcqGHRaWkcaaIXaaacaGLOaGaayzkaaGaey4kaSIa aG4maiaaykW7cqaH4oqCdaahaaqabKqbGeaacaaIYaaaaKqbakaad2 eadaahaaqabKqbGeaacaaIYaaaaKqbakabgUcaRiaaiAdacaaMc8+a aeWaaeaacqaH4oqCcaWGnbGaey4kaSIaaGymaaGaayjkaiaawMcaaa Gaay5Eaiaaw2haaiaadwgadaahaaqabKqbGeaacqGHsislcqaH4oqC caaMc8UaamytaaaaaKqbagaacqaH4oqCdaqadaqaaiabeI7aXnaaCa aabeqcfasaaiaaiodaaaqcfaOaey4kaSIaaGOmaaGaayjkaiaawMca aaaacqGHsislcqaH8oqBaaa@716F@                                         (7.6)

Order Statistics

Let X 1 , X 2 ,..., X n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiwam aaBaaabaGaaGymaaqabaGaaiilaiaaykW7caWGybWaaSbaaeaacaaI YaaabeaacaGGSaGaaGPaVlaac6cacaGGUaGaaiOlaiaacYcacaaMc8 UaamiwamaaBaaabaGaamOBaaqabaaaaa@44AF@  be a random sample of size n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOBaa aa@3777@  from Ishita distribution (3.5). Let X ( 1 ) < X ( 2 ) <...< X ( n ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiwam aaBaaajuaibaqcfa4aaeWaaKqbGeaacaaIXaaacaGLOaGaayzkaaaa juaGbeaacqGH8aapcaWGybWaaSbaaKqbGeaajuaGdaqadaqcfasaai aaikdaaiaawIcacaGLPaaaaKqbagqaaiabgYda8iaaykW7caaMc8Ua aiOlaiaac6cacaGGUaGaaGPaVlaaykW7cqGH8aapcaWGybWaaSbaaK qbGeaajuaGdaqadaqcfasaaiaad6gaaiaawIcacaGLPaaaaKqbagqa aaaa@5039@ denote the corresponding order statistics. The p.d.f. and the c.d.f. of the k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Aaaaa@36E6@ th order statistic, say Y= X ( k ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamywai abg2da9iaadIfadaWgaaqcfasaaKqbaoaabmaajuaibaGaam4AaaGa ayjkaiaawMcaaaqcfayabaaaaa@3D57@ are given by

f Y ( y )= n! ( k1 )!( nk )! [ F( y ) ] k1 [ 1F( y ) ] nk f( y ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aaBaaajuaibaGaamywaaqcfayabaWaaeWaaeaacaWG5baacaGLOaGa ayzkaaGaeyypa0ZaaSaaaeaacaWGUbGaaiyiaaqaamaabmaabaGaam 4AaiabgkHiTiaaigdaaiaawIcacaGLPaaacaGGHaGaaGPaVpaabmaa baGaamOBaiabgkHiTiaadUgaaiaawIcacaGLPaaacaGGHaaaaiaayk W7daWadaqaaiaadAeadaqadaqaaiaadMhaaiaawIcacaGLPaaaaiaa wUfacaGLDbaadaahaaqabKqbGeaacaWGRbGaeyOeI0IaaGymaaaaju aGdaWadaqaaiaaigdacqGHsislcaWGgbWaaeWaaeaacaWG5baacaGL OaGaayzkaaaacaGLBbGaayzxaaWaaWbaaeqajuaibaGaamOBaiabgk HiTiaadUgaaaqcfaOaamOzamaabmaabaGaamyEaaGaayjkaiaawMca aaaa@61FF@

= n! ( k1 )!( nk )! l=0 nk ( nk l ) ( 1 ) l [ F( y ) ] k+l1 f( y ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyypa0 ZaaSaaaeaacaWGUbGaaiyiaaqaamaabmaabaGaam4AaiabgkHiTiaa igdaaiaawIcacaGLPaaacaGGHaGaaGPaVpaabmaabaGaamOBaiabgk HiTiaadUgaaiaawIcacaGLPaaacaGGHaaaaiaaykW7daaeWbqaamaa bmaabaqbaeqabiqaaaqaaiaad6gacqGHsislcaWGRbaabaGaamiBaa aaaiaawIcacaGLPaaaaKqbGeaacaWGSbGaeyypa0JaaGimaaqaaiaa d6gacqGHsislcaWGRbaajuaGcqGHris5amaabmaabaGaeyOeI0IaaG ymaaGaayjkaiaawMcaamaaCaaabeqcfasaaiaadYgaaaqcfa4aamWa aeaacaWGgbWaaeWaaeaacaWG5baacaGLOaGaayzkaaaacaGLBbGaay zxaaWaaWbaaeqajuaibaGaam4AaiabgUcaRiaadYgacqGHsislcaaI XaaaaKqbakaadAgadaqadaqaaiaadMhaaiaawIcacaGLPaaaaaa@66C9@

and

F Y ( y )= j=k n ( n j ) [ F( y ) ] j [ 1F( y ) ] nj MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOram aaBaaajuaibaGaamywaaqcfayabaWaaeWaaeaacaWG5baacaGLOaGa ayzkaaGaeyypa0ZaaabCaeaadaqadaqaauaabeqaceaaaeaacaWGUb aabaGaamOAaaaaaiaawIcacaGLPaaaaKqbGeaacaWGQbGaeyypa0Ja am4Aaaqaaiaad6gaaKqbakabggHiLdGaaGPaVpaadmaabaGaamOram aabmaabaGaamyEaaGaayjkaiaawMcaaaGaay5waiaaw2faamaaCaaa beqcfasaaiaadQgaaaqcfa4aamWaaeaacaaIXaGaeyOeI0IaamOram aabmaabaGaamyEaaGaayjkaiaawMcaaaGaay5waiaaw2faamaaCaaa beqcfasaaiaad6gacqGHsislcaWGQbaaaaaa@5974@

= j=k n l=0 nj ( n j ) ( nj l ) ( 1 ) l [ F( y ) ] j+l MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyypa0 ZaaabCaeaadaaeWbqaamaabmaabaqbaeqabiqaaaqaaiaad6gaaeaa caWGQbaaaaGaayjkaiaawMcaaaqcfasaaiaadYgacqGH9aqpcaaIWa aabaGaamOBaiabgkHiTiaadQgaaKqbakabggHiLdWaaeWaaeaafaqa beGabaaabaGaamOBaiabgkHiTiaadQgaaeaacaWGSbaaaaGaayjkai aawMcaaaqcfasaaiaadQgacqGH9aqpcaWGRbaabaGaamOBaaqcfaOa eyyeIuoacaaMc8+aaeWaaeaacqGHsislcaaIXaaacaGLOaGaayzkaa WaaWbaaeqajuaibaGaamiBaaaajuaGdaWadaqaaiaadAeadaqadaqa aiaadMhaaiaawIcacaGLPaaaaiaawUfacaGLDbaadaahaaqabKqbGe aacaWGQbGaey4kaSIaamiBaaaaaaa@5E5D@ ,

respectively, for k=1,2,3,...,n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4Aai abg2da9iaaigdacaGGSaGaaGOmaiaacYcacaaIZaGaaiilaiaac6ca caGGUaGaaiOlaiaacYcacaWGUbaaaa@4077@ .

 Thus, the p.d.f. and the c.d.f of k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4Aaa aa@3774@ th order statistic of Ishita distribution (3.5) are given by

f Y ( y )= n! θ 3 ( θ+ x 2 ) e θx ( θ 3 +2 )( k1 )!( nk )! l=0 nk ( nk l ) × [ 1 θx( θx+2 )+( θ 3 +2 ) θ 3 +2 e θx ] k+l1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aaBaaajuaibaGaamywaaqcfayabaWaaeWaaeaacaWG5baacaGLOaGa ayzkaaGaeyypa0ZaaSaaaeaacaWGUbGaaiyiaiabeI7aXnaaCaaabe qcfasaaiaaiodaaaqcfa4aaeWaaeaacqaH4oqCcqGHRaWkcaWG4bWa aWbaaeqajuaibaGaaGOmaaaaaKqbakaawIcacaGLPaaacaWGLbWaaW baaeqajuaibaGaeyOeI0IaeqiUdeNaamiEaaaaaKqbagaadaqadaqa aiabeI7aXnaaCaaabeqcfasaaiaaiodaaaqcfaOaey4kaSIaaGOmaa GaayjkaiaawMcaamaabmaabaGaam4AaiabgkHiTiaaigdaaiaawIca caGLPaaacaGGHaGaaGPaVpaabmaabaGaamOBaiabgkHiTiaadUgaai aawIcacaGLPaaacaGGHaaaaiaaykW7daaeWbqaamaabmaabaqbaeqa biqaaaqaaiaad6gacqGHsislcaWGRbaabaGaamiBaaaaaiaawIcaca GLPaaaaKqbGeaacaWGSbGaeyypa0JaaGimaaqaaiaad6gacqGHsisl caWGRbaajuaGcqGHris5aiabgEna0oaadmaabaGaaGymaiabgkHiTm aalaaabaGaeqiUdeNaamiEamaabmaabaGaeqiUdeNaamiEaiabgUca RiaaikdaaiaawIcacaGLPaaacqGHRaWkdaqadaqaaiabeI7aXnaaCa aabeqcfasaaiaaiodaaaqcfaOaey4kaSIaaGOmaaGaayjkaiaawMca aaqaaiabeI7aXnaaCaaabeqcfasaaiaaiodaaaqcfaOaey4kaSIaaG OmaaaacaWGLbWaaWbaaeqajuaibaGaeyOeI0IaeqiUdeNaamiEaaaa aKqbakaawUfacaGLDbaadaahaaqabKqbGeaacaWGRbGaey4kaSIaam iBaiabgkHiTiaaigdaaaaaaa@939A@

and

F Y ( y )= j=k n l=0 nj ( n j ) ( nj l ) ( 1 ) l [ 1 θx( θx+2 )+( θ 3 +2 ) θ 3 +2 e θx ] j+l MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOram aaBaaajuaibaGaamywaaqcfayabaWaaeWaaeaacaWG5baacaGLOaGa ayzkaaGaeyypa0ZaaabCaeaadaaeWbqaamaabmaabaqbaeqabiqaaa qaaiaad6gaaeaacaWGQbaaaaGaayjkaiaawMcaaaqcfasaaiaadYga cqGH9aqpcaaIWaaabaGaamOBaiabgkHiTiaadQgaaKqbakabggHiLd WaaeWaaeaafaqabeGabaaabaGaamOBaiabgkHiTiaadQgaaeaacaWG SbaaaaGaayjkaiaawMcaaaqcfasaaiaadQgacqGH9aqpcaWGRbaaba GaamOBaaqcfaOaeyyeIuoacaaMc8+aaeWaaeaacqGHsislcaaIXaaa caGLOaGaayzkaaWaaWbaaeqajuaibaGaamiBaaaajuaGdaWadaqaai aaigdacqGHsisldaWcaaqaaiabeI7aXjaadIhadaqadaqaaiabeI7a XjaadIhacqGHRaWkcaaIYaaacaGLOaGaayzkaaGaey4kaSYaaeWaae aacqaH4oqCdaahaaqabKqbGeaacaaIZaaaaKqbakabgUcaRiaaikda aiaawIcacaGLPaaaaeaacqaH4oqCdaahaaqabKqbGeaacaaIZaaaaK qbakabgUcaRiaaikdaaaGaamyzamaaCaaabeqcfasaaiabgkHiTiab eI7aXjaadIhaaaaajuaGcaGLBbGaayzxaaWaaWbaaeqajuaibaGaam OAaiabgUcaRiaadYgaaaaaaa@7C0E@

Bonferroni And Lorenz Curves

The Bonferroni and Lorenz curves Bonferroni [19] and Bonferroni and Gini indices have applications not only in economics to study income and poverty, but also in other fields like reliability, demography, insurance and medicine. The Bonferroni and Lorenz curves are defined as

B( p )= 1 pμ 0 q xf( x ) dx= 1 pμ [ 0 xf( x )dx q xf( x ) dx ]= 1 pμ [ μ q xf( x ) dx ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOqam aabmaabaGaamiCaaGaayjkaiaawMcaaiabg2da9maalaaabaGaaGym aaqaaiaadchacqaH8oqBaaWaa8qCaeaacaWG4bGaaGPaVlaadAgada qadaqaaiaadIhaaiaawIcacaGLPaaacaaMc8oajuaibaGaaGimaaqa aiaadghaaKqbakabgUIiYdGaamizaiaadIhacqGH9aqpdaWcaaqaai aaigdaaeaacaWGWbGaeqiVd0gaamaadmaabaWaa8qCaeaacaWG4bGa aGPaVlaadAgadaqadaqaaiaadIhaaiaawIcacaGLPaaacaWGKbGaam iEaiabgkHiTaqcfasaaiaaicdaaeaacqGHEisPaKqbakabgUIiYdWa a8qCaeaacaWG4bGaaGPaVlaadAgadaqadaqaaiaadIhaaiaawIcaca GLPaaaaKqbGeaacaWGXbaabaGaeyOhIukajuaGcqGHRiI8aiaaykW7 caWGKbGaamiEaaGaay5waiaaw2faaiabg2da9maalaaabaGaaGymaa qaaiaadchacqaH8oqBaaWaamWaaeaacqaH8oqBcqGHsisldaWdXbqa aiaadIhacaaMc8UaamOzamaabmaabaGaamiEaaGaayjkaiaawMcaaa qcfasaaiaadghaaeaacqGHEisPaKqbakabgUIiYdGaaGPaVlaadsga caWG4baacaGLBbGaayzxaaaaaa@87AB@          (9.1)

and L( p )= 1 μ 0 q xf( x ) dx= 1 μ [ 0 xf( x )dx q xf( x ) dx ]= 1 μ [ μ q xf( x ) dx ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamitam aabmaabaGaamiCaaGaayjkaiaawMcaaiabg2da9maalaaabaGaaGym aaqaaiabeY7aTbaadaWdXbqaaiaadIhacaaMc8UaamOzamaabmaaba GaamiEaaGaayjkaiaawMcaaaqcfasaaiaaicdaaeaacaWGXbaajuaG cqGHRiI8aiaaykW7caWGKbGaamiEaiabg2da9maalaaabaGaaGymaa qaaiabeY7aTbaadaWadaqaamaapehabaGaamiEaiaaykW7caWGMbWa aeWaaeaacaWG4baacaGLOaGaayzkaaGaamizaiaadIhacqGHsislaK qbGeaacaaIWaaabaGaeyOhIukajuaGcqGHRiI8amaapehabaGaamiE aiaaykW7caWGMbWaaeWaaeaacaWG4baacaGLOaGaayzkaaGaaGPaVd qcfasaaiaadghaaeaacqGHEisPaKqbakabgUIiYdGaamizaiaadIha aiaawUfacaGLDbaacqGH9aqpdaWcaaqaaiaaigdaaeaacqaH8oqBaa WaamWaaeaacqaH8oqBcqGHsisldaWdXbqaaiaadIhacaaMc8UaamOz amaabmaabaGaamiEaaGaayjkaiaawMcaaaqcfasaaiaadghaaeaacq GHEisPaKqbakabgUIiYdGaaGPaVlaadsgacaWG4baacaGLBbGaayzx aaaaaa@84D6@  9.2)

respectively or equivalently

< B( p )= 1 pμ 0 p F 1 ( x ) dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOqam aabmaabaGaamiCaaGaayjkaiaawMcaaiabg2da9maalaaabaGaaGym aaqaaiaadchacqaH8oqBaaWaa8qCaeaacaWGgbWaaWbaaeqajuaiba GaeyOeI0IaaGymaaaajuaGdaqadaqaaiaadIhaaiaawIcacaGLPaaa aKqbGeaacaaIWaaabaGaamiCaaqcfaOaey4kIipacaaMc8Uaamizai aadIhaaaa@4C49@                               (9.3)

and                                             L( p )= 1 μ 0 p F 1 ( x ) dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamitam aabmaabaGaamiCaaGaayjkaiaawMcaaiabg2da9maalaaabaGaaGym aaqaaiabeY7aTbaadaWdXbqaaiaadAeadaahaaqabKqbGeaacqGHsi slcaaIXaaaaKqbaoaabmaabaGaamiEaaGaayjkaiaawMcaaaqcfasa aiaaicdaaeaacaWGWbaajuaGcqGHRiI8aiaaykW7caWGKbGaamiEaa aa@4B5E@           (9.4)

respectively, where μ=E( X ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 Maeyypa0JaamyramaabmaabaGaamiwaaGaayjkaiaawMcaaaaa@3C70@  and q= F 1 ( p ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyCai abg2da9iaadAeadaahaaqabKqbGeaacqGHsislcaaIXaaaaKqbaoaa bmaabaGaamiCaaGaayjkaiaawMcaaaaa@3E4F@ .

The Bonferroni and Gini indices are thus defined as

B=1 0 1 B( p ) dp MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOqai abg2da9iaaigdacqGHsisldaWdXbqaaiaadkeadaqadaqaaiaadcha aiaawIcacaGLPaaaaKqbGeaacaaIWaaabaGaaGymaaqcfaOaey4kIi pacaaMc8Uaamizaiaadchaaaa@4529@                        (9.5)

And                                              G=12 0 1 L( p ) dp MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4rai abg2da9iaaigdacqGHsislcaaIYaWaa8qCaeaacaWGmbWaaeWaaeaa caWGWbaacaGLOaGaayzkaaGaaGPaVdqcfasaaiaaicdaaeaacaaIXa aajuaGcqGHRiI8aiaadsgacaWGWbaaaa@45F4@             (9.6)

respectively.

Using p.d.f. (1.5), we get

q x f 3 ( x,θ ) dx= { θ 4 q+ θ 3 ( q 3 +1 )+3 θ 2 q 2 +6( θq+1 ) } e θq θ( θ 3 +2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aa8qCae aacaWG4bGaaGPaVlaadAgadaWgaaqcfasaaiaaiodaaKqbagqaamaa bmaabaGaamiEaiaacYcacqaH4oqCaiaawIcacaGLPaaaaKqbGeaaca WGXbaabaGaeyOhIukajuaGcqGHRiI8aiaaykW7caWGKbGaamiEaiab g2da9maalaaabaWaaiWaaeaacqaH4oqCdaahaaqabKqbGeaacaaI0a aaaKqbakaadghacqGHRaWkcqaH4oqCdaahaaqabKqbGeaacaaIZaaa aKqbaoaabmaabaGaamyCamaaCaaabeqcfasaaiaaiodaaaqcfaOaey 4kaSIaaGymaaGaayjkaiaawMcaaiabgUcaRiaaiodacqaH4oqCdaah aaqabKqbGeaacaaIYaaaaKqbakaadghadaahaaqabKqbGeaacaaIYa aaaKqbakabgUcaRiaaiAdadaqadaqaaiabeI7aXjaadghacqGHRaWk caaIXaaacaGLOaGaayzkaaaacaGL7bGaayzFaaGaamyzamaaCaaabe qcfasaaiabgkHiTiabeI7aXjaadghaaaaajuaGbaGaeqiUde3aaeWa aeaacqaH4oqCdaahaaqabKqbGeaacaaIZaaaaKqbakabgUcaRiaaik daaiaawIcacaGLPaaaaaaaaa@775D@                    (9.7)

Now using equation (7.7) in (7.1) and (7.2), we get

B( p )= 1 p [ 1 { θ 4 q+ θ 3 ( q 3 +1 )+3 θ 2 q 2 +6( θq+1 ) } e θq θ 3 +6 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOqam aabmaabaGaamiCaaGaayjkaiaawMcaaiabg2da9maalaaabaGaaGym aaqaaiaadchaaaWaamWaaeaacaaIXaGaeyOeI0YaaSaaaeaadaGada qaaiabeI7aXnaaCaaabeqcfasaaiaaisdaaaqcfaOaamyCaiabgUca RiabeI7aXnaaCaaabeqcfasaaiaaiodaaaqcfa4aaeWaaeaacaWGXb WaaWbaaeqajuaibaGaaG4maaaajuaGcqGHRaWkcaaIXaaacaGLOaGa ayzkaaGaey4kaSIaaG4maiabeI7aXnaaCaaabeqcfasaaiaaikdaaa qcfaOaamyCamaaCaaabeqcfasaaiaaikdaaaqcfaOaey4kaSIaaGOn amaabmaabaGaeqiUdeNaamyCaiabgUcaRiaaigdaaiaawIcacaGLPa aaaiaawUhacaGL9baacaWGLbWaaWbaaeqajuaibaGaeyOeI0IaeqiU deNaamyCaaaaaKqbagaacqaH4oqCdaahaaqabKqbGeaacaaIZaaaaK qbakabgUcaRiaaiAdaaaaacaGLBbGaayzxaaaaaa@69E3@                   (9.8)

and                    L( p )=1 { θ 4 q+ θ 3 ( q 3 +1 )+3 θ 2 q 2 +6( θq+1 ) } e θq θ 3 +6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamitam aabmaabaGaamiCaaGaayjkaiaawMcaaiabg2da9iaaigdacqGHsisl daWcaaqaamaacmaabaGaeqiUde3aaWbaaeqajuaibaGaaGinaaaaju aGcaWGXbGaey4kaSIaeqiUde3aaWbaaeqajuaibaGaaG4maaaajuaG daqadaqaaiaadghadaahaaqabKqbGeaacaaIZaaaaKqbakabgUcaRi aaigdaaiaawIcacaGLPaaacqGHRaWkcaaIZaGaeqiUde3aaWbaaeqa juaibaGaaGOmaaaajuaGcaWGXbWaaWbaaeqajuaibaGaaGOmaaaaju aGcqGHRaWkcaaI2aWaaeWaaeaacqaH4oqCcaWGXbGaey4kaSIaaGym aaGaayjkaiaawMcaaaGaay5Eaiaaw2haaiaadwgadaahaaqabKqbGe aacqGHsislcqaH4oqCcaWGXbaaaaqcfayaaiabeI7aXnaaCaaabeqc fasaaiaaiodaaaqcfaOaey4kaSIaaGOnaaaaaaa@663B@                                    (9.9)

Now using equations (7.8) and (7.9) in (7.5) and (7.6), the Bonferroni and Gini indices of Ishita distribution (1.5) are obtained as

B=1 { θ 4 q+ θ 3 ( q 3 +1 )+3 θ 2 q 2 +6( θq+1 ) } e θq θ 3 +6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOqai abg2da9iaaigdacqGHsisldaWcaaqaamaacmaabaGaeqiUde3aaWba aeqajuaibaGaaGinaaaajuaGcaWGXbGaey4kaSIaeqiUde3aaWbaae qajuaibaGaaG4maaaajuaGdaqadaqaaiaadghadaahaaqabKqbGeaa caaIZaaaaKqbakabgUcaRiaaigdaaiaawIcacaGLPaaacqGHRaWkca aIZaGaeqiUde3aaWbaaeqajuaibaGaaGOmaaaajuaGcaWGXbWaaWba aeqajuaibaGaaGOmaaaajuaGcqGHRaWkcaaI2aWaaeWaaeaacqaH4o qCcaWGXbGaey4kaSIaaGymaaGaayjkaiaawMcaaaGaay5Eaiaaw2ha aiaadwgadaahaaqabKqbGeaacqGHsislcqaH4oqCcaWGXbaaaaqcfa yaaiabeI7aXnaaCaaabeqcfasaaiaaiodaaaqcfaOaey4kaSIaaGOn aaaaaaa@63B3@         (9.10)

G= 2{ θ 4 q+ θ 3 ( q 3 +1 )+3 θ 2 q 2 +6( θq+1 ) } e θq θ 2 +6 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4rai abg2da9maalaaabaGaaGOmamaacmaabaGaeqiUde3aaWbaaeqajuai baGaaGinaaaajuaGcaWGXbGaey4kaSIaeqiUde3aaWbaaeqajuaiba GaaG4maaaajuaGdaqadaqaaiaadghadaahaaqabKqbGeaacaaIZaaa aKqbakabgUcaRiaaigdaaiaawIcacaGLPaaacqGHRaWkcaaIZaGaeq iUde3aaWbaaeqajuaibaGaaGOmaaaajuaGcaWGXbWaaWbaaeqajuai baGaaGOmaaaajuaGcqGHRaWkcaaI2aWaaeWaaeaacqaH4oqCcaWGXb Gaey4kaSIaaGymaaGaayjkaiaawMcaaaGaay5Eaiaaw2haaiaadwga daahaaqabKqbGeaacqGHsislcqaH4oqCcaWGXbaaaaqcfayaaiabeI 7aXnaaCaaabeqcfasaaiaaikdaaaqcfaOaey4kaSIaaGOnaaaacqGH sislcaaIXaaaaa@6473@                                    (9.11)

Renyi Entropy

An entropy of a random variable X MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiwaa aa@3761@ is a measure of variation of uncertainty. A popular entropy measure is Renyi entropy [20]. If X MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiwaa aa@3761@ is a continuous random variable having probability density function f( . ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aabmaabaGaaiOlaaGaayjkaiaawMcaaaaa@39AA@ , then Renyi entropy is defined as

T R ( γ )= 1 1γ log{ f γ ( x )dx } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaBaaajuaibaGaamOuaaqcfayabaWaaeWaaeaacqaHZoWzaiaawIca caGLPaaacqGH9aqpdaWcaaqaaiaaigdaaeaacaaIXaGaeyOeI0Iaeq 4SdCgaaiGacYgacaGGVbGaai4zamaacmaabaWaa8qaaeaacaWGMbWa aWbaaeqajuaibaGaeq4SdCgaaKqbaoaabmaabaGaamiEaaGaayjkai aawMcaaiaadsgacaWG4baabeqabiabgUIiYdaacaGL7bGaayzFaaaa aa@5021@

where γ>0andγ1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4SdC MaeyOpa4JaaGimaiaaykW7caaMc8UaaGPaVlaabggacaqGUbGaaeiz aiaaykW7caaMc8UaaGPaVlabeo7aNjabgcMi5kaaigdaaaa@4A14@ .

Thus, the Renyi entropy for the Ishita distribution (3.5) can be obtained as

                                     T R ( γ )= 1 1γ log[ 0 θ 3γ ( θ 3 +2 ) γ ( θ+ x 2 ) γ e θγx dx ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaBaaajuaibaGaamOuaaqcfayabaWaaeWaaeaacqaHZoWzaiaawIca caGLPaaacqGH9aqpdaWcaaqaaiaaigdaaeaacaaIXaGaeyOeI0Iaeq 4SdCgaaiGacYgacaGGVbGaai4zamaadmaabaWaa8qCaeaadaWcaaqa aiabeI7aXnaaCaaabeqcfasaaiaaiodacqaHZoWzaaaajuaGbaWaae WaaeaacqaH4oqCdaahaaqabKqbGeaacaaIZaaaaKqbakabgUcaRiaa ikdaaiaawIcacaGLPaaadaahaaqabKqbGeaacqaHZoWzaaaaaKqbao aabmaabaGaeqiUdeNaey4kaSIaamiEamaaCaaabeqcfasaaiaaikda aaaajuaGcaGLOaGaayzkaaWaaWbaaeqajuaibaGaeq4SdCgaaKqbak aadwgadaahaaqabKqbGeaacqGHsislcqaH4oqCcaaMc8Uaeq4SdCMa aGPaVlaadIhaaaqcfaOaamizaiaadIhaaKqbGeaacaaIWaaabaGaey OhIukajuaGcqGHRiI8aaGaay5waiaaw2faaaaa@6EA8@

= 1 1γ log[ 0 θ 3γ ( θ 3 +2 ) γ θ γ ( 1+ x 2 θ ) γ e θγx dx ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyypa0 ZaaSaaaeaacaaIXaaabaGaaGymaiabgkHiTiabeo7aNbaaciGGSbGa ai4BaiaacEgadaWadaqaamaapehabaWaaSaaaeaacqaH4oqCdaahaa qabKqbGeaacaaIZaGaeq4SdCgaaaqcfayaamaabmaabaGaeqiUde3a aWbaaeqajuaibaGaaG4maaaajuaGcqGHRaWkcaaIYaaacaGLOaGaay zkaaWaaWbaaeqajuaibaGaeq4SdCgaaaaajuaGcqaH4oqCdaahaaqa bKqbGeaacqaHZoWzaaqcfa4aaeWaaeaacaaIXaGaey4kaSYaaSaaae aacaWG4bWaaWbaaeqajuaibaGaaGOmaaaaaKqbagaacqaH4oqCaaaa caGLOaGaayzkaaWaaWbaaeqajuaibaGaeq4SdCgaaKqbakaadwgada ahaaqabKqbGeaacqGHsislcqaH4oqCcaaMc8Uaeq4SdCMaaGPaVlaa dIhaaaqcfaOaamizaiaadIhaaKqbGeaacaaIWaaabaGaeyOhIukaju aGcqGHRiI8aaGaay5waiaaw2faaaaa@6DF1@

= 1 1γ log[ 0 θ 4γ ( θ 3 +2 ) γ j=0 ( γ j ) ( x 2 θ ) j e θγx dx ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyypa0 ZaaSaaaeaacaaIXaaabaGaaGymaiabgkHiTiabeo7aNbaaciGGSbGa ai4BaiaacEgadaWadaqaamaapehabaWaaSaaaeaacqaH4oqCdaahaa qabKqbGeaacaaI0aGaeq4SdCgaaaqcfayaamaabmaabaGaeqiUde3a aWbaaeqajuaibaGaaG4maaaajuaGcqGHRaWkcaaIYaaacaGLOaGaay zkaaWaaWbaaeqajuaibaGaeq4SdCgaaaaajuaGdaaeWbqaamaabmaa baqbaeqabiqaaaqaaiabeo7aNbqaaiaadQgaaaaacaGLOaGaayzkaa aajuaibaGaamOAaiabg2da9iaaicdaaeaacqGHEisPaKqbakabggHi LdWaaeWaaeaadaWcaaqaaiaadIhadaahaaqabKqbGeaacaaIYaaaaa qcfayaaiabeI7aXbaaaiaawIcacaGLPaaadaahaaqabKqbGeaacaWG QbaaaKqbakaadwgadaahaaqabKqbGeaacqGHsislcqaH4oqCcaaMc8 Uaeq4SdCMaaGPaVlaadIhaaaqcfaOaamizaiaadIhaaKqbGeaacaaI WaaabaGaeyOhIukajuaGcqGHRiI8aaGaay5waiaaw2faaaaa@7294@

= 1 1γ log[ 0 θ 4γj ( θ 3 +2 ) γ j=0 ( γ j ) x 2j e θγx dx ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyypa0 ZaaSaaaeaacaaIXaaabaGaaGymaiabgkHiTiabeo7aNbaaciGGSbGa ai4BaiaacEgadaWadaqaamaapehabaWaaSaaaeaacqaH4oqCdaahaa qabKqbGeaacaaI0aGaeq4SdCMaeyOeI0IaamOAaaaaaKqbagaadaqa daqaaiabeI7aXnaaCaaabeqcfasaaiaaiodaaaqcfaOaey4kaSIaaG OmaaGaayjkaiaawMcaamaaCaaabeqcfasaaiabeo7aNbaaaaqcfa4a aabCaeaadaqadaqaauaabeqaceaaaeaacqaHZoWzaeaacaWGQbaaaa GaayjkaiaawMcaaaqcfasaaiaadQgacqGH9aqpcaaIWaaabaGaeyOh IukajuaGcqGHris5aiaaykW7caWG4bWaaWbaaeqajuaibaGaaGOmai aadQgaaaqcfaOaamyzamaaCaaabeqcfasaaiabgkHiTiabeI7aXjaa ykW7cqaHZoWzcaaMc8UaamiEaaaajuaGcaWGKbGaamiEaaqcfasaai aaicdaaeaacqGHEisPaKqbakabgUIiYdaacaGLBbGaayzxaaaaaa@71CE@

= 1 1γ log[ j=0 ( γ j ) θ 4γj ( θ 3 +2 ) γ 0 e θγx x 2j+11 dx ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyypa0 ZaaSaaaeaacaaIXaaabaGaaGymaiabgkHiTiabeo7aNbaaciGGSbGa ai4BaiaacEgadaWadaqaamaaqahabaWaaeWaaeaafaqabeGabaaaba Gaeq4SdCgabaGaamOAaaaaaiaawIcacaGLPaaaaKqbGeaacaWGQbGa eyypa0JaaGimaaqaaiabg6HiLcqcfaOaeyyeIuoadaWcaaqaaiabeI 7aXnaaCaaabeqcfasaaiaaisdacqaHZoWzcqGHsislcaWGQbaaaaqc fayaamaabmaabaGaeqiUde3aaWbaaeqajuaibaGaaG4maaaajuaGcq GHRaWkcaaIYaaacaGLOaGaayzkaaWaaWbaaeqajuaibaGaeq4SdCga aaaajuaGdaWdXbqaaiaadwgadaahaaqabKqbGeaacqGHsislcqaH4o qCcaaMc8Uaeq4SdCMaaGPaVlaadIhaaaaabaGaaGimaaqaaiabg6Hi LcqcfaOaey4kIipacaWG4bWaaWbaaeqajuaibaGaaGOmaiaadQgacq GHRaWkcaaIXaGaeyOeI0IaaGymaaaajuaGcaWGKbGaamiEaaGaay5w aiaaw2faaaaa@72CC@

= 1 1γ log[ j=0 ( γ j ) θ 4γj ( θ 3 +2 ) γ Γ( 2j+1 ) ( θγ ) 2j+1 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyypa0 ZaaSaaaeaacaaIXaaabaGaaGymaiabgkHiTiabeo7aNbaaciGGSbGa ai4BaiaacEgadaWadaqaamaaqahabaWaaeWaaeaafaqabeGabaaaba Gaeq4SdCgabaGaamOAaaaaaiaawIcacaGLPaaaaKqbGeaacaWGQbGa eyypa0JaaGimaaqaaiabg6HiLcqcfaOaeyyeIuoadaWcaaqaaiabeI 7aXnaaCaaabeqcfasaaiaaisdacqaHZoWzcqGHsislcaWGQbaaaaqc fayaamaabmaabaGaeqiUde3aaWbaaeqajuaibaGaaG4maaaajuaGcq GHRaWkcaaIYaaacaGLOaGaayzkaaWaaWbaaeqajuaibaGaeq4SdCga aaaajuaGdaWcaaqaaiabfo5ahnaabmaabaGaaGOmaiaadQgacqGHRa WkcaaIXaaacaGLOaGaayzkaaaabaWaaeWaaeaacqaH4oqCcqaHZoWz aiaawIcacaGLPaaadaahaaqabKqbGeaacaaIYaGaamOAaiabgUcaRi aaigdaaaaaaaqcfaOaay5waiaaw2faaaaa@6ACF@

= 1 1γ log[ j=0 ( γ j ) θ 4γ3j1 ( θ 3 +2 ) γ Γ( 2j+1 ) ( γ ) 2j+1 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyypa0 ZaaSaaaeaacaaIXaaabaGaaGymaiabgkHiTiabeo7aNbaaciGGSbGa ai4BaiaacEgadaWadaqaamaaqahabaWaaeWaaeaafaqabeGabaaaba Gaeq4SdCgabaGaamOAaaaaaiaawIcacaGLPaaaaKqbGeaacaWGQbGa eyypa0JaaGimaaqaaiabg6HiLcqcfaOaeyyeIuoadaWcaaqaaiabeI 7aXnaaCaaabeqcfasaaiaaisdacqaHZoWzcqGHsislcaaIZaGaamOA aiabgkHiTiaaigdaaaaajuaGbaWaaeWaaeaacqaH4oqCdaahaaqabK qbGeaacaaIZaaaaKqbakabgUcaRiaaikdaaiaawIcacaGLPaaadaah aaqabKqbGeaacqaHZoWzaaaaaKqbaoaalaaabaGaeu4KdC0aaeWaae aacaaIYaGaamOAaiabgUcaRiaaigdaaiaawIcacaGLPaaaaeaadaqa daqaaiabeo7aNbGaayjkaiaawMcaamaaCaaabeqcfasaaiaaikdaca WGQbGaey4kaSIaaGymaaaaaaaajuaGcaGLBbGaayzxaaaaaa@6B7E@

Stress-Strength Reliability

 The stress- strength reliability describes the life of a component which has random strength X MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiwaa aa@3761@ that is subjected to a random stress Y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamywaa aa@3762@ . When the stress applied to it exceeds the strength, the component fails instantly and the component will function satisfactorily till X>Y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiwai abg6da+iaadMfaaaa@3947@ . Therefore, R=P( Y<X ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOuai abg2da9iaadcfadaqadaqaaiaadMfacqGH8aapcaWGybaacaGLOaGa ayzkaaaaaa@3D7E@ is a measure of component reliability and in statistical literature it is known as stress-strength parameter. It has wide applications in almost all areas of knowledge especially in engineering such as structures, deterioration of rocket motors, static fatigue of ceramic components, aging of concrete pressure vessels etc.

Let X MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiwaa aa@3761@ and Y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamywaa aa@3762@ be independent strength and stress random variables having Ishita distribution (3.5) with parameter θ 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde 3aaSbaaeaacaaIXaaabeaaaaa@3916@  and θ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde 3aaSbaaeaacaaIYaaabeaaaaa@3917@  respectively. Then the stress-strength reliability R MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOuaa aa@375B@  of Ishita distribution can be obtained as

R=P( Y<X )= 0 P( Y<X|X=x ) f X ( x )dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOuai abg2da9iaadcfadaqadaqaaiaadMfacqGH8aapcaWGybaacaGLOaGa ayzkaaGaeyypa0Zaa8qCaeaacaWGqbWaaeWaaeaacaWGzbGaeyipaW JaamiwaiaacYhacaWGybGaeyypa0JaamiEaaGaayjkaiaawMcaaaqc fasaaiaaicdaaeaacqGHEisPaKqbakabgUIiYdGaamOzamaaBaaaju aibaGaamiwaaqcfayabaWaaeWaaeaacaWG4baacaGLOaGaayzkaaGa amizaiaadIhaaaa@53CA@

= 0 f 3 ( x, θ 1 ) F 3 ( x, θ 2 )dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyypa0 Zaa8qCaeaacaWGMbWaaSbaaKqbGeaacaaIZaaajuaGbeaadaqadaqa aiaadIhacaGGSaGaeqiUde3aaSbaaKqbGeaacaaIXaaajuaGbeaaai aawIcacaGLPaaaaKqbGeaacaaIWaaabaGaeyOhIukajuaGcqGHRiI8 aiaaykW7caaMc8UaamOramaaBaaajuaibaGaaG4maaqcfayabaWaae WaaeaacaWG4bGaaiilaiabeI7aXnaaBaaajuaibaGaaGOmaaqcfaya baaacaGLOaGaayzkaaGaamizaiaadIhaaaa@53B1@

=1 θ 1 3 [ 24 θ 2 2 +12 θ 2 ( θ 1 + θ 2 )+2( θ 2 3 + θ 1 θ 2 2 +2 ) ( θ 1 + θ 2 ) 2 +2 θ 1 θ 2 ( θ 1 + θ 2 ) 3 + θ 1 ( θ 2 3 +2 ) ( θ 1 + θ 2 ) 4 ] ( θ 1 3 +2 )( θ 2 3 +2 ) ( θ 1 + θ 2 ) 5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyypa0 JaaGymaiabgkHiTmaalaaabaGaeqiUde3aaSbaaKqbGeaacaaIXaaa juaGbeaadaahaaqabKqbGeaacaaIZaaaaKqbaoaadmaaeaqabeaaca aIYaGaaGinaiabeI7aXnaaBaaajuaibaGaaGOmaaqcfayabaWaaWba aeqajuaibaGaaGOmaaaajuaGcqGHRaWkcaaIXaGaaGOmaiabeI7aXn aaBaaajuaibaGaaGOmaaqcfayabaWaaeWaaeaacqaH4oqCdaWgaaqc fasaaiaaigdaaKqbagqaaiabgUcaRiabeI7aXnaaBaaajuaibaGaaG OmaaqcfayabaaacaGLOaGaayzkaaGaey4kaSIaaGOmamaabmaabaGa eqiUde3aaSbaaKqbGeaacaaIYaaajuaGbeaadaahaaqabKqbGeaaca aIZaaaaKqbakabgUcaRiabeI7aXnaaBaaajuaibaGaaGymaaqcfaya baGaeqiUde3aaSbaaKqbGeaacaaIYaaajuaGbeaadaahaaqabKqbGe aacaaIYaaaaKqbakabgUcaRiaaikdaaiaawIcacaGLPaaadaqadaqa aiabeI7aXnaaBaaajqwba+FaaiaaigdaaKqbagqaaiabgUcaRiabeI 7aXnaaBaaajuaibaGaaGOmaaqcfayabaaacaGLOaGaayzkaaWaaWba aeqajuaibaGaaGOmaaaaaKqbagaacqGHRaWkcaaIYaGaeqiUde3aaS baaKqbGeaacaaIXaaajuaGbeaacqaH4oqCdaWgaaqcfasaaiaaikda aKqbagqaamaabmaabaGaeqiUde3aaSbaaKqbGeaacaaIXaaajuaGbe aacqGHRaWkcqaH4oqCdaWgaaqcfasaaiaaikdaaKqbagqaaaGaayjk aiaawMcaamaaCaaabeqcfasaaiaaiodaaaqcfaOaey4kaSIaeqiUde 3aaSbaaKqbGeaacaaIXaaajuaGbeaadaqadaqaaiabeI7aXnaaBaaa juaibaGaaGOmaaqcfayabaWaaWbaaeqajuaibaGaaG4maaaajuaGcq GHRaWkcaaIYaaacaGLOaGaayzkaaWaaeWaaeaacqaH4oqCdaWgaaqc fasaaiaaigdaaKqbagqaaiabgUcaRiabeI7aXnaaBaaajuaibaGaaG OmaaqcfayabaaacaGLOaGaayzkaaWaaWbaaeqajuaibaGaaGinaaaa aaqcfaOaay5waiaaw2faaaqaamaabmaabaGaeqiUde3aaSbaaKqbGe aacaaIXaaajuaGbeaadaahaaqabKqbGeaacaaIZaaaaKqbakabgUca RiaaikdaaiaawIcacaGLPaaadaqadaqaaiabeI7aXnaaBaaajuaiba GaaGOmaaqcfayabaWaaWbaaeqajuaibaGaaG4maaaajuaGcqGHRaWk caaIYaaacaGLOaGaayzkaaWaaeWaaeaacqaH4oqCdaWgaaqcfasaai aaigdaaKqbagqaaiabgUcaRiabeI7aXnaaBaaajuaibaGaaGOmaaqc fayabaaacaGLOaGaayzkaaWaaWbaaeqajuaibaGaaGynaaaaaaaaaa@B85F@ .

Estimation of Parameter

12.1. Maximum likelihood estimate (MLE)
Let ( x 1 , x 2 , x 3 ,..., x n ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacaWG4bWaaSbaaeaacaaIXaaabeaacaGGSaGaaGPaVlaadIhadaWg aaqaaiaaikdaaeqaaiaacYcacaaMc8UaamiEamaaBaaabaGaaG4maa qabaGaaiilaiaaykW7caaMc8UaaiOlaiaac6cacaGGUaGaaGPaVlaa ykW7caGGSaGaamiEamaaBaaabaGaamOBaaqabaaacaGLOaGaayzkaa aaaa@4DC4@  be a random sample from Ishita distribution (3.5). The likelihood function, L MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamitaa aa@3755@ of (3.5) is given by

L= ( θ 3 θ 3 +2 ) n i=1 n ( θ+ x i 2 ) e nθ x ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamitai abg2da9maabmaabaWaaSaaaeaacqaH4oqCdaahaaqabKqbGeaacaaI ZaaaaaqcfayaaiabeI7aXnaaCaaabeqcfasaaiaaiodaaaqcfaOaey 4kaSIaaGOmaaaaaiaawIcacaGLPaaadaahaaqabKqbGeaacaWGUbaa aKqbaoaarahabaWaaeWaaeaacqaH4oqCcqGHRaWkcaWG4bWaaSbaae aacaWGPbaabeaadaahaaqabKqbGeaacaaIYaaaaaqcfaOaayjkaiaa wMcaaaqcfasaaiaadMgacqGH9aqpcaaIXaaabaGaamOBaaqcfaOaey 4dIunacaaMc8UaamyzamaaCaaabeqcfasaaiabgkHiTiaad6gacaaM c8UaeqiUdeNaaGPaVlqadIhagaqeaaaaaaa@5CC9@

The natural log likelihood function is thus obtained as

lnL=nln( θ 3 θ 3 +2 )+ i=1 n ln( θ+ x i 2 ) nθ x ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaciiBai aac6gacaWGmbGaeyypa0JaamOBaiGacYgacaGGUbWaaeWaaeaadaWc aaqaaiabeI7aXnaaCaaabeqcfasaaiaaiodaaaaajuaGbaGaeqiUde 3aaWbaaeqajuaibaGaaG4maaaajuaGcqGHRaWkcaaIYaaaaaGaayjk aiaawMcaaiabgUcaRmaaqahabaGaciiBaiaac6gadaqadaqaaiabeI 7aXjabgUcaRiaadIhadaWgaaqaaiaadMgaaeqaamaaCaaabeqcfasa aiaaikdaaaaajuaGcaGLOaGaayzkaaaajuaibaGaamyAaiabg2da9i aaigdaaeaacaWGUbaajuaGcqGHris5aiabgkHiTiaad6gacaaMc8Ua eqiUdeNaaGPaVlqadIhagaqeaaaa@5FC5@

Now    dlnL dθ = 6n θ( θ 3 +2 ) + i=1 n 1 θ+ x i 2 n x ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacaWGKbGaciiBaiaac6gacaWGmbaabaGaamizaiabeI7aXbaacqGH 9aqpdaWcaaqaaiaaiAdacaWGUbaabaGaeqiUde3aaeWaaeaacqaH4o qCdaahaaqabKqbGeaacaaIZaaaaKqbakabgUcaRiaaikdaaiaawIca caGLPaaaaaGaey4kaSYaaabCaeaadaWcaaqaaiaaigdaaeaacqaH4o qCcqGHRaWkcaWG4bWaaSbaaeaacaWGPbaabeaadaahaaqabKqbGeaa caaIYaaaaaaaaeaacaWGPbGaeyypa0JaaGymaaqaaiaad6gaaKqbak abggHiLdGaeyOeI0IaamOBaiaaykW7ceWG4bGbaebaaaa@59FF@

where x ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOabmiEay aaraaaaa@3799@ is the sample mean.

The maximum likelihood estimate, θ ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaaaaa@384A@  of θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde haaa@383A@  is the solution of the equation dlnL dθ =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacaWGKbGaciiBaiaac6gacaWGmbaabaGaamizaiabeI7aXbaacqGH 9aqpcaaIWaaaaa@3E91@  and it can be obtained by solving the following non-linear equation

6n θ( θ 3 +2 ) + i=1 n 1 θ+ x i 2 n x ¯ =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacaaI2aGaamOBaaqaaiabeI7aXnaabmaabaGaeqiUde3aaWbaaeqa juaibaGaaG4maaaajuaGcqGHRaWkcaaIYaaacaGLOaGaayzkaaaaai abgUcaRmaaqahabaWaaSaaaeaacaaIXaaabaGaeqiUdeNaey4kaSIa amiEamaaBaaabaGaamyAaaqabaWaaWbaaeqajuaibaGaaGOmaaaaaa aabaGaamyAaiabg2da9iaaigdaaeaacaWGUbaajuaGcqGHris5aiab gkHiTiaad6gacaaMc8UabmiEayaaraGaeyypa0JaaGimaaaa@546C@ .

Method of moment estimate (MOME)

Equating the population mean of the Ishita distribution (3.5) to the corresponding sample mean, the method of moment estimate (MOME) θ ˜ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaGaaaaa@3849@  of θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde haaa@383A@  is the solution of the following non-linear equation

x ¯ θ 4 θ 3 +2θ x ¯ 6=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOabmiEay aaraGaaGPaVlabeI7aXnaaCaaabeqcfasaaiaaisdaaaqcfaOaeyOe I0IaeqiUde3aaWbaaeqajuaibaGaaG4maaaajuaGcqGHRaWkcaaIYa GaeqiUdeNabmiEayaaraGaeyOeI0IaaGOnaiabg2da9iaaicdaaaa@488A@ , where x ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOabmiEay aaraaaaa@3799@ is the sample mean.

Goodness of Fit of Ishita Distribution

The goodness of fit of Ishita distribution has been done on several lifetime data sets. In this section, we present the goodness of fit of Ishita distribution using maximum likelihood estimate of the parameter on two data sets and the fit has been compared with Akash, Lindley and exponential distributions. For testing the goodness of fit of Ishita distribution over exponential, Lindley and Akash distributions, following two data sets have been considered.

Data set 1: The second data set is the strength data of glass of the aircraft window reported by Fuller et al. [21]
18.83, 20.80, 21.657, 23.03, 23.23, 24.05, 24.321, 25.50, 25.52, 25.80, 26.69, 26.77, 26.78, 27.05, 27.67, 29.90, 31.11, 33.20, 33.73, 33.76, 33.89, 34.76, 35.75, 35.91, 36.98, 37.08, 37.09, 39.58, 44.045, 45.29, 45.381

Data Set 2: The following data represent the tensile strength, measured in GPa, of 69 carbon fibers tested under tension at gauge lengths of 20mm, Bader and Priest [22]

1.312
1.314
1.479
1.552
1.7
1.803
1.861
1.865
1.944
1.958
1.966
1.997
2.006
2.021
2.027
2.055
2.063
2.098
2.14
2.179
2.224
2.24
2.253
2.27
2.272
2.274
2.301
2.301
2.359
2.382
2.382
2.426
2.434
2.435
2.478
2.49
2.511
2.514
2.535
2.554
2.566
2.57
2.586
2.629
2.633
2.642
2.648
2.684
2.697
2.726
2.77
2.773
2.8
2.809
2.818
2.821
2.848
2.88
2.954
3.012
3.067
3.084
3.09
3.096
3.128
3.233
3.433
3.585
3.585

In order to compare Ishita, Akash, Lindley and exponential distributions, values of 2lnL MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyOeI0 IaaGOmaiGacYgacaGGUbGaamitaaaa@3AE2@ , AIC (Akaike Information Criterion), AICC (Akaike Information Criterion Corrected), BIC (Bayesian Information Criterion) and K-S Statistic ( Kolmogorov-Smirnov Statistic) for two real data sets have been computed and presented in table 2. The formulae for computing AIC, AICC, BIC, and K-S Statistic are as follows:

AIC=2lnL+2k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyqai aadMeacaWGdbGaeyypa0JaeyOeI0IaaGOmaiGacYgacaGGUbGaamit aiabgUcaRiaaikdacaWGRbaaaa@40D2@ , AICC=AIC+ 2k( k+1 ) ( nk1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyqai aadMeacaWGdbGaam4qaiabg2da9iaadgeacaWGjbGaam4qaiabgUca RmaalaaabaGaaGOmaiaadUgadaqadaqaaiaadUgacqGHRaWkcaaIXa aacaGLOaGaayzkaaaabaWaaeWaaeaacaWGUbGaeyOeI0Iaam4Aaiab gkHiTiaaigdaaiaawIcacaGLPaaaaaaaaa@49BF@ , BIC=2lnL+klnn MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOqai aadMeacaWGdbGaeyypa0JaeyOeI0IaaGOmaiGacYgacaGGUbGaamit aiabgUcaRiaadUgaciGGSbGaaiOBaiaaykW7caWGUbaaaa@4479@  and

K-S = Sup x | F n ( x ) F 0 ( x ) | MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyypa0 ZaaCbeaeaacaqGtbGaaeyDaiaabchaaKqbGeaacaWG4baajuaGbeaa daabdaqaaiaadAeadaWgaaqcfasaaiaad6gaaKqbagqaamaabmaaba GaamiEaaGaayjkaiaawMcaaiabgkHiTiaadAeadaWgaaqcfasaaiaa icdaaKqbagqaamaabmaabaGaamiEaaGaayjkaiaawMcaaaGaay5bSl aawIa7aaaa@4A4A@ , where k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4Aaa aa@3774@  = the number of parameters, n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOBaa aa@3777@  = the sample size and F n ( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOram aaBaaajuaibaGaamOBaaqcfayabaWaaeWaaeaacaWG4baacaGLOaGa ayzkaaaaaa@3BA5@ is the empirical distribution function.

The best distribution corresponds to lower values of 2lnL MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyOeI0 IaaGOmaiGacYgacaGGUbGaamitaaaa@3AE2@ , AIC, AICC, BIC, and K-S statistic.

It can be easily seen from above table that Ishita distribution gives better fit than exponential, Lindley and Akash distribution and hence Ishita distribution should be preferred to exponential, Lindley and Akash distributions for modeling lifetime data from biomedical science and engineering.

 

Distributions

MLE of θ ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaaaaa@384A@

S.E( θ ^ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4uai aac6cacaWGfbWaaeWaaeaacuaH4oqCgaqcaaGaayjkaiaawMcaaaaa @3C27@

2lnL MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyOeI0 IaaGOmaiGacYgacaGGUbGaamitaaaa@3AE2@

AIC

AICC

BIC

K-S

Data 1

Ishita

0.0973

0.0100

240.48

242.48

242.62

243.91

0.297

Akash

0.0971

0.0101

240.68

242.68

242.82

244.11

0.298

Lindley

0.0630

0.0080

253.98

255.98

256.12

257.41

0.365

Exponential

0.0324

0.0058

274.52

276.52

276.66

277.95

0.458

Data 2

Ishita

0.9315

0.0560

223.14

225.14

225.20

227.37

0.331

Akash

0.9647

0.0646

224.27

226.27

226.33

228.50

0.362

Lindley

0.6545

0.0580

238.38

240.38

240.44

242.61

0.401

Exponential

0.4079

0.0491

261.73

263.73

263.79

265.96

0.448

Table 2: MLE,s of θ ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaaaaa@384A@ , S.E. ( θ ^ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacuaH4oqCgaqcaaGaayjkaiaawMcaaaaa@39D3@ , -2ln L, AIC, AICC, BIC, and K-S Statistic of the fitted distributions of data set 1 and 2.

Concluding Remarks

A lifetime distribution named, “Ishita distribution” for modeling lifetime data from biomedical science and engineering has been proposed and its various statistical and mathematical properties including its shape, moments, skewness, kurtosis, hazard rate function, mean residual life function, stochastic ordering, mean deviations, order statistics, Bonferroni and Lorenz curves, Renyi entropy measure and stress-strength reliability have been studied. The conditions of over-dispersed, equi-dispersed, and under-dispersed of Ishita distribution has been presented along with Akash, Lindley and exponential distributions. The estimation of parameter has been discussed using both maximum likelihood estimation and method of moments. The goodness of fit of Ishita distribution has been discussed and illustrated with two real lifetime data sets and it has been shown that it gives better fit than exponential, Lindley and Akash distributions.

NOTE: The paper is named Ishita distribution in the name of Ishita Shukla, a lovely daughter of second author Dr. Kamlesh Kumar Shukla, Department of Statistics, Eritrea Institute of Technology, Asmara, Eritrea.

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