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eISSN: 2378-315X

Biometrics & Biostatistics International Journal

Research Article Volume 7 Issue 3

A two–parameter Sujatha distribution

Mussie Tesfay, Rama Shanker

Department of Statistics, Eritrea Institute of Technology Asmara, Eritrea

Correspondence:

Received: March 15, 2018 | Published: May 10, 2018

Citation: Tesfay M, Shanker R. A two–parameter Sujatha distribution. Biom Biostat Int J. 2018;7(3):188–197. DOI: 10.15406/bbij.2018.07.00208

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Abstract

This paper proposes a two parameter Sujatha distribution (TPSD). This includes size–biased Lindley distribution and Sujatha distribution as particular cases. It’s important statistical properties including its shapes for varying values of parameters, coefficient of variation, skewness, kurtosis, index of dispersion, hazard rate function, mean residual life function, stochastic ordering ,mean deviations, Bonferroni and Lorenz curves, and stress–strength reliability have been discussed. The estimation of parameters has been discussed using the method of moments and the method of maximum likelihood. Application of the distribution has been discussed with a real lifetime data.

Keywords: Sujatha distribution, moments, statistical properties, estimation of parameters, application

Introduction

The statistical analysis and modeling of lifetime data are crucial for statisticians working in various field of knowledge including medical science, engineering, social science, behavioral science, insurance, finance, among others. The classical one parameter lifetime distribution in statistics which were popular for modeling lifetime data are exponential distribution and Lindley distribution proposed by Lindley.1 Shanker et al.2 have detailed critical study on applications of lifetime data from engineering and biomedical science and observed that exponential and Lindley distributions are not always suitable due to theoretical or applied point of view and presence of single parameter. In search for a lifetime distribution which gives a better fit than exponential and Lindley distributions Shanker3 has proposed a new lifetime distribution named Sujatha distribution defined by its probability density function (pdf) and cumulative distribution function (cdf).

f 1 (x;θ)= θ 3 θ 2 +θ+2 ( 1+x+ x 2 ) e θx ;x>0,θ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamOzaS WaaSbaaKqaGeaajugWaiaaigdaaKqaGeqaaKqzGeGaaiikaiaadIha caaMc8Uaai4oaiaaykW7cqaH4oqCcaGGPaGaeyypa0tcfa4aaSaaaO qaaKqzGeGaeqiUde3cdaahaaqcbasabeaajugWaiaaiodaaaaakeaa jugibiabeI7aXTWaaWbaaKqaGeqabaqcLbmacaaIYaaaaKqzGeGaey 4kaSIaeqiUdeNaey4kaSIaaGOmaaaajuaGdaqadaGcbaqcLbsacaaI XaGaey4kaSIaamiEaiabgUcaRiaadIhajuaGdaahaaWcbeqcbasaaK qzadGaaGOmaaaaaOGaayjkaiaawMcaaKqzGeGaamyzaSWaaWbaaKqa GeqabaqcLbmacqGHsislcqaH4oqCcaWG4baaaKqzGeGaaGPaVlaacU dacaaMc8UaaGPaVlaadIhacqGH+aGpcaaIWaGaaiilaiabeI7aXjab g6da+iaaicdacaaMc8oaaa@70C9@ (1.1)

F 1 ( x;θ;α )=1[ 1+ θx( θx+θ+2 ) α θ 2 +θ+2 ] e θx ;x>0,θ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamOraK qbaoaaBaaajeaibaqcLbmacaaIXaaaleqaaKqbaoaabmaakeaajugi biaadIhacaaMc8Uaai4oaiaaykW7cqaH4oqCcaGG7aGaeqySdegaki aawIcacaGLPaaajugibiabg2da9iaaigdacqGHsisljuaGdaWadaGc baqcLbsacaaIXaGaey4kaSscfa4aaSaaaOqaaKqzGeGaeqiUdeNaam iEaKqbaoaabmaakeaajugibiabeI7aXjaadIhacqGHRaWkcqaH4oqC cqGHRaWkcaaIYaaakiaawIcacaGLPaaaaeaajugibiabeg7aHjabeI 7aXLqbaoaaCaaaleqabaqcLbsacaaIYaaaaiabgUcaRiabeI7aXjab gUcaRiaaikdaaaaakiaawUfacaGLDbaajugibiaadwgajuaGdaahaa WcbeqcbasaaKqzadGaeyOeI0IaeqiUdeNaamiEaaaajugibiaacUda caaMc8UaaGPaVlaadIhacqGH+aGpcaaIWaGaaiilaiabeI7aXjabg6 da+iaaicdacaaMc8oaaa@795E@ (1.2)

where θ is a scale parameter. It has been shown by Shanker3 that Sujatha distribution is a convex combination of exponential (θ) distribution, a gamma (2, θ) distribution and a gamma (3, θ) distribution. The first four moments about origin and central moments of Sujatha distribution obtained by Shanker3 are

μ 1 = θ 2 +2θ+6 θ( θ 2 +θ+2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH8o qBlmaaBaaajeaibaqcLbmacaaIXaaajeaibeaalmaaCaaajeaibeqa aKqzadGamai4gkdiIcaajugibiabg2da9Kqbaoaalaaakeaajugibi abeI7aXLqbaoaaCaaaleqajeaibaqcLbmacaaIYaaaaKqzGeGaey4k aSIaaGOmaiabeI7aXjabgUcaRiaaiAdaaOqaaKqzGeGaeqiUdexcfa 4aaeWaaOqaaKqzGeGaeqiUde3cdaahaaqcbasabeaajugWaiaaikda aaqcLbsacqGHRaWkcqaH4oqCcqGHRaWkcaaIYaaakiaawIcacaGLPa aaaaaaaa@59AC@ μ 2 = 2( θ 2 +3θ+12 ) θ 2 ( θ 2 +θ+2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH8o qBjuaGdaWgaaqcbasaaKqzadGaaGOmaaqcbasabaqcfa4aaWbaaKqa GeqabaqcLbmacWaGGBOmGikaaKqzGeGaeyypa0tcfa4aaSaaaOqaaK qzGeGaaGOmaKqbaoaabmaakeaajugibiabeI7aXLqbaoaaCaaajeai beqaaKqzadGaaGOmaaaajugibiabgUcaRiaaiodacqaH4oqCcqGHRa WkcaaIXaGaaGOmaaGccaGLOaGaayzkaaaabaqcLbsacqaH4oqCjuaG daahaaWcbeqaaKqzGeGaaGOmaaaajuaGdaqadaGcbaqcLbsacqaH4o qCjuaGdaahaaqcbasabeaajugWaiaaikdaaaqcLbsacqGHRaWkcqaH 4oqCcqGHRaWkcaaIYaaakiaawIcacaGLPaaaaaaaaa@6155@

μ 3 = 6( θ 2 +4θ+20 ) θ 3 ( θ 2 +θ+2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH8o qBlmaaBaaajeaibaqcLbmacaaIZaaajeaibeaalmaaCaaajeaibeqa aKqzadGamai4gkdiIcaajugibiabg2da9Kqbaoaalaaakeaajugibi aaiAdajuaGdaqadaGcbaqcLbsacqaH4oqCjuaGdaahaaWcbeqcbasa aKqzadGaaGOmaaaajugibiabgUcaRiaaisdacqaH4oqCcqGHRaWkca aIYaGaaGimaaGccaGLOaGaayzkaaaabaqcLbsacqaH4oqClmaaCaaa jeaibeqaaKqzadGaaG4maaaajuaGdaqadaGcbaqcLbsacqaH4oqClm aaCaaajeaibeqaaKqzadGaaGOmaaaajugibiabgUcaRiabeI7aXjab gUcaRiaaikdaaOGaayjkaiaawMcaaaaaaaa@6018@ μ 4 = 24( θ 2 +5θ+30 ) θ 4 ( θ 2 +θ+2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH8o qBlmaaBaaajeaibaqcLbmacaaI0aaajeaibeaalmaaCaaajeaibeqa aKqzadGamai4gkdiIcaajugibiabg2da9Kqbaoaalaaakeaajugibi aaikdacaaI0aqcfa4aaeWaaOqaaKqzGeGaeqiUde3cdaahaaqcbasa beaajugWaiaaikdaaaqcLbsacqGHRaWkcaaI1aGaeqiUdeNaey4kaS IaaG4maiaaicdaaOGaayjkaiaawMcaaaqaaKqzGeGaeqiUdexcfa4a aWbaaSqabeaajugibiaaisdaaaqcfa4aaeWaaOqaaKqzGeGaeqiUde 3cdaahaaqcbasabeaajugWaiaaikdaaaqcLbsacqGHRaWkcqaH4oqC cqGHRaWkcaaIYaaakiaawIcacaGLPaaaaaaaaa@600D@

μ 2 = θ 4 +4 θ 3 +18 θ 2 +12θ+12 θ 2 ( θ 2 +θ+2 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH8o qBjuaGdaWgaaqcbasaaKqzadGaaGOmaaWcbeaajugibiabg2da9Kqb aoaalaaakeaajugibiabeI7aXLqbaoaaCaaaleqajeaibaqcLbmaca aI0aaaaKqzGeGaey4kaSIaaGinaiabeI7aXLqbaoaaCaaaleqajeai baqcLbmacaaIZaaaaKqzGeGaey4kaSIaaGymaiaaiIdacqaH4oqCju aGdaahaaWcbeqcbasaaKqzadGaaGOmaaaajugibiabgUcaRiaaigda caaIYaGaeqiUdeNaey4kaSIaaGymaiaaikdaaOqaaKqzGeGaeqiUde xcfa4aaWbaaSqabKqaGeaajugWaiaaikdaaaqcfa4aaeWaaOqaaKqz GeGaeqiUdexcfa4aaWbaaSqabKqaGeaajugWaiaaikdaaaqcLbsacq GHRaWkcqaH4oqCcqGHRaWkcaaIYaaakiaawIcacaGLPaaajuaGdaah aaWcbeqcbasaaKqzadGaaGOmaaaaaaaaaa@6B6C@

μ 3 = 2( θ 6 +6 θ 5 +36 θ 4 +44 θ 3 +54 θ 2 +36θ+24 ) θ 3 ( θ 2 +θ+2 ) 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH8o qBlmaaBaaajeaibaqcLbmacaaIZaaajeaibeaajugibiabg2da9Kqb aoaalaaakeaajugibiaaikdajuaGdaqadaGcbaqcLbsacqaH4oqCju aGdaahaaWcbeqcbasaaKqzadGaaGOnaaaajugibiabgUcaRiaaiAda cqaH4oqCjuaGdaahaaWcbeqcbasaaKqzadGaaGynaaaajugibiabgU caRiaaiodacaaI2aGaeqiUdexcfa4aaWbaaSqabKqaGeaajugWaiaa isdaaaqcLbsacqGHRaWkcaaI0aGaaGinaiabeI7aXLqbaoaaCaaale qajeaibaqcLbmacaaIZaaaaKqzGeGaey4kaSIaaGynaiaaisdacqaH 4oqClmaaCaaajeaibeqaaKqzadGaaGOmaaaajugibiabgUcaRiaaio dacaaI2aGaeqiUdeNaey4kaSIaaGOmaiaaisdaaOGaayjkaiaawMca aaqaaKqzGeGaeqiUdexcfa4aaWbaaSqabKqaGeaajugWaiaaiodaaa qcfa4aaeWaaOqaaKqzGeGaeqiUde3cdaahaaqcbasabeaajugWaiaa ikdaaaqcLbsacqGHRaWkcqaH4oqCcqGHRaWkcaaIYaaakiaawIcaca GLPaaalmaaCaaajeaibeqaaKqzadGaaG4maaaaaaaaaa@7BC4@

μ 4 = 3( 3 θ 8 +24 θ 7 +172 θ 6 +376 θ 5 +736 θ 4 +864 θ 3 +912 θ 2 +480θ+240 ) θ 4 ( θ 2 +θ+2 ) 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH8o qBjuaGdaWgaaqcbasaaKqzadGaaGinaaWcbeaajugibiabg2da9Kqb aoaalaaakeaajugibiaaiodajuaGdaqadaGcbaqcLbsacaaIZaGaeq iUdexcfa4aaWbaaSqabKqaGeaajugWaiaaiIdaaaqcLbsacqGHRaWk caaIYaGaaGinaiabeI7aXLqbaoaaCaaaleqajeaibaqcLbmacaaI3a aaaKqzGeGaey4kaSIaaGymaiaaiEdacaaIYaGaeqiUdexcfa4aaWba aSqabKqaGeaajugWaiaaiAdaaaqcLbsacqGHRaWkcaaIZaGaaG4nai aaiAdacqaH4oqCjuaGdaahaaWcbeqcbasaaKqzadGaaGynaaaajugi biabgUcaRiaaiEdacaaIZaGaaGOnaiabeI7aXLqbaoaaCaaaleqaje aibaqcLbmacaaI0aaaaKqzGeGaey4kaSIaaGioaiaaiAdacaaI0aGa eqiUdexcfa4aaWbaaSqabKqaGeaajugWaiaaiodaaaqcLbsacqGHRa WkcaaI5aGaaGymaiaaikdacqaH4oqCjuaGdaahaaWcbeqcbasaaKqz adGaaGOmaaaajugibiabgUcaRiaaisdacaaI4aGaaGimaiabeI7aXj abgUcaRiaaikdacaaI0aGaaGimaaGccaGLOaGaayzkaaaabaqcLbsa cqaH4oqCjuaGdaahaaWcbeqcbasaaKqzadGaaGinaaaajuaGdaqada GcbaqcLbsacqaH4oqCjuaGdaahaaWcbeqcbasaaKqzadGaaGOmaaaa jugibiabgUcaRiabeI7aXjabgUcaRiaaikdaaOGaayjkaiaawMcaaK qbaoaaCaaaleqajeaibaqcLbmacaaI0aaaaaaaaaa@9373@

Shanker3 has discussed its important properties including shapes of density function for varying values of parameters, hazard rate function, mean residual life function, stochastic ordering, mean deviations, Bonferroni and Lorenz curves, and stress–strength reliability. Shanker3 discussed the maximum likelihood estimation of parameter and showed applications of Sujatha distribution to model lifetime data from biomedical science and engineering. Shanker4 has introduced Poisson–Sujatha distribution (PSD), a Poisson mixture of Sujatha distribution, and studied its properties, estimation of parameter and applications to model count data. Shanker & Hagos5 have discussed zero–truncated Poisson– Sujatha distribution (ZTPSD) and applications for modeling count data excluding zero counts. Shanker & Hagos6 have also studied size–biased Poisson– Sujatha distribution and its applications for count data excluding zero counts.

The Lindley distribution and a size–biased Lindley distribution (SBLD) having parameter  are defined by their pdf

f 2 (x;θ)= θ 2 θ+1 ( 1+x ) e θx ;x>0,θ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamOzaK qbaoaaBaaajeaibaqcLbmacaaIYaaaleqaaKqzGeGaaiikaiaadIha caaMc8Uaai4oaiabeI7aXjaacMcacqGH9aqpjuaGdaWcaaGcbaqcLb sacqaH4oqCjuaGdaahaaWcbeqcbasaaKqzadGaaGOmaaaaaOqaaKqz GeGaeqiUdeNaey4kaSIaaGymaaaajuaGdaqadaGcbaqcLbsacaaIXa Gaey4kaSIaamiEaaGccaGLOaGaayzkaaqcLbsacaWGLbWcdaahaaqc basabeaajugWaiabgkHiTiabeI7aXjaadIhaaaGaaGPaVNqzGeGaai 4oaiaaykW7caaMc8UaamiEaiabg6da+iaaicdacaGGSaGaeqiUdeNa eyOpa4JaaGimaiaaykW7aaa@6619@ (1.3)

f 3 (x;θ)= θ 3 θ+2 x( 1+x ) e θx ;x>0,θ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamOzaK qbaoaaBaaajeaibaqcLbmacaaIZaaaleqaaKqzGeGaaiikaiaadIha caaMc8Uaai4oaiaaykW7cqaH4oqCcaGGPaGaeyypa0tcfa4aaSaaaO qaaKqzGeGaeqiUdexcfa4aaWbaaSqabKqaGeaajugWaiaaiodaaaaa keaajugibiabeI7aXjabgUcaRiaaikdaaaGaaGPaVlaaykW7caWG4b qcfa4aaeWaaOqaaKqzGeGaaGymaiabgUcaRiaadIhaaOGaayjkaiaa wMcaaKqzGeGaamyzaKqbaoaaCaaaleqajeaibaqcLbmacqGHsislcq aH4oqCcaWG4baaaKqzGeGaaGPaVlaacUdacaaMc8UaamiEaiabg6da +iaaicdacaGGSaGaeqiUdeNaeyOpa4JaaGimaiaaykW7aaa@6ABD@ (1.4)

Ghitany et al.7 have discussed various statistical and mathematical properties, estimation of parameter and application of Lindley distribution to model waiting time data in a bank and it has been showed that Lindley distribution provides better fit than exponential distribution.

In this paper, a two– parameter Sujatha distribution (TPSD), which includes size–biased Lindley distribution and Sujatha distribution as particular cases, has been proposed. It’s important statistical properties including coefficient of variation, skewness, kurtosis, index of dispersion, hazard rate function, mean residual life function, stochastic ordering, mean deviations, Bonferroni and Lorenz curves, stress–strength reliability have been discussed. The estimation of the parameters has been discussed using maximum likelihood estimation. A numerical example has been given to test the goodness of fit of TPSD over Lindley and Sujatha distributions.

A two–parameter Sujatha distribution

A Two parameter Sujatha distribution (TPSD) having parameters θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqiUde haaa@3830@  and α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqySde gaaa@3819@  is defined by its pdf

f 4 (x;θ,α)= θ 3 α θ 2 +θ+2 ( α+x+ x 2 ) e θx ;x>0,θ>0,α0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamOzaK qbaoaaBaaajqwaG9FaaKqzadGaaGinaaWcbeaajugibiaacIcacaWG 4bGaaGPaVlaacUdacaaMc8UaeqiUdeNaaGPaVlaacYcacqaHXoqyca GGPaGaeyypa0tcfa4aaSaaaOqaaKqzGeGaeqiUdexcfa4aaWbaaSqa bKazba2=baqcLbmacaaIZaaaaaGcbaqcLbsacqaHXoqycqaH4oqCju aGdaahaaWcbeqcKfay=haajugWaiaaikdaaaqcLbsacqGHRaWkcqaH 4oqCcqGHRaWkcaaIYaaaaKqbaoaabmaakeaajugibiabeg7aHjabgU caRiaadIhacqGHRaWkcaWG4bqcfa4aaWbaaSqabKazba2=baqcLbma caaIYaaaaaGccaGLOaGaayzkaaqcLbsacaWGLbqcfa4aaWbaaSqabK azba2=baqcLbmacqGHsislcqaH4oqCcaWG4baaaKqzGeGaai4oaiaa dIhacqGH+aGpcaaIWaGaaiilaiabeI7aXjabg6da+iaaicdacaGGSa GaeqySdeMaeyyzImRaaGimaaaa@8009@     (2.1)

where θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqiUde haaa@3830@ is a scale parameter and is α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqySde gaaa@3819@  is a shape parameter. It can be easily verified that (2.1) reduces to Sujatha distribution (1.1) and SBLD (1.4) for α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqySde gaaa@3819@ = 1 and α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqySde gaaa@3819@ = 0 respectively.

Like Sujatha distribution (1.1), TPSD (2.1) is also a convex combination of exponential ( θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqiUde haaa@3830@ ), gamma (2, θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqiUde haaa@3830@ ) and gamma (3, θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqiUde haaa@3830@ ) distributions. We have

f 4 ( x;θ,α )= p 1 g 1 ( x,θ )+ p 2 g 2 ( x,θ )+( 1 p 1 p 2 ) g 3 ( x,θ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamOzaK qbaoaaBaaajqwaa+FaaKqzadGaaGinaaqcbawabaqcfa4aaeWaaKaa GfaajugibiaadIhacaaMc8Uaai4oaiabeI7aXjaacYcacqaHXoqyaK aaGjaawIcacaGLPaaajugibiabg2da9iaadchajuaGdaWgaaqcKfaG =haajugWaiaaigdaaKqaGfqaaKqzGeGaaGPaVlaadEgalmaaBaaajq waa+FaaKqzadGaaGymaaqcKfaG=hqaaKqbaoaabmaajaaybaqcLbsa caWG4bGaaiilaiabeI7aXbqcaaMaayjkaiaawMcaaKqzGeGaey4kaS IaamiCaKqbaoaaBaaajqwaa+FaaKqzadGaaGOmaaqcbawabaqcLbsa caaMc8Uaam4zaSWaaSbaaKazba4=baqcLbmacaaIYaaajqwaa+Faba qcfa4aaeWaaKaaGfaajugibiaadIhacaGGSaGaeqiUdehajaaycaGL OaGaayzkaaqcLbsacqGHRaWkjuaGdaqadaqcaawaaKqzGeGaaGymai abgkHiTiaadchalmaaBaaajqwaa+FaaKqzadGaaGymaaqcKfaG=hqa aKqzGeGaeyOeI0IaamiCaKqbaoaaBaaajqwaa+FaaKqzadGaaGOmaa qcbawabaaajaaycaGLOaGaayzkaaqcLbsacaWGNbqcfa4aaSbaaKaz ba4=baqcLbmacaaIZaaajeaybeaajuaGdaqadaqcaawaaKqzGeGaam iEaiaacYcacqaH4oqCaKaaGjaawIcacaGLPaaajugibiaaykW7aaa@99C2@     (2.2)

where p 1 = α θ 2 α θ 2 +θ+2 , p 2 = θ α θ 2 +θ+2 , g 1 ( x,θ )=θ e θx ;x>0,θ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamiCaK qbaoaaBaaajeaibaqcLbmacaaIXaaaleqaaKqzGeGaeyypa0tcfa4a aSaaaOqaaKqzGeGaeqySdeMaeqiUdexcfa4aaWbaaSqabKqaGeaaju gWaiaaikdaaaaakeaajugibiabeg7aHjabeI7aXLqbaoaaCaaaleqa jeaibaqcLbmacaaIYaaaaKqzGeGaey4kaSIaeqiUdeNaey4kaSIaaG OmaaaacaGGSaGaaGPaVlaaykW7caWGWbqcfa4aaSbaaKqaGeaajugW aiaaikdaaSqabaqcLbsacqGH9aqpjuaGdaWcaaGcbaqcLbsacqaH4o qCaOqaaKqzGeGaeqySdeMaeqiUdexcfa4aaWbaaSqabKqaGeaajugW aiaaikdaaaqcLbsacqGHRaWkcqaH4oqCcqGHRaWkcaaIYaaaaiaacY cacaaMc8UaaGPaVlaadEgajuaGdaWgaaqcbasaaKqzadGaaGymaaWc beaajuaGdaqadaGcbaqcLbsacaWG4bGaaiilaiabeI7aXbGccaGLOa GaayzkaaqcLbsacqGH9aqpcqaH4oqCcaWGLbqcfa4aaWbaaSqabKqa GeaajugWaiabgkHiTiabeI7aXjaadIhaaaqcLbsacaaMc8Uaai4oai aaykW7caaMc8UaamiEaiabg6da+iaaicdacaGGSaGaaGPaVlaaykW7 cqaH4oqCcqGH+aGpcaaIWaaaaa@8DAB@

g 2 ( x,θ )= θ 2 Γ( 2 ) e θx x 21 ;x>0,θ>0, g 3 ( x,θ )= θ 3 Γ( 3 ) e θx x 31 ;x>0,θ>0. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaam4zaK qbaoaaBaaajeaibaqcLbmacaaIYaaaleqaaKqbaoaabmaakeaajugi biaadIhacaGGSaGaeqiUdehakiaawIcacaGLPaaajugibiabg2da9K qbaoaalaaakeaajugibiabeI7aXLqbaoaaCaaaleqajeaibaqcLbma caaIYaaaaaGcbaqcLbsacqqHtoWrjuaGdaqadaGcbaqcLbsacaaIYa aakiaawIcacaGLPaaaaaqcLbsacaWGLbqcfa4aaWbaaSqabKqaGeaa jugWaiabgkHiTiabeI7aXjaadIhaaaqcLbsacaWG4bqcfa4aaWbaaS qabKqaGeaajugWaiaaikdacqGHsislcaaIXaaaaKqzGeGaai4oaiaa dIhacqGH+aGpcaaIWaGaaiilaiaaykW7caaMc8UaeqiUdeNaeyOpa4 JaaGimaiaaykW7caGGSaGaaGPaVlaaykW7caWGNbqcfa4aaSbaaKqa GeaajugWaiaaiodaaSqabaqcfa4aaeWaaOqaaKqzGeGaamiEaiaacY cacqaH4oqCaOGaayjkaiaawMcaaKqzGeGaeyypa0tcfa4aaSaaaOqa aKqzGeGaeqiUdexcfa4aaWbaaSqabKqaGeaajugWaiaaiodaaaaake aajugibiabfo5ahLqbaoaabmaakeaajugibiaaiodaaOGaayjkaiaa wMcaaaaajugibiaadwgajuaGdaahaaWcbeqcbasaaKqzadGaeyOeI0 IaeqiUdeNaamiEaaaajugibiaadIhajuaGdaahaaWcbeqcbasaaKqz adGaaG4maiabgkHiTiaaigdaaaqcLbsacaGG7aGaaGPaVlaaykW7ca WG4bGaeyOpa4JaaGimaiaacYcacaaMc8UaaGPaVlabeI7aXjabg6da +iaaicdacaGGUaaaaa@9DC9@

The corresponding cdf of TPSD (2.1) can be obtained as

F 2 ( x;θ,α )=1[ 1+ θx( θx+θ+2 ) α θ 2 +θ+2 ] e θx ;x>0,θ>0,α>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamOraK qbaoaaBaaajeaibaqcLbmacaaIYaaaleqaaKqbaoaabmaakeaajugi biaadIhacaaMc8Uaai4oaiaaykW7cqaH4oqCcaGGSaGaeqySdegaki aawIcacaGLPaaajugibiabg2da9iaaigdacqGHsisljuaGdaWadaGc baqcLbsacaaIXaGaey4kaSscfa4aaSaaaOqaaKqzGeGaeqiUdeNaam iEaKqbaoaabmaakeaajugibiabeI7aXjaadIhacqGHRaWkcqaH4oqC cqGHRaWkcaaIYaaakiaawIcacaGLPaaaaeaajugibiabeg7aHjabeI 7aXLqbaoaaCaaaleqajeaibaqcLbmacaaIYaaaaKqzGeGaey4kaSIa eqiUdeNaey4kaSIaaGOmaaaaaOGaay5waiaaw2faaKqzGeGaamyzaK qbaoaaCaaaleqajeaibaqcLbmacqGHsislcqaH4oqCcaWG4baaaKqz GeGaaGPaVlaacUdacaaMc8UaaGPaVlaadIhacqGH+aGpcaaIWaGaai ilaiabeI7aXjabg6da+iaaicdacaGGSaGaeqySdeMaeyOpa4JaaGim aaaa@7EB9@    (2.3)

Behavior of the pdf and the cdf of TPSD for varying values of parameter  and α are shown in Figures 1 & 2 respectively.

Figure 1 Behavior of the pdf of TPSD for varying values of parameter θ and α.
Figure 2 Behavior of the cdf of TPSD for varying values of parameter θ and α.

Moments and related measures

The moment generating function of TPSD (2.1) can be obtained as

M X ( t )= θ 3 α θ 2 +θ+2 0 e ( θt )x ( α+x+ x 2 ) dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamytaK qbaoaaBaaajeaibaqcLbmacaWGybaaleqaaKqbaoaabmaakeaajugi biaadshaaOGaayjkaiaawMcaaKqzGeGaeyypa0tcfa4aaSaaaOqaaK qzGeGaeqiUdexcfa4aaWbaaSqabKqaGeaajugWaiaaiodaaaaakeaa jugibiabeg7aHjabeI7aXLqbaoaaCaaaleqajeaibaqcLbmacaaIYa aaaKqzGeGaey4kaSIaeqiUdeNaey4kaSIaaGOmaaaajuaGdaWdXaGc baqcLbsacaWGLbqcfa4aaWbaaSqabKqaGeaajugWaiabgkHiTSWaae WaaKqaGeaajugWaiabeI7aXjabgkHiTiaadshaaKqaGiaawIcacaGL PaaajugWaiaadIhaaaqcfa4aaeWaaOqaaKqzGeGaeqySdeMaey4kaS IaamiEaiabgUcaRiaadIhajuaGdaahaaWcbeqcbasaaKqzadGaaGOm aaaaaOGaayjkaiaawMcaaaqcbasaaKqzadGaaGimaaqcbasaaKqzad GaeyOhIukajugibiabgUIiYdGaaGPaVlaaykW7caWGKbGaamiEaaaa @760D@

= θ 3 α θ 2 +θ+2 [ α ( θt ) + 1 ( θt ) 2 + 2 ( θt ) 3 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeyypa0 tcfa4aaSaaaOqaaKqzGeGaeqiUdexcfa4aaWbaaSqabKqaGeaajugW aiaaiodaaaaakeaajugibiabeg7aHjabeI7aXLqbaoaaCaaaleqaje aibaqcLbmacaaIYaaaaKqzGeGaey4kaSIaeqiUdeNaey4kaSIaaGOm aaaajuaGdaWadaGcbaqcfa4aaSaaaOqaaKqzGeGaeqySdegakeaaju aGdaqadaGcbaqcLbsacqaH4oqCcqGHsislcaWG0baakiaawIcacaGL PaaaaaqcLbsacqGHRaWkjuaGdaWcaaGcbaqcLbsacaaIXaaakeaaju aGdaqadaGcbaqcLbsacqaH4oqCcqGHsislcaWG0baakiaawIcacaGL PaaajuaGdaahaaWcbeqcbasaaKqzadGaaGOmaaaaaaqcLbsacqGHRa WkjuaGdaWcaaGcbaqcLbsacaaIYaaakeaajuaGdaqadaGcbaqcLbsa cqaH4oqCcqGHsislcaWG0baakiaawIcacaGLPaaajuaGdaahaaWcbe qcbasaaKqzadGaaG4maaaaaaaakiaawUfacaGLDbaaaaa@6DAF@

= θ 3 α θ 2 +θ+2 [ α θ k=0 ( t θ ) k + 1 θ 2 k=0 ( k k+1 ) ( t θ ) k + 2 θ 3 k=0 ( k k+2 ) ( t θ ) k ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeyypa0 tcfa4aaSaaaOqaaKqzGeGaeqiUdexcfa4aaWbaaSqabKqaGeaajugW aiaaiodaaaaakeaajugibiabeg7aHjabeI7aXLqbaoaaCaaaleqaje aibaqcLbmacaaIYaaaaKqzGeGaey4kaSIaeqiUdeNaey4kaSIaaGOm aaaajuaGdaWadaGcbaqcfa4aaSaaaOqaaKqzGeGaeqySdegakeaaju gibiabeI7aXbaajuaGdaaeWbGcbaqcfa4aaeWaaOqaaKqbaoaalaaa keaajugibiaadshaaOqaaKqzGeGaeqiUdehaaaGccaGLOaGaayzkaa qcfa4aaWbaaSqabKqaGeaajugWaiaadUgaaaqcLbsacqGHRaWkjuaG daWcaaGcbaqcLbsacaaIXaaakeaajugibiabeI7aXLqbaoaaCaaale qajeaibaqcLbmacaaIYaaaaaaajuaGdaaeWbGcbaqcfa4aaeWaaOqa aKqbaoaaxadakeaaaKqaGeaajugWaiaadUgaaKqaGeaajugWaiaadU gacqGHRaWkcaaIXaaaaaGccaGLOaGaayzkaaaajeaibaqcLbmacaWG RbGaeyypa0JaaGimaaqcbasaaKqzadGaeyOhIukajugibiabggHiLd qcfa4aaeWaaOqaaKqbaoaalaaakeaajugibiaadshaaOqaaKqzGeGa eqiUdehaaaGccaGLOaGaayzkaaaajeaibaqcLbmacaWGRbGaeyypa0 JaaGimaaqcbasaaKqzadGaeyOhIukajugibiabggHiLdqcfa4aaWba aSqabKqaGeaajugWaiaadUgaaaqcLbsacqGHRaWkjuaGdaWcaaGcba qcLbsacaaIYaaakeaajugibiabeI7aXLqbaoaaCaaaleqajeaibaqc LbmacaaIZaaaaaaajuaGdaaeWbGcbaqcfa4aaeWaaOqaaKqbaoaaxa dakeaaaKqaGeaajugWaiaadUgaaKqaGeaajugWaiaadUgacqGHRaWk caaIYaaaaaGccaGLOaGaayzkaaaaleaajugibiaadUgacqGH9aqpca aIWaaaleaajugibiabg6HiLcGaeyyeIuoajuaGdaqadaGcbaqcfa4a aSaaaOqaaKqzGeGaamiDaaGcbaqcLbsacqaH4oqCaaaakiaawIcaca GLPaaajuaGdaahaaWcbeqcbasaaKqzadGaam4AaaaaaOGaay5waiaa w2faaaaa@AD59@

= k=0 α θ 2 +θ( k+1 )+( k+1 )( k+2 ) α θ 2 +θ+2 ( t θ ) k . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeyypa0 tcfa4aaabCaOqaaKqbaoaalaaakeaajugibiabeg7aHjabeI7aXLqb aoaaCaaaleqajeaibaqcLbmacaaIYaaaaKqzGeGaey4kaSIaeqiUde xcfa4aaeWaaOqaaKqzGeGaam4AaiabgUcaRiaaigdaaOGaayjkaiaa wMcaaKqzGeGaey4kaSscfa4aaeWaaOqaaKqzGeGaam4AaiabgUcaRi aaigdaaOGaayjkaiaawMcaaKqbaoaabmaakeaajugibiaadUgacqGH RaWkcaaIYaaakiaawIcacaGLPaaaaeaajugibiabeg7aHjabeI7aXL qbaoaaCaaaleqajeaibaqcLbmacaaIYaaaaKqzGeGaey4kaSIaeqiU deNaey4kaSIaaGOmaaaajuaGdaqadaGcbaqcfa4aaSaaaOqaaKqzGe GaamiDaaGcbaqcLbsacqaH4oqCaaaakiaawIcacaGLPaaajuaGdaah aaWcbeqcbasaaKqzadGaam4AaaaaaKqaGeaajugWaiaadUgacqGH9a qpcaaIWaaajeaibaqcLbmacqGHEisPaKqzGeGaeyyeIuoacaGGUaaa aa@7342@

Thus, the rth moment about origin of TPSD (2.1), obtained as the coefficient of t r r! MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaake aajugibiaadshajuaGdaahaaWcbeqcbasaaKqzadGaamOCaaaaaOqa aKqzGeGaamOCaiaacgcaaaaaaa@3D5A@  in M X ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamytaK qbaoaaBaaajeaibaqcLbmacaWGybaaleqaaKqbaoaabmaakeaajugi biaadshaaOGaayjkaiaawMcaaaaa@3DEE@ , is given by

μ r / = r!{ α θ 2 +θ( r+1 )+( r+1 )( r+2 ) } θ r ( α θ 2 +θ+2 ) ;r=1,2,3,... MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqiVd0 wcfa4aa0baaKqaGeaajugWaiaadkhaaKqaGeaajugWaiaac+caaaqc LbsacqGH9aqpjuaGdaWcaaGcbaqcLbsacaWGYbGaaiyiaKqbaoaacm aakeaajugibiabeg7aHjabeI7aXLqbaoaaCaaaleqajeaibaqcLbma caaIYaaaaKqzGeGaey4kaSIaeqiUdexcfa4aaeWaaOqaaKqzGeGaam OCaiabgUcaRiaaigdaaOGaayjkaiaawMcaaKqzGeGaey4kaSscfa4a aeWaaOqaaKqzGeGaamOCaiabgUcaRiaaigdaaOGaayjkaiaawMcaaK qbaoaabmaakeaajugibiaadkhacqGHRaWkcaaIYaaakiaawIcacaGL PaaaaiaawUhacaGL9baaaeaajugibiabeI7aXLqbaoaaCaaaleqaje aibaqcLbmacaWGYbaaaKqbaoaabmaakeaajugibiabeg7aHjabeI7a XLqbaoaaCaaaleqajeaibaqcLbmacaaIYaaaaKqzGeGaey4kaSIaeq iUdeNaey4kaSIaaGOmaaGccaGLOaGaayzkaaaaaKqzGeGaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caGG7aGaaGPaVlaaykW7caWGYbGaey ypa0JaaGymaiaacYcacaaIYaGaaiilaiaaiodacaGGSaGaaiOlaiaa c6cacaGGUaaaaa@8705@   (3.1)

The first four moments about origin of TPSD are obtained as

μ 1 / = α θ 2 +2θ+6 θ( α θ 2 +θ+2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqiVd0 2cdaqhaaqcbasaaKqzadGaaGymaaqcbasaaKqzadGaai4laaaajugi biabg2da9Kqbaoaalaaakeaajugibiabeg7aHjabeI7aXLqbaoaaCa aaleqajeaibaqcLbmacaaIYaaaaKqzGeGaey4kaSIaaGOmaiabeI7a XjabgUcaRiaaiAdaaOqaaKqzGeGaeqiUdexcfa4aaeWaaOqaaKqzGe GaeqySdeMaeqiUdexcfa4aaWbaaSqabKqaGeaajugWaiaaikdaaaqc LbsacqGHRaWkcqaH4oqCcqGHRaWkcaaIYaaakiaawIcacaGLPaaaaa aaaa@5AE2@     μ 2 / = 2(α θ 2 +3θ+12) θ 2 ( α θ 2 +θ+2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqiVd0 2cdaqhaaqcbasaaKqzadGaaGOmaaqcbasaaKqzadGaai4laaaajugi biabg2da9KqbaoaalaaakeaajugibiaaikdacaGGOaGaeqySdeMaeq iUdexcfa4aaWbaaSqabKqaGeaajugWaiaaikdaaaqcLbsacqGHRaWk caaIZaGaeqiUdeNaey4kaSIaaGymaiaaikdacaGGPaaakeaajugibi abeI7aXLqbaoaaCaaaleqabaqcLbsacaaIYaaaaKqbaoaabmaakeaa jugibiabeg7aHjabeI7aXLqbaoaaCaaaleqajeaibaqcLbmacaaIYa aaaKqzGeGaey4kaSIaeqiUdeNaey4kaSIaaGOmaaGccaGLOaGaayzk aaaaaaaa@5FB6@

μ 3 / = 6(α θ 2 +4θ+20) θ 3 ( α θ 2 +θ+2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqiVd0 2cdaqhaaqcbasaaKqzadGaaG4maaqcbasaaKqzadGaai4laaaajugi biabg2da9KqbaoaalaaakeaajugibiaaiAdacaGGOaGaeqySdeMaeq iUdexcfa4aaWbaaSqabKqaGeaajugWaiaaikdaaaqcLbsacqGHRaWk caaI0aGaeqiUdeNaey4kaSIaaGOmaiaaicdacaGGPaaakeaajugibi abeI7aXLqbaoaaCaaaleqajeaibaqcLbmacaaIZaaaaKqbaoaabmaa keaajugibiabeg7aHjabeI7aXLqbaoaaCaaaleqajeaibaqcLbmaca aIYaaaaKqzGeGaey4kaSIaeqiUdeNaey4kaSIaaGOmaaGccaGLOaGa ayzkaaaaaaaa@6085@         μ 4 / = 24(α θ 2 +5θ+30) θ 4 ( α θ 2 +θ+2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqiVd0 2cdaqhaaqcbasaaKqzadGaaGinaaqcbasaaKqzadGaai4laaaajugi biabg2da9KqbaoaalaaakeaajugibiaaikdacaaI0aGaaiikaiabeg 7aHjabeI7aXTWaaWbaaKqaGeqabaqcLbmacaaIYaaaaKqzGeGaey4k aSIaaGynaiabeI7aXjabgUcaRiaaiodacaaIWaGaaiykaaGcbaqcLb sacqaH4oqCjuaGdaahaaWcbeqcbasaaKqzadGaaGinaaaajuaGdaqa daGcbaqcLbsacqaHXoqycqaH4oqCjuaGdaahaaWcbeqcbasaaKqzad GaaGOmaaaajugibiabgUcaRiabeI7aXjabgUcaRiaaikdaaOGaayjk aiaawMcaaaaaaaa@60B5@

Using the relationship between moments about the mean and moments about the origin, the moments about mean of TPSD are obtained as

μ 2 = α 2 θ 4 +4α θ 3 +16α θ 2 +2 θ 2 +12θ+12 θ 2 ( α θ 2 +θ+2 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqiVd0 wcfa4aaSbaaKqaGeaajugWaiaaikdaaSqabaqcLbsacqGH9aqpjuaG daWcaaGcbaqcLbsacqaHXoqyjuaGdaahaaWcbeqcbasaaKqzadGaaG OmaaaajugibiabeI7aXLqbaoaaCaaaleqajeaibaqcLbmacaaI0aaa aKqzGeGaey4kaSIaaGinaiabeg7aHjabeI7aXLqbaoaaCaaaleqaje aibaqcLbmacaaIZaaaaKqzGeGaey4kaSIaaGymaiaaiAdacqaHXoqy cqaH4oqCjuaGdaahaaWcbeqcbasaaKqzadGaaGOmaaaajugibiabgU caRiaaikdacqaH4oqCjuaGdaahaaWcbeqcbasaaKqzadGaaGOmaaaa jugibiabgUcaRiaaigdacaaIYaGaeqiUdeNaey4kaSIaaGymaiaaik daaOqaaKqzGeGaeqiUdexcfa4aaWbaaSqabKqaGeaajugWaiaaikda aaqcfa4aaeWaaOqaaKqzGeGaeqySdeMaeqiUdexcfa4aaWbaaSqabK qaGeaajugWaiaaikdaaaqcLbsacqGHRaWkcqaH4oqCcqGHRaWkcaaI YaaakiaawIcacaGLPaaajuaGdaahaaWcbeqcbasaaKqzadGaaGOmaa aaaaaaaa@7BEB@

μ 3 = 2( α 3 θ 6 +6 α 2 θ 5 +30 α 2 θ 4 +6α θ 4 +42α θ 3 +36α θ 2 +2 θ 3 +18 θ 2 +36θ+24 ) θ 3 ( α θ 2 +θ+2 ) 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqiVd0 wcfa4aaSbaaKqaGeaajugWaiaaiodaaSqabaqcLbsacqGH9aqpjuaG daWcaaGcbaqcLbsacaaIYaqcfa4aaeWaaOqaaKqzGeGaeqySdewcfa 4aaWbaaSqabKqaGeaajugWaiaaiodaaaqcLbsacqaH4oqCjuaGdaah aaWcbeqcbasaaKqzadGaaGOnaaaajugibiabgUcaRiaaiAdacqaHXo qyjuaGdaahaaWcbeqcbasaaKqzadGaaGOmaaaajugibiabeI7aXLqb aoaaCaaaleqajeaibaqcLbmacaaI1aaaaKqzGeGaey4kaSIaaG4mai aaicdacqaHXoqyjuaGdaahaaWcbeqcbasaaKqzadGaaGOmaaaajugi biabeI7aXLqbaoaaCaaaleqajeaibaqcLbmacaaI0aaaaKqzGeGaey 4kaSIaaGOnaiabeg7aHjabeI7aXLqbaoaaCaaaleqajeaqbaqcLboa caaI0aaaaKqzGeGaey4kaSIaaGinaiaaikdacqaHXoqycqaH4oqCju aGdaahaaWcbeqcbasaaKqzadGaaG4maaaajugibiabgUcaRiaaioda caaI2aGaeqySdeMaeqiUdexcfa4aaWbaaSqabKqaGeaajugWaiaaik daaaqcLbsacqGHRaWkcaaIYaGaeqiUdexcfa4aaWbaaSqabKqaGeaa jugWaiaaiodaaaqcLbsacqGHRaWkcaaIXaGaaGioaiabeI7aXLqbao aaCaaaleqajeaibaqcLbmacaaIYaaaaKqzGeGaey4kaSIaaG4maiaa iAdacqaH4oqCcqGHRaWkcaaIYaGaaGinaaGccaGLOaGaayzkaaaaba qcLbsacqaH4oqCjuaGdaahaaWcbeqcbasaaKqzadGaaG4maaaajuaG daqadaGcbaqcLbsacqaHXoqycqaH4oqCjuaGdaahaaWcbeqcbasaaK qzadGaaGOmaaaajugibiabgUcaRiabeI7aXjabgUcaRiaaikdaaOGa ayjkaiaawMcaaKqbaoaaCaaaleqajeaibaqcLbmacaaIZaaaaaaaaa a@A851@

μ 4 = 3( 3 α 4 θ 8 +24 α 3 θ 7 +128 α 3 θ 6 +44 α 2 θ 6 +344 α 2 θ 5 +408 α 2 θ 4 +32α θ 5 +320α θ 4 +768α θ 3 +8 θ 4 +576α θ 2 +96 θ 3 +336 θ 2 +480θ+240 ) θ 4 ( α θ 2 +θ+2 ) 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGamGjGeY 7aTLqbaoacyc4gaaqcbasaiGjGjugWaiacyciI0aaaleqcyciajugi biadycOH9aqpjuaGdaWcaaGcbaqcLbsacaaIZaqcfa4aaeWaaKqzGe abaeqakeaajugibiaaiodacqaHXoqyjuaGdaahaaWcbeqcbasaaKqz adGaaGinaaaajugibiabeI7aXLqbaoaaCaaaleqajeaibaqcLbmaca aI4aaaaKqzGeGaey4kaSIaaGOmaiaaisdacqaHXoqyjuaGdaahaaWc beqcbasaaKqzadGaaG4maaaajugibiabeI7aXLqbaoaaCaaaleqaje aibaqcLbmacaaI3aaaaKqzGeGaey4kaSIaaGymaiaaikdacaaI4aGa eqySdewcfa4aaWbaaSqabKqaGeaajugWaiaaiodaaaqcLbsacqaH4o qCjuaGdaahaaWcbeqcbasaaKqzadGaaGOnaaaajugibiabgUcaRiaa isdacaaI0aGaeqySdewcfa4aaWbaaSqabKqaGeaajugWaiaaikdaaa qcLbsacqaH4oqCjuaGdaahaaWcbeqcbasaaKqzadGaaGOnaaaajugi biabgUcaRiaaiodacaaI0aGaaGinaiabeg7aHLqbaoaaCaaaleqaje aibaqcLbmacaaIYaaaaKqzGeGaeqiUdexcfa4aaWbaaSqabKqaGeaa jugWaiaaiwdaaaqcLbsacqGHRaWkcaaI0aGaaGimaiaaiIdacqaHXo qyjuaGdaahaaWcbeqcbasaaKqzadGaaGOmaaaajugibiabeI7aXLqb aoaaCaaaleqajeaibaqcLbmacaaI0aaaaKqzGeGaey4kaSIaaG4mai aaikdacqaHXoqycqaH4oqCjuaGdaahaaWcbeqcbasaaKqzadGaaGyn aaaajugibiabgUcaRiaaiodacaaIYaGaaGimaiabeg7aHjabeI7aXL qbaoaaCaaaleqajeaibaqcLbmacaaI0aaaaaGcbaqcLbsacqGHRaWk caaI3aGaaGOnaiaaiIdacqaHXoqycqaH4oqCjuaGdaahaaWcbeqcba saaKqzadGaaG4maaaajugibiabgUcaRiaaiIdacqaH4oqCjuaGdaah aaWcbeqcbasaaKqzadGaaGinaaaajugibiabgUcaRiaaiwdacaaI3a GaaGOnaiabeg7aHjabeI7aXLqbaoaaCaaaleqajeaibaqcLbmacaaI YaaaaKqzGeGaey4kaSIaaGyoaiaaiAdacqaH4oqCjuaGdaahaaWcbe qcbasaaKqzadGaaG4maaaajugibiabgUcaRiaaiodacaaIZaGaaGOn aiabeI7aXLqbaoaaCaaaleqajeaibaqcLbmacaaIYaaaaKqzGeGaey 4kaSIaaGinaiaaiIdacaaIWaGaeqiUdeNaey4kaSIaaGOmaiaaisda caaIWaaaaOGaayjkaiaawMcaaaqaaKqzGeGaeqiUdexcfa4aaWbaaS qabKqaafaajug4aiaaisdaaaqcfa4aaeWaaOqaaKqzGeGaeqySdeMa eqiUdexcfa4aaWbaaSqabKqaGeaajugWaiaaikdaaaqcLbsacqGHRa WkcqaH4oqCcqGHRaWkcaaIYaaakiaawIcacaGLPaaajuaGdaahaaWc beqcbasaaKqzadGaaGinaaaaaaaaaa@EE8D@

The coefficient of variation (C.V), coefficient of skewness ( β 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaake aajuaGdaGcaaGcbaqcLbsacqaHYoGyjuaGdaWgaaWcbaqcLbsacaaI XaaaleqaaaqabaaakiaawIcacaGLPaaaaaa@3CFD@ , coefficient of kurtosis ( β 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaake aajugibiabek7aILqbaoaaBaaaleaajugibiaaikdaaSqabaaakiaa wIcacaGLPaaaaaa@3C56@  and index of dispersion ( γ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaake aajugibiabeo7aNbGccaGLOaGaayzkaaaaaa@3A4C@  of TPSD are given by

C.V= σ μ 1 ' = α 2 θ 4 +4α θ 3 +16α θ 2 +2 θ 2 +12θ+12 α θ 2 +2θ+6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaam4qai aac6cacaWGwbGaeyypa0tcfa4aaSaaaOqaaKqzGeGaeq4Wdmhakeaa jugibiabeY7aTTWaa0baaKqaGeaajugWaiaaigdaaKqaGeaajugWai aacEcaaaaaaKqzGeGaeyypa0tcfa4aaSaaaOqaaKqbaoaakaaakeaa jugibiabeg7aHLqbaoaaCaaaleqajeaibaqcLbmacaaIYaaaaKqzGe GaeqiUdexcfa4aaWbaaSqabKqaGeaajugWaiaaisdaaaqcLbsacqGH RaWkcaaI0aGaeqySdeMaeqiUdexcfa4aaWbaaSqabKqaGeaajugWai aaiodaaaqcLbsacqGHRaWkcaaIXaGaaGOnaiabeg7aHjabeI7aXLqb aoaaCaaaleqajeaibaqcLbmacaaIYaaaaKqzGeGaey4kaSIaaGOmai abeI7aXLqbaoaaCaaaleqajeaibaqcLbmacaaIYaaaaKqzGeGaey4k aSIaaGymaiaaikdacqaH4oqCcqGHRaWkcaaIXaGaaGOmaaWcbeaaaO qaaKqzGeGaeqySdeMaeqiUdexcfa4aaWbaaSqabKqaGeaajugWaiaa ikdaaaqcLbsacqGHRaWkcaaIYaGaeqiUdeNaey4kaSIaaGOnaaaaaa a@7BB3@

β 1 = μ 3 μ 2 3/2 = 2( α 3 θ 6 +6 α 2 θ 5 +30 α 2 θ 4 +6α θ 4 +42α θ 3 +36α θ 2 +2 θ 3 +18 θ 2 +36θ+24 ) ( α 2 θ 4 +4α θ 3 +16α θ 2 +2 θ 2 +12θ+12 ) 3/2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaakaaake aajugibiabek7aITWaaSbaaKqaGeaajugWaiaaigdaaKqaGeqaaaWc beaajugibiabg2da9KqbaoaalaaakeaajugibiabeY7aTLqbaoaaBa aajeaibaqcLbmacaaIZaaaleqaaaGcbaqcLbsacqaH8oqBlmaaDaaa jeaibaqcLbmacaaIYaaajeaibaqcLbmacaaIZaGaai4laiaaikdaaa aaaKqzGeGaeyypa0tcfa4aaSaaaOqaaKqzGeGaaGOmaKqbaoaabmaa keaajugibiabeg7aHLqbaoaaCaaaleqajeaibaqcLbmacaaIZaaaaK qzGeGaeqiUdexcfa4aaWbaaSqabKqaGeaajugWaiaaiAdaaaqcLbsa cqGHRaWkcaaI2aGaeqySdewcfa4aaWbaaSqabKqaGeaajugWaiaaik daaaqcLbsacqaH4oqCjuaGdaahaaWcbeqcbasaaKqzadGaaGynaaaa jugibiabgUcaRiaaiodacaaIWaGaeqySdewcfa4aaWbaaSqabKqaGe aajugWaiaaikdaaaqcLbsacqaH4oqCjuaGdaahaaWcbeqcbasaaKqz adGaaGinaaaajugibiabgUcaRiaaiAdacqaHXoqycqaH4oqCjuaGda ahaaWcbeqcbasaaKqzadGaaGinaaaajugibiabgUcaRiaaisdacaaI YaGaeqySdeMaeqiUdexcfa4aaWbaaSqabKqaGeaajugWaiaaiodaaa qcLbsacqGHRaWkcaaIZaGaaGOnaiabeg7aHjabeI7aXLqbaoaaCaaa leqajeaibaqcLbmacaaIYaaaaKqzGeGaey4kaSIaaGOmaiabeI7aXL qbaoaaCaaaleqajeaibaqcLbmacaaIZaaaaKqzGeGaey4kaSIaaGym aiaaiIdacqaH4oqCjuaGdaahaaWcbeqcbasaaKqzadGaaGOmaaaaju gibiabgUcaRiaaiodacaaI2aGaeqiUdeNaey4kaSIaaGOmaiaaisda aOGaayjkaiaawMcaaaqaaKqbaoaabmaakeaajugibiabeg7aHLqbao aaCaaaleqajeaibaqcLbmacaaIYaaaaKqzGeGaeqiUdexcfa4aaWba aSqabKqaGeaajugWaiaaisdaaaqcLbsacqGHRaWkcaaI0aGaeqySde MaeqiUdexcfa4aaWbaaSqabKqaGeaajugWaiaaiodaaaqcLbsacqGH RaWkcaaIXaGaaGOnaiabeg7aHjabeI7aXLqbaoaaCaaaleqajeaiba qcLbmacaaIYaaaaKqzGeGaey4kaSIaaGOmaiabeI7aXLqbaoaaCaaa leqajeaibaqcLbmacaaIYaaaaKqzGeGaey4kaSIaaGymaiaaikdacq aH4oqCcqGHRaWkcaaIXaGaaGOmaaGccaGLOaGaayzkaaqcfa4aaWba aSqabKqaGeaajugWaiaaiodacaGGVaGaaGOmaaaaaaaaaa@D1B0@

β 2 = μ 4 μ 2 2 = 3( 3 α 4 θ 8 +24 α 3 θ 7 +128 α 3 θ 6 +44 α 2 θ 6 +344 α 2 θ 5 +408 α 2 θ 4 +32α θ 5 +320α θ 4 +768α θ 3 +8 θ 4 +576α θ 2 +96 θ 3 +336 θ 2 +480θ+240 ) ( α 2 θ 4 +4α θ 3 +16α θ 2 +2 θ 2 +12θ+12 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqOSdi wcfa4aaSbaaKqaGeaajugWaiaaikdaaSqabaqcLbsacWaMaAypa0Ja aGPaVNqbaoaalaaakeaajugibiabeY7aTLqbaoaaBaaajeaibaqcLb macaaI0aaaleqaaaGcbaqcLbsacqaH8oqBlmaaDaaajeaibaqcLbma caaIYaaajeaibaqcLbmacaaIYaaaaaaajugibiabg2da9Kqbaoaala aakeaajugibiaaiodajuaGdaqadaqcLbsaeaqabOqaaKqzGeGaaG4m aiabeg7aHLqbaoaaCaaaleqajeaibaqcLbmacaaI0aaaaKqzGeGaeq iUdexcfa4aaWbaaSqabKqaGeaajugWaiaaiIdaaaqcLbsacqGHRaWk caaIYaGaaGinaiabeg7aHLqbaoaaCaaaleqajeaibaqcLbmacaaIZa aaaKqzGeGaeqiUdexcfa4aaWbaaSqabKqaGeaajugWaiaaiEdaaaqc LbsacqGHRaWkcaaIXaGaaGOmaiaaiIdacqaHXoqyjuaGdaahaaWcbe qcbasaaKqzadGaaG4maaaajugibiabeI7aXLqbaoaaCaaaleqajeai baqcLbmacaaI2aaaaKqzGeGaey4kaSIaaGinaiaaisdacqaHXoqyju aGdaahaaWcbeqcbasaaKqzadGaaGOmaaaajugibiabeI7aXLqbaoaa CaaaleqajeaibaqcLbmacaaI2aaaaKqzGeGaey4kaSIaaG4maiaais dacaaI0aGaeqySdewcfa4aaWbaaSqabKqaGeaajugWaiaaikdaaaqc LbsacqaH4oqCjuaGdaahaaWcbeqcbasaaKqzadGaaGynaaaajugibi abgUcaRiaaisdacaaIWaGaaGioaiabeg7aHLqbaoaaCaaaleqajeai baqcLbmacaaIYaaaaKqzGeGaeqiUdexcfa4aaWbaaSqabKqaGeaaju gWaiaaisdaaaqcLbsacqGHRaWkcaaIZaGaaGOmaiabeg7aHjabeI7a XLqbaoaaCaaaleqajeaibaqcLbmacaaI1aaaaKqzGeGaey4kaSIaaG 4maiaaikdacaaIWaGaeqySdeMaeqiUdexcfa4aaWbaaSqabKqaafaa jug4aiaaisdaaaaakeaajugibiabgUcaRiaaiEdacaaI2aGaaGioai abeg7aHjabeI7aXLqbaoaaCaaaleqajeaibaqcLbmacaaIZaaaaKqz GeGaey4kaSIaaGioaiabeI7aXLqbaoaaCaaaleqajeaibaqcLbmaca aI0aaaaKqzGeGaey4kaSIaaGynaiaaiEdacaaI2aGaeqySdeMaeqiU dexcfa4aaWbaaSqabKqaGeaajugWaiaaikdaaaqcLbsacqGHRaWkca aI5aGaaGOnaiabeI7aXLqbaoaaCaaaleqajeaibaqcLbmacaaIZaaa aKqzGeGaey4kaSIaaG4maiaaiodacaaI2aGaeqiUdexcfa4aaWbaaS qabeaajugibiaaikdaaaGaey4kaSIaaGinaiaaiIdacaaIWaGaeqiU deNaey4kaSIaaGOmaiaaisdacaaIWaaaaOGaayjkaiaawMcaaaqaaK qbaoaabmaakeaajugibiabeg7aHLqbaoaaCaaaleqajeaibaqcLbma caaIYaaaaKqzGeGaeqiUdexcfa4aaWbaaSqabKqaGeaajugWaiaais daaaqcLbsacqGHRaWkcaaI0aGaeqySdeMaeqiUdexcfa4aaWbaaSqa bKqaGeaajugWaiaaiodaaaqcLbsacqGHRaWkcaaIXaGaaGOnaiabeg 7aHjabeI7aXLqbaoaaCaaaleqajeaibaqcLbmacaaIYaaaaKqzGeGa ey4kaSIaaGOmaiabeI7aXLqbaoaaCaaaleqajeaibaqcLbmacaaIYa aaaKqzGeGaey4kaSIaaGymaiaaikdacqaH4oqCcqGHRaWkcaaIXaGa aGOmaaGccaGLOaGaayzkaaqcfa4aaWbaaSqabKqaGeaajugWaiaaik daaaaaaaaa@0F8F@

γ= σ 2 μ 1 ' = α 2 θ 4 +4α θ 3 +16α θ 2 +2 θ 2 +12θ+12 θ( α θ 2 +θ+2 )( α θ 2 +2θ+6 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeq4SdC Maeyypa0tcfa4aaSaaaOqaaKqzGeGaeq4Wdmxcfa4aaWbaaSqabKqa GeaajugWaiaaikdaaaaakeaajugibiabeY7aTTWaa0baaKqaGeaaju gWaiaaigdaaKqaGeaajugWaiaacEcaaaaaaKqzGeGaeyypa0tcfa4a aSaaaOqaaKqzGeGaeqySdewcfa4aaWbaaSqabKqaGeaajugWaiaaik daaaqcLbsacqaH4oqCjuaGdaahaaWcbeqcbasaaKqzadGaaGinaaaa jugibiabgUcaRiaaisdacqaHXoqycqaH4oqCjuaGdaahaaWcbeqcba uaaKqzGdGaaG4maaaajugibiabgUcaRiaaigdacaaI2aGaeqySdeMa eqiUdexcfa4aaWbaaSqabKqaGeaajugWaiaaikdaaaqcLbsacqGHRa WkcaaIYaGaeqiUdexcfa4aaWbaaSqabKqaGeaajugWaiaaikdaaaqc LbsacqGHRaWkcaaIXaGaaGOmaiabeI7aXjabgUcaRiaaigdacaaIYa aakeaajugibiabeI7aXLqbaoaabmaakeaajugibiabeg7aHjabeI7a XLqbaoaaCaaaleqajeaibaqcLbmacaaIYaaaaKqzGeGaey4kaSIaeq iUdeNaey4kaSIaaGOmaaGccaGLOaGaayzkaaqcfa4aaeWaaOqaaKqz GeGaeqySdeMaeqiUdexcfa4aaWbaaSqabKqaafaajug4aiaaikdaaa qcLbsacqGHRaWkcaaIYaGaeqiUdeNaey4kaSIaaGOnaaGccaGLOaGa ayzkaaaaaaaa@8FB4@

It can be easily verified that these statistical constants of TPSD reduce to the corresponding statistical constants of Sujatha distribution and SBLD at α=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqySde Maeyypa0JaaGymaaaa@39DA@  and α=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqySde Maeyypa0JaaGimaaaa@39D9@ respectively. To study the behavior of C.V., β 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaakaaake aajugibiabek7aITWaaSbaaKqaafaajug4aiaaigdaaKqaafqaaaWc beaaaaa@3B97@ , β 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqOSdi wcfa4aaSbaaKqaGeaajugWaiaaikdaaSqabaaaaa@3AE9@  and γ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeq4SdC gaaa@3821@ , their values for varying values of the parameters θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqiUde haaa@3830@  and α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqySde gaaa@3819@ have been computed and presented in Tables 1–4.

      θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaaKTacceqcLb sacqWF4oqCaaa@38A8@
α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaaKTacceqcLb sacqWFXoqyaaa@3891@  

0.2

0.5

1

2

3

4

5

0.2

0.59658

0.624798

0.668399

0.739814

0.792609

0.8317

0.861102

0.5

0.599565

0.639569

0.708329

0.816497

0.882958

0.922627

0.946881

1

0.604466

0.662392

0.761739

0.892143

0.95119

0.977525

0.989835

2

0.614004

0.702377

0.83666

0.96225

0.996661

1.005655

1.007547

3

0.623205

0.736304

0.886072

0.991701

1.009814

1.011382

1.009973

4

0.632091

0.765466

0.920447

1.0059

1.014222

1.012415

1.009836

5

0.640678

0.790787

0.945247

1.013246

1.015576

1.012149

1.009163

Table 1 CV of TPSD for varying values of parameters and
For a given value of , C.V increases as the value of increases .But for values , CV decreases as the value of increases.

 θ
α 

0.2

0.5

1

2

3

4

5

0.2

1.156092

1.164414

1.193838

1.288579

1.40832

1.544566

1.694179

0.5

1.151692

1.153618

1.202728

1.394848

1.600302

1.785072

1.947347

1

1.145006

1.145839

1.247611

1.535588

1.733747

1.848046

1.912879

2

1.133828

1.153526

1.352316

1.647373

1.698737

1.653874

1.586127

3

1.125191

1.176753

1.43637

1.643895

1.562899

1.429794

1.310578

4

1.118703

1.206238

1.496066

1.59473

1.421347

1.244821

1.108

5

1.114041

1.237609

1.535958

1.528385

1.293862

1.097469

0.956984

Table 2 Coefficient of skewness of TPSD for varying values of parameters and
Since , TPSD is always positively skewed, and this means that TPSD is a suitable model for positively skewed lifetime data.

        θ 
α 

0.2

0.5

1

2

3

4

5

0.2

5.003116

5.022048

5.093943

5.346882

5.661781

5.979645

6.275987

0.5

4.991667

4.984856

5.082378

5.625

6.28542

6.865586

7.326691

1

4.973635

4.944566

5.170213

6.21499

7.193906

7.868405

8.297711

2

4.94128

4.924032

5.510204

7.2144

8.270528

8.774988

9.001011

3

4.913483

4.956867

5.903269

7.900925

8.799663

9.113171

9.206366

4

4.889821

5.022933

6.283795

8.3676

9.077558

9.253624

9.271966

5

4.869916

5.109996

6.633262

8.690336

9.230612

9.313262

9.289003

Table 3 Coefficient of kurtosis of TPSD for varying values of parameters and
Since TPSD is always leptokurtic, and this means that TPSD is more peaked than the normal curve. Thus TPSD is suitable for lifetime data which are leptokurtic.

θ

α

0.2

0.5

1

2

3

4

5

0.2

5.164536

2.158531

1.144817

0.615741

0.424979

0.323306

0.259524

0.5

5.197861

2.220551

1.218487

0.666667

0.451356

0.334416

0.262078

1

5.252329

2.31348

1.305556

0.696429

0.452381

0.325758

0.251067

2

5.357375

2.466667

1.4

0.694444

0.431884

0.306064

0.235088

3

5.457478

2.585608

1.439394

0.676136

0.414263

0.293608

0.2264

4

5.552914

2.678571

1.452381

0.657692

0.401423

0.285531

0.221109

5

5.643939

2.751515

1.451923

0.641667

0.39193

0.279936

0.217569

Table 4 Index of dispersion of TPSD for varying values of parameters and
As long as and , the nature of TPSD is over dispersed and for and , the nature of TPSD is over dispersed

The behavior of C.V., β 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaakaaake aajugibiabek7aILqbaoaaBaaajeaibaqcLbmacaaIXaaaleqaaaqa baaaaa@3B90@ , β 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqOSdi wcfa4aaSbaaKqaGeaajugWaiaaikdaaSqabaaaaa@3AE9@  and γ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeo7aNbaa@3792@ , for selected values of the parameters θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqiUde haaa@3830@  and α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqySde gaaa@3819@ are shown in Figure 3.

Figure 3Behavior of C.V., , and , for varying values of the parameters θ and α.

Statistical properties

In this section, statistical properties of TPSD including hazard rate function, mean residual life function, stochastic ordering, mean deviation, Bonferroni and Lorenz curves and stress–strength reliability have been discussed.

Hazard rate function and mean residual life function

Let X be a continuous random variable with pdf f( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamOzaK qbaoaabmaakeaajugibiaadIhaaOGaayjkaiaawMcaaaaa@3B1C@  and cdf F( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadAeadaqada qaaiaadIhaaiaawIcacaGLPaaaaaa@393C@ .The hazard rate function (also known as failure rate function), h( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamiAaK qbaoaabmaakeaajugibiaadIhaaOGaayjkaiaawMcaaaaa@3B1E@  and the mean residual function, m( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaad2gadaqada qaaiaadIhaaiaawIcacaGLPaaaaaa@3963@  of X 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 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamiAaK qbaoaabmaakeaajugibiaadIhaaOGaayjkaiaawMcaaKqzGeGaeyyp a0tcfa4aaCbeaOqaaKqzGeGaciiBaiaacMgacaGGTbaajqwaG9FaaK qzadGaeuiLdqKaamiEaiabgkziUkaaicdaaSqabaqcfa4aaSaaaOqa aKqzGeGaamiCaKqbaoaabmaakeaajugibiaadIfacqGH8aapcaWG4b Gaey4kaSIaeuiLdqKaamiEaiaacYhacaWGybGaeyOpa4JaamiEaaGc caGLOaGaayzkaaaabaqcLbsacqqHuoarcaWG4baaaiabg2da9Kqbao aalaaakeaajugibiaadAgajuaGdaqadaGcbaqcLbsacaWG4baakiaa wIcacaGLPaaaaeaajugibiaaigdacqGHsislcaWGgbqcfa4aaeWaaO qaaKqzGeGaamiEaaGccaGLOaGaayzkaaaaaaaa@685D@

and m( x )=E[ Xx|X>x ]= 1 1F( x ) x [ 1F( t ) ] dt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamyBaK qbaoaabmaakeaajugibiaadIhaaOGaayjkaiaawMcaaKqzGeGaeyyp a0JaamyraKqbaoaadmaakeaajugibiaadIfacqGHsislcaWG4bGaai iFaiaadIfacqGH+aGpcaWG4baakiaawUfacaGLDbaajugibiabg2da 9KqbaoaalaaakeaajugibiaaigdaaOqaaKqzGeGaaGymaiabgkHiTi aadAeajuaGdaqadaGcbaqcLbsacaWG4baakiaawIcacaGLPaaaaaqc fa4aa8qmaOqaaKqbaoaadmaakeaajugibiaaigdacqGHsislcaWGgb qcfa4aaeWaaOqaaKqzGeGaamiDaaGccaGLOaGaayzkaaaacaGLBbGa ayzxaaaajeaibaqcLbmacaWG4baajeaqbaqcLboacqGHEisPaKqzGe Gaey4kIipacaWGKbGaamiDaaaa@654F@

 The corresponding hazard rate function, h( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamiAaK qbaoaabmaakeaajugibiaadIhaaOGaayjkaiaawMcaaaaa@3B1E@  and the mean residual function, m( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamyBaK qbaoaabmaakeaajugibiaadIhaaOGaayjkaiaawMcaaaaa@3B23@ of TPSD (2.1) are thus obtained as

h( x )= θ 3 ( α+x+ x 2 ) θ 2 ( α+x+ x 2 )+2θx+θ+2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamiAaK qbaoaabmaakeaajugibiaadIhaaOGaayjkaiaawMcaaKqzGeGaeyyp a0tcfa4aaSaaaOqaaKqzGeGaeqiUdexcfa4aaWbaaSqabKqaGeaaju gWaiaaiodaaaqcfa4aaeWaaOqaaKqzGeGaeqySdeMaey4kaSIaamiE aiabgUcaRiaadIhajuaGdaahaaWcbeqcbasaaKqzadGaaGOmaaaaaO GaayjkaiaawMcaaaqaaKqzGeGaeqiUdexcfa4aaWbaaSqabKqaGeaa jugWaiaaikdaaaqcfa4aaeWaaOqaaKqzGeGaeqySdeMaey4kaSIaam iEaiabgUcaRiaadIhajuaGdaahaaWcbeqcbasaaKqzGcGaaGOmaaaa aOGaayjkaiaawMcaaKqzGeGaey4kaSIaaGOmaiabeI7aXjaadIhacq GHRaWkcqaH4oqCcqGHRaWkcaaIYaaaaaaa@6646@

And m( x )= α θ 2 +θ+2 [ ( α θ 2 +θ+2 )+θx( θx+θ+2 ) ] e θx x [ 1+ θt( θt+θ+2 ) α θ 2 +θ+2 ] e θt dt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamyBaK qbaoaabmaakeaajugibiaadIhaaOGaayjkaiaawMcaaKqzGeGaeyyp a0tcfa4aaSaaaOqaaKqzGeGaeqySdeMaeqiUdexcfa4aaWbaaSqabK qaGeaajugWaiaaikdaaaqcLbsacqGHRaWkcqaH4oqCcqGHRaWkcaaI YaaakeaajuaGdaWadaGcbaqcfa4aaeWaaOqaaKqzGeGaeqySdeMaeq iUdexcfa4aaWbaaSqabKqaGeaajugWaiaaikdaaaqcLbsacqGHRaWk cqaH4oqCcqGHRaWkcaaIYaaakiaawIcacaGLPaaajugibiabgUcaRi abeI7aXjaadIhajuaGdaqadaGcbaqcLbsacqaH4oqCcaWG4bGaey4k aSIaeqiUdeNaey4kaSIaaGOmaaGccaGLOaGaayzkaaaacaGLBbGaay zxaaqcLbsacaWGLbqcfa4aaWbaaSqabeaajugibiabgkHiTiabeI7a XjaadIhaaaaaaKqbaoaapedakeaajuaGdaWadaGcbaqcLbsacaaIXa Gaey4kaSscfa4aaSaaaOqaaKqzGeGaeqiUdeNaamiDaKqbaoaabmaa keaajugibiabeI7aXjaadshacqGHRaWkcqaH4oqCcqGHRaWkcaaIYa aakiaawIcacaGLPaaaaeaajugibiabeg7aHjabeI7aXLqbaoaaCaaa leqabaqcLbsacaaIYaaaaiabgUcaRiabeI7aXjabgUcaRiaaikdaaa aakiaawUfacaGLDbaaaKqaGeaajugWaiaadIhaaKqaGeaajugWaiab g6HiLcqcLbsacqGHRiI8aiaaykW7caWGLbqcfa4aaWbaaSqabKqaGe aajugWaiabgkHiTiabeI7aXjaadshaaaqcLbsacaWGKbGaamiDaaaa @9C89@

= θ 2 ( α+x+ x 2 )+2θ( 2x+1 )+6 θ[ ( α θ 2 +θ+2 )+θx( θx+θ+2 ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeyypa0 tcfa4aaSaaaOqaaKqzGeGaeqiUdexcfa4aaWbaaSqabKqaGeaajugW aiaaikdaaaqcfa4aaeWaaOqaaKqzGeGaeqySdeMaey4kaSIaamiEai abgUcaRiaadIhajuaGdaahaaWcbeqcbasaaKqzadGaaGOmaaaaaOGa ayjkaiaawMcaaKqzGeGaey4kaSIaaGOmaiabeI7aXLqbaoaabmaake aajugibiaaikdacaWG4bGaey4kaSIaaGymaaGccaGLOaGaayzkaaqc LbsacqGHRaWkcaaI2aaakeaajugibiabeI7aXLqbaoaadmaakeaaju aGdaqadaGcbaqcLbsacqaHXoqycqaH4oqCjuaGdaahaaWcbeqcbasa aKqzadGaaGOmaaaajugibiabgUcaRiabeI7aXjabgUcaRiaaikdaaO GaayjkaiaawMcaaKqzGeGaey4kaSIaeqiUdeNaamiEaKqbaoaabmaa keaajugibiabeI7aXjaadIhacqGHRaWkcqaH4oqCcqGHRaWkcaaIYa aakiaawIcacaGLPaaaaiaawUfacaGLDbaaaaaaaa@7489@

= θ 2 x 2 +θ( θ+4 )x+( α θ 2 +2θ+6 ) θ[ θ 2 x 2 +θ( θ+2 )x+( α θ 2 +θ+2 ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeyypa0 tcfa4aaSaaaOqaaKqzGeGaeqiUdexcfa4aaWbaaSqabKqaGeaajugW aiaaikdaaaqcLbsacaWG4bqcfa4aaWbaaSqabKqaGeaajugWaiaaik daaaqcLbsacqGHRaWkcqaH4oqCjuaGdaqadaGcbaqcLbsacqaH4oqC cqGHRaWkcaaI0aaakiaawIcacaGLPaaajugibiaadIhacqGHRaWkju aGdaqadaGcbaqcLbsacqaHXoqycqaH4oqCjuaGdaahaaWcbeqcbasa aKqzadGaaGOmaaaajugibiabgUcaRiaaikdacqaH4oqCcqGHRaWkca aI2aaakiaawIcacaGLPaaaaeaajugibiabeI7aXLqbaoaadmaakeaa jugibiabeI7aXLqbaoaaCaaaleqajeaibaqcLbmacaaIYaaaaKqzGe GaamiEaKqbaoaaCaaaleqajeaibaqcLbmacaaIYaaaaKqzGeGaey4k aSIaeqiUdexcfa4aaeWaaOqaaKqzGeGaeqiUdeNaey4kaSIaaGOmaa GccaGLOaGaayzkaaqcLbsacaWG4bGaey4kaSscfa4aaeWaaOqaaKqz GeGaeqySdeMaeqiUdexcfa4aaWbaaSqabKqaGeaajugWaiaaikdaaa qcLbsacqGHRaWkcqaH4oqCcqGHRaWkcaaIYaaakiaawIcacaGLPaaa aiaawUfacaGLDbaaaaaaaa@8323@

It can be easily verified that h( 0 )= α θ 3 α θ 2 +θ+2 =f( 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamiAaK qbaoaabmaakeaajugibiaaicdaaOGaayjkaiaawMcaaKqzGeGaeyyp a0tcfa4aaSaaaOqaaKqzGeGaeqySdeMaeqiUdexcfa4aaWbaaSqabK qaGeaajugWaiaaiodaaaaakeaajugibiabeg7aHjabeI7aXLqbaoaa CaaaleqajeaibaqcLbmacaaIYaaaaKqzGeGaey4kaSIaeqiUdeNaey 4kaSIaaGOmaaaacqGH9aqpcaWGMbqcfa4aaeWaaOqaaKqzGeGaaGim aaGccaGLOaGaayzkaaaaaa@54B3@  and m( 0 )= α θ 2 +2θ+6 θ(α θ 2 +θ+2) = μ 1 / . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamyBaK qbaoaabmaakeaajugibiaaicdaaOGaayjkaiaawMcaaKqzGeGaeyyp a0tcfa4aaSaaaOqaaKqzGeGaeqySdeMaeqiUdexcfa4aaWbaaSqabK qaGeaajugWaiaaikdaaaqcLbsacqGHRaWkcaaIYaGaeqiUdeNaey4k aSIaaGOnaaGcbaqcLbsacqaH4oqCcaGGOaGaeqySdeMaeqiUdexcfa 4aaWbaaSqabKqaGeaajugWaiaaikdaaaqcLbsacqGHRaWkcqaH4oqC cqGHRaWkcaaIYaGaaiykaaaacqGH9aqpcqaH8oqBjuaGdaqhaaqcba saaKqzadGaaGymaaqcbasaaKqzadGaai4laaaajugibiaac6caaaa@60B1@

It can also be easily verified that the expression of h(x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacaWGObGaaiikaiaadIhacaGGPaaaaa@39DD@ and m(x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacaWGTbGaaiikaiaadIhacaGGPaaaaa@39E2@ of TPSD reduce to the corresponding h(x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacaWGObGaaiikaiaadIhacaGGPaaaaa@39DD@  and of Sujatha distribution at α=1. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqySde Maeyypa0JaaGymaiaac6caaaa@3A8C@

The behavior of h(x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacaWGObGaaiikaiaadIhacaGGPaaaaa@39DD@  and m(x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacaWGTbGaaiikaiaadIhacaGGPaaaaa@39E2@ of TPSD (2.1) for different values of its parameters are shown in Figures 4 & 5 respectively.

Figure 4Behavior of of TPSD for selected values of parameters θ and α.
Figure 5Behavior of of TPSD for selected values of parameters θ and α.

It is clearly seen from the graphs of h(x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacaWGObGaaiikaiaadIhacaGGPaaaaa@39DD@ and m(x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacaWGTbGaaiikaiaadIhacaGGPaaaaa@39E2@  that h(x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacaWGObGaaiikaiaadIhacaGGPaaaaa@39DD@ is monotonically increasing function of x,θandα MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamiEai aacYcacqaH4oqCcaaMc8Uaaeyyaiaab6gacaqGKbGaaGPaVlabeg7a Hbaa@414E@ where as m(x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacaWGTbGaaiikaiaadIhacaGGPaaaaa@39E2@  is monotonically decreasing function of x,θandα MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamiEai aacYcacqaH4oqCcaaMc8Uaaeyyaiaab6gacaqGKbGaaGPaVlabeg7a Hbaa@414E@ .

Stochastic ordering

Stochastic ordering of positive continuous random variable is an important tool for judging the comparative behavior of continuous distributions. A random variable X is said to be smaller than a random variable Y in the

  1. Stochastic order ( X st Y ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaake aajugibiaadIfacqGHKjYOjuaGdaWgaaqcbasaaKqzadGaam4Caiaa dshaaSqabaqcLbsacaWGzbaakiaawIcacaGLPaaaaaa@40A7@  if F x ( x ) F y ( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamOraK qbaoaaBaaajeaibaqcLbmacaWG4baaleqaaKqbaoaabmaakeaajugi biaadIhaaOGaayjkaiaawMcaaKqzGeGaeyyzImRaamOraKqbaoaaBa aajeaibaqcLbmacaWG5baaleqaaKqbaoaabmaakeaajugibiaadIha aOGaayjkaiaawMcaaaaa@47F2@  for all x
  2. Hazard rate order ( X hr Y ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaake aajugibiaadIfacqGHKjYOjuaGdaWgaaqcbasaaKqzadGaamiAaiaa dkhaaSqabaqcLbsacaWGzbaakiaawIcacaGLPaaaaaa@409A@ if h x ( x ) h y ( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamiAaK qbaoaaBaaajeaibaqcLbmacaWG4baaleqaaKqbaoaabmaakeaajugi biaadIhaaOGaayjkaiaawMcaaKqzGeGaeyyzImRaamiAaKqbaoaaBa aajeaibaqcLbmacaWG5baaleqaaKqbaoaabmaakeaajugibiaadIha aOGaayjkaiaawMcaaaaa@4836@  for all x
  3. Mean residual life order ( X mrl Y ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaake aajugibiaadIfacqGHKjYOjuaGdaWgaaqcbasaaKqzadGaamyBaiaa dkhacaWGSbaaleqaaKqzGeGaamywaaGccaGLOaGaayzkaaaaaa@4190@  if m x ( x ) m y ( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamyBaK qbaoaaBaaajeaibaqcLbmacaWG4baaleqaaKqbaoaabmaakeaajugi biaadIhaaOGaayjkaiaawMcaaKqzGeGaeyizImQaamyBaKqbaoaaBa aajeaibaqcLbmacaWG5baaleqaaKqbaoaabmaakeaajugibiaadIha aOGaayjkaiaawMcaaaaa@482F@  for all x
  4. Likelihood ratio order ( X lr Y ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaake aajugibiaadIfacqGHKjYOjuaGdaWgaaqcbasaaKqzadGaamiBaiaa dkhaaSqabaqcLbsacaWGzbaakiaawIcacaGLPaaaaaa@409E@  if f x ( x ) f y ( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaake aajugibiaadAgajuaGdaWgaaqcbasaaKqzadGaamiEaaWcbeaajuaG daqadaGcbaqcLbsacaWG4baakiaawIcacaGLPaaaaeaajugibiaadA gajuaGdaWgaaqcbasaaKqzadGaamyEaaWcbeaajuaGdaqadaGcbaqc LbsacaWG4baakiaawIcacaGLPaaaaaaaaa@4714@  decreases in x

The following results due to Shaked & Shanthikumar8 are well known for establishing stochastic ordering of distributions

X lr YX hr x st y YX mlr Y. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamiwai abgsMiJMqbaoaaBaaajeaibaqcLbsacaWGSbGaamOCaaWcbeaajugi biaadMfacqGHshI3caWGybqcfa4aaCbeaOqaaKqzGeGaeyizImAcfa 4aaSbaaKazba2=baqcLbmacaWGObGaamOCaaWcbeaaaKqaGeaalmaa wafajeaibeqccasaaKqzadGaamiEaiabgsMiJUWaaSbaaKGaGeaaju gWaiaadohacaWG0baajiaibeaajugWaiaadMhaaKGaGeqajqaibaqc LbmacqGHthY3aaaaleqaaKqzGeGaamywaiabgkDiElaadIfacqGHKj YOjuaGdaWgaaqcbasaaKqzadGaamyBaiaadYgacaWGYbaaleqaaKqz GeGaamywaiaaykW7caaMc8UaaiOlaaaa@68C4@

The TPSD (2.1) is ordered with respect to the strongest “likelihood ratio” ordering as shown in the following theorem:

Theorem: Let X~TPSD( θ 1 , α 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamiwai aac6hacaqGubGaaeiuaiaabofacaqGebqcfa4aaeWaaOqaaKqzGeGa eqiUdexcfa4aaSbaaKqaGeaajugWaiaaigdaaSqabaqcLbsacaGGSa GaeqySdewcfa4aaSbaaKqaGeaajugWaiaaigdaaSqabaaakiaawIca caGLPaaaaaa@4888@  and Y~TPSD( θ 2 , α 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamywai aac6hacaqGubGaaeiuaiaabofacaqGebqcfa4aaeWaaOqaaKqzGeGa eqiUdexcfa4aaSbaaKqaGeaajugWaiaaikdaaSqabaqcLbsacaGGSa GaeqySdewcfa4aaSbaaKqaGeaajugWaiaaikdaaSqabaaakiaawIca caGLPaaaaaa@488B@ .If θ 1 > θ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqiUde xcfa4aaSbaaKqaGeaajugWaiaaigdaaSqabaqcLbsacqGH+aGpcqaH 4oqCjuaGdaWgaaqcbasaaKqzadGaaGOmaaWcbeaaaaa@4118@  and α 1 = α 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqySde wcfa4aaSbaaKqaGeaajugWaiaaigdaaSqabaqcLbsacqGH9aqpcqaH XoqyjuaGdaWgaaqcbasaaKqzadGaaGOmaaWcbeaaaaa@40E8@ (or θ 1 = θ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqiUde xcfa4aaSbaaKqaGeaajugWaiaaigdaaSqabaqcLbsacqGH9aqpcqaH 4oqCjuaGdaWgaaqcbasaaKqzadGaaGOmaaWcbeaaaaa@4116@  and α 1 α 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqySde wcfa4aaSbaaKqaGeaajugWaiaaigdaaSqabaqcLbsacqGHLjYScqaH XoqyjuaGdaWgaaqcbasaaKqzadGaaGOmaaWcbeaaaaa@41A8@ ) then X lr YandhenceX hr Y,X mrl YandX st Y. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamiwai abgsMiJMqbaoaaBaaajeaibaqcLbmacaWGSbGaamOCaaWcbeaajugi biaadMfacaaMc8UaaGPaVlaabggacaqGUbGaaeizaiaaykW7caqGOb Gaaeyzaiaab6gacaqGJbGaaeyzaiaaykW7caWGybGaeyizImAcfa4a aSbaaKqaGeaajugWaiaadIgacaWGYbaaleqaaKqzGeGaamywaiaacY cacaWGybGaeyizImAcfa4aaSbaaKqaGeaajugWaiaad2gacaWGYbGa amiBaaWcbeaajugibiaadMfacaaMc8Uaaeyyaiaab6gacaqGKbGaaG PaVlaadIfacqGHKjYOjuaGdaWgaaqcbasaaKqzadGaam4Caiaadsha aSqabaqcLbsacaWGzbGaaiOlaaaa@6C03@  

Proof: We have

f X ( x; θ 1 , α 1 ) f Y ( x; θ 2 , α 2 ) = θ 1 3 ( α 2 θ 2 2 + θ 2 +2 ) θ 2 3 ( α 1 θ 1 2 + θ 1 +2 ) ( α 1 +x+ x 2 α 2 +x+ x 2 ) e ( θ 1 θ 2 )x ;x>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaake aajugibiaadAgajuaGdaWgaaqcbasaaKqzadGaamiwaaWcbeaajuaG daqadaGcbaqcLbsacaWG4bGaai4oaiabeI7aXLqbaoaaBaaajeaiba qcLbmacaaIXaaaleqaaKqzGeGaaiilaiabeg7aHLqbaoaaBaaajeai baqcLbmacaaIXaaaleqaaaGccaGLOaGaayzkaaaabaqcLbsacaWGMb qcfa4aaSbaaKqaGeaajugWaiaadMfaaSqabaqcfa4aaeWaaOqaaKqz GeGaamiEaiaacUdacqaH4oqCjuaGdaWgaaqcbasaaKqzadGaaGOmaa WcbeaajugibiaacYcacqaHXoqyjuaGdaWgaaqcbauaaKqzGdGaaGOm aaWcbeaaaOGaayjkaiaawMcaaaaajugibiabg2da9Kqbaoaalaaake aajugibiabeI7aXLqbaoaaBaaajeaibaqcLbmacaaIXaaaleqaaKqb aoaaCaaaleqajeaibaqcLbmacaaIZaaaaKqbaoaabmaakeaajugibi abeg7aHLqbaoaaBaaajeaibaqcLbmacaaIYaaaleqaaKqzGeGaeqiU dexcfa4aaSbaaKqaGeaajugWaiaaikdaaSqabaqcfa4aaWbaaSqabK qaGeaajugWaiaaikdaaaqcLbsacqGHRaWkcqaH4oqCjuaGdaWgaaqc basaaKqzadGaaGOmaaWcbeaajugibiabgUcaRiaaikdaaOGaayjkai aawMcaaaqaaKqzGeGaeqiUdexcfa4aaSbaaKqaGeaajugWaiaaikda aSqabaqcfa4aaWbaaSqabKqaGeaajugWaiaaiodaaaqcfa4aaeWaaO qaaKqzGeGaeqySdewcfa4aaSbaaKqaGeaajugWaiaaigdaaSqabaqc LbsacqaH4oqCjuaGdaWgaaqcbasaaKqzadGaaGymaaWcbeaajuaGda ahaaWcbeqcbasaaKqzadGaaGOmaaaajugibiabgUcaRiabeI7aXLqb aoaaBaaajeaibaqcLbmacaaIXaaaleqaaKqzGeGaey4kaSIaaGOmaa GccaGLOaGaayzkaaaaaKqbaoaabmaakeaajuaGdaWcaaGcbaqcLbsa cqaHXoqyjuaGdaWgaaqcbasaaKqzadGaaGymaaWcbeaajugibiabgU caRiaadIhacqGHRaWkcaWG4bqcfa4aaWbaaSqabKqaGeaajugWaiaa ikdaaaaakeaajugibiabeg7aHLqbaoaaBaaajeaibaqcLbmacaaIYa aaleqaaKqzGeGaey4kaSIaamiEaiabgUcaRiaadIhajuaGdaahaaWc beqcbasaaKqzadGaaGOmaaaaaaaakiaawIcacaGLPaaajugibiaadw gajuaGdaahaaqcbasabeaajugWaiabgkHiTKqbaoaabmaajeaibaqc LbmacqaH4oqCjuaGdaWgaaqccasaaKqzadGaaGymaaqccasabaqcLb macqGHsislcqaH4oqCjuaGdaWgaaqccasaaKqzadGaaGOmaaqccasa baaajeaicaGLOaGaayzkaaqcLbmacaWG4baaaKqzGeGaai4oaiaadI hacqGH+aGpcaaIWaaaaa@D1F4@

ln f X ( x; θ 1 , α 1 ) f Y ( x; θ 2 , α 2 ) =ln[ θ 1 3 ( α 2 θ 2 2 + θ 2 +2 ) θ 2 3 ( α 1 θ 1 2 + θ 1 +2 ) ]+ln( α 1 +x+ x 2 α 2 +x+ x 2 )( θ 1 θ 2 )x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaciiBai aac6gajuaGdaWcaaGcbaqcLbsacaWGMbqcfa4aaSbaaKqaGeaajugW aiaadIfaaSqabaqcfa4aaeWaaOqaaKqzGeGaamiEaiaacUdacqaH4o qCjuaGdaWgaaqcbasaaKqzadGaaGymaaWcbeaajugibiaacYcacqaH XoqyjuaGdaWgaaqcbasaaKqzadGaaGymaaWcbeaaaOGaayjkaiaawM caaaqaaKqzGeGaamOzaKqbaoaaBaaajeaibaqcLbmacaWGzbaaleqa aKqbaoaabmaakeaajugibiaadIhacaGG7aGaeqiUdexcfa4aaSbaaK qaGeaajugWaiaaikdaaSqabaqcLbsacaGGSaGaeqySdewcfa4aaSba aKqaGeaajugWaiaaikdaaSqabaaakiaawIcacaGLPaaaaaqcLbsacq GH9aqpciGGSbGaaiOBaKqbaoaadmaakeaajuaGdaWcaaGcbaqcLbsa cqaH4oqCjuaGdaWgaaqcbasaaKqzadGaaGymaaWcbeaajuaGdaahaa WcbeqcbasaaKqzadGaaG4maaaajuaGdaqadaGcbaqcLbsacqaHXoqy juaGdaWgaaqcbasaaKqzadGaaGOmaaWcbeaajugibiabeI7aXLqbao aaBaaajeaibaqcLbmacaaIYaaaleqaaKqbaoaaCaaaleqajeaibaqc LbmacaaIYaaaaKqzGeGaey4kaSIaeqiUdexcfa4aaSbaaKqaGeaaju gWaiaaikdaaSqabaqcLbsacqGHRaWkcaaIYaaakiaawIcacaGLPaaa aeaajugibiabeI7aXLqbaoaaBaaajeaibaqcLbmacaaIYaaaleqaaK qbaoaaCaaaleqajeaibaqcLbmacaaIZaaaaKqbaoaabmaakeaajugi biabeg7aHLqbaoaaBaaajeaibaqcLbmacaaIXaaaleqaaKqzGeGaeq iUdexcfa4aaSbaaKqaGeaajugWaiaaigdaaSqabaqcfa4aaWbaaSqa bKqaGeaajugWaiaaikdaaaqcLbsacqGHRaWkcqaH4oqCjuaGdaWgaa qcKfaG=haajugOaiaaigdaaSqabaqcLbsacqGHRaWkcaaIYaaakiaa wIcacaGLPaaaaaaacaGLBbGaayzxaaqcLbsacqGHRaWkciGGSbGaai OBaKqbaoaabmaakeaajuaGdaWcaaGcbaqcLbsacqaHXoqyjuaGdaWg aaqcbasaaKqzadGaaGymaaWcbeaajugibiabgUcaRiaadIhacqGHRa WkcaWG4bqcfa4aaWbaaSqabKqaGeaajugWaiaaikdaaaaakeaajugi biabeg7aHLqbaoaaBaaajeaibaqcLbmacaaIYaaaleqaaKqzGeGaey 4kaSIaamiEaiabgUcaRiaadIhajuaGdaahaaWcbeqcbauaaKqzGdGa aGOmaaaaaaaakiaawIcacaGLPaaajugibiabgkHiTKqbaoaabmaake aajugibiabeI7aXLqbaoaaBaaajeaibaqcLbmacaaIXaaaleqaaKqz GeGaeyOeI0IaeqiUdexcfa4aaSbaaKqaGeaajugWaiaaikdaaSqaba aakiaawIcacaGLPaaajugibiaadIhaaaa@D46F@

This gives d dx ln f X ( x; θ 1 , α 1 ) f Y ( x; θ 2 , α 2 ) = ( α 2 α 1 )+2( α 2 α 1 )x ( α 1 +x+ x 2 )( α 2 +x+ x 2 ) ( θ 1 θ 2 ). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaake aajugibiaadsgaaOqaaKqzGeGaamizaiaadIhaaaGaciiBaiaac6ga juaGdaWcaaGcbaqcLbsacaWGMbqcfa4aaSbaaKqaGeaajugWaiaadI faaSqabaqcfa4aaeWaaOqaaKqzGeGaamiEaiaacUdacqaH4oqCjuaG daWgaaqcbasaaKqzadGaaGymaaWcbeaajugibiaacYcacqaHXoqyju aGdaWgaaqcbasaaKqzadGaaGymaaWcbeaaaOGaayjkaiaawMcaaaqa aKqzGeGaamOzaKqbaoaaBaaajeaibaqcLbmacaWGzbaaleqaaKqbao aabmaakeaajugibiaadIhacaGG7aGaeqiUdexcfa4aaSbaaKqaGeaa jugWaiaaikdaaSqabaqcLbsacaGGSaGaeqySdewcfa4aaSbaaKqaGe aajugWaiaaikdaaSqabaaakiaawIcacaGLPaaaaaqcLbsacqGH9aqp juaGdaWcaaGcbaqcfa4aaeWaaOqaaKqzGeGaeqySdewcfa4aaSbaaK qaGeaajugWaiaaikdaaSqabaqcLbsacqGHsislcqaHXoqyjuaGdaWg aaqcbasaaKqzadGaaGymaaWcbeaaaOGaayjkaiaawMcaaKqzGeGaey 4kaSIaaGOmaKqbaoaabmaakeaajugibiabeg7aHLqbaoaaBaaajeai baqcLbmacaaIYaaaleqaaKqzGeGaeyOeI0IaeqySdewcfa4aaSbaaK qaGeaajugWaiaaigdaaSqabaaakiaawIcacaGLPaaajugibiaadIha aOqaaKqbaoaabmaakeaajugibiabeg7aHLqbaoaaBaaajeaibaqcLb macaaIXaaaleqaaKqzGeGaey4kaSIaamiEaiabgUcaRiaadIhajuaG daahaaWcbeqcbasaaKqzadGaaGOmaaaaaOGaayjkaiaawMcaaKqbao aabmaakeaajugibiabeg7aHLqbaoaaBaaajeaibaqcLbmacaaIYaaa leqaaKqzGeGaey4kaSIaamiEaiabgUcaRiaadIhajuaGdaahaaWcbe qcbasaaKqzadGaaGOmaaaaaOGaayjkaiaawMcaaaaajugibiabgkHi TKqbaoaabmaakeaajugibiabeI7aXLqbaoaaBaaajeaibaqcLbmaca aIXaaaleqaaKqzGeGaeyOeI0IaeqiUdexcfa4aaSbaaKqaGeaajugW aiaaikdaaSqabaaakiaawIcacaGLPaaajugibiaac6caaaa@AFD0@

Thus, for ( θ 1 > θ 2 and α 1 = α 2 )or( α 1 α 2 and θ 1 = θ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaake aajugibiabeI7aXLqbaoaaBaaajeaibaqcLbmacaaIXaaaleqaaKqz GeGaeyOpa4JaeqiUdexcfa4aaSbaaKqaGeaajugWaiaaikdaaSqaba qcLbsacaaMc8Uaaeyyaiaab6gacaqGKbGaaGPaVlabeg7aHLqbaoaa BaaajeaibaqcLbmacaaIXaaaleqaaKqzGeGaeyypa0JaeqySdewcfa 4aaSbaaKqaGeaajugWaiaaikdaaSqabaaakiaawIcacaGLPaaajugi biaaykW7caaMc8Uaae4BaiaabkhacaaMc8UaaGPaVNqbaoaabmaake aajugibiabeg7aHLqbaoaaBaaajeaibaqcLbmacaaIXaaaleqaaKqz GeGaeyyzImRaeqySdewcfa4aaSbaaKqaGeaajugWaiaaikdaaSqaba qcLbsacaaMc8Uaaeyyaiaab6gacaqGKbGaaGPaVlaaykW7cqaH4oqC juaGdaWgaaqcbasaaKqzadGaaGymaaWcbeaajugibiabg2da9iabeI 7aXLqbaoaaBaaajeaibaqcLbmacaaIYaaaleqaaaGccaGLOaGaayzk aaqcLbsacaaMc8UaaGPaVdaa@80C9@ , d dx ln f X ( x; θ 1 , α 1 ) f Y ( x; θ 2 , α 2 ) <0. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaake aajugibiaadsgaaOqaaKqzGeGaamizaiaadIhaaaGaciiBaiaac6ga juaGdaWcaaGcbaqcLbsacaWGMbqcfa4aaSbaaKqaGeaajugWaiaadI faaSqabaqcfa4aaeWaaOqaaKqzGeGaamiEaiaacUdacqaH4oqCjuaG daWgaaqcbasaaKqzadGaaGymaaWcbeaajugibiaacYcacqaHXoqyju aGdaWgaaqcbasaaKqzadGaaGymaaWcbeaaaOGaayjkaiaawMcaaaqa aKqzGeGaamOzaKqbaoaaBaaajeaibaqcLbmacaWGzbaaleqaaKqbao aabmaakeaajugibiaadIhacaGG7aGaeqiUdexcfa4aaSbaaKqaGeaa jugWaiaaikdaaSqabaqcLbsacaGGSaGaeqySdewcfa4aaSbaaKqaGe aajugWaiaaikdaaSqabaaakiaawIcacaGLPaaaaaqcLbsacqGH8aap caaIWaGaaGPaVlaaykW7caGGUaaaaa@6948@

This means that X lr YandhenceX hr Y,X mrl YandX st Y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamiwai abgsMiJMqbaoaaBaaajeaibaqcLbmacaWGSbGaamOCaaWcbeaajugi biaadMfacaaMc8UaaGPaVlaabggacaqGUbGaaeizaiaaykW7caaMc8 UaaeiAaiaabwgacaqGUbGaae4yaiaabwgacaaMc8UaaGPaVlaadIfa cqGHKjYOjuaGdaWgaaqcbasaaKqzadGaamiAaiaadkhaaSqabaqcLb sacaWGzbGaaiilaiaadIfacqGHKjYOjuaGdaWgaaqcbasaaKqzadGa amyBaiaadkhacaWGSbaaleqaaKqzGeGaamywaiaaykW7caaMc8Uaae yyaiaab6gacaqGKbGaaGPaVlaaykW7caWGybGaeyizImAcfa4aaSba aKqaafaajug4aiaadohacaWG0baaleqaaKqzGeGaamywaaaa@71BD@ . This shows flexibility of TPSD over Sujatha distribution.

Mean deviations

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

  δ 1 ( x )= 0 | xμ | f( x )dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqiTdq wcfa4aaSbaaKqaGeaajugWaiaaigdaaSqabaqcfa4aaeWaaOqaaKqz GeGaamiEaaGccaGLOaGaayzkaaqcLbsacqGH9aqpjuaGdaWdXaGcba qcfa4aaqWaaOqaaKqzGeGaamiEaiabgkHiTiabeY7aTbGccaGLhWUa ayjcSdaajeaibaqcLbmacaaIWaaajeaibaqcLbmacqGHEisPaKqzGe Gaey4kIipacaWGMbqcfa4aaeWaaOqaaKqzGeGaamiEaaGccaGLOaGa ayzkaaqcLbsacaWGKbGaamiEaaaa@5755@  and δ 2 ( x )= 0 | xM | f( x )dx, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqiTdq wcfa4aaSbaaKqaGeaajugWaiaaikdaaSqabaqcfa4aaeWaaOqaaKqz GeGaamiEaaGccaGLOaGaayzkaaqcLbsacqGH9aqpjuaGdaWdXaGcba qcfa4aaqWaaOqaaKqzGeGaamiEaiabgkHiTiaad2eaaOGaay5bSlaa wIa7aaqcbasaaKqzadGaaGimaaqcbasaaKqzadGaeyOhIukajugibi abgUIiYdGaamOzaKqbaoaabmaakeaajugibiaadIhaaOGaayjkaiaa wMcaaKqzGeGaamizaiaadIhacaGGSaaaaa@5722@ respectively,

where μ=E( X ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqiVd0 Maeyypa0JaamyraKqbaoaabmaakeaajugibiaadIfaaOGaayjkaiaa wMcaaaaa@3D97@  and M=Median( X ). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamytai abg2da9iaad2eacaWGLbGaamizaiaadMgacaWGHbGaamOBaKqbaoaa bmaakeaajugibiaadIfaaOGaayjkaiaawMcaaKqzGeGaaGPaVlaac6 caaaa@4421@  The measures δ 1 ( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqiTdq wcfa4aaSbaaKqaGeaajugWaiaaigdaaSqabaqcfa4aaeWaaOqaaKqz GeGaamiEaaGccaGLOaGaayzkaaaaaa@3EA3@  and δ 2 ( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqiTdq wcfa4aaSbaaKqaGeaajugWaiaaikdaaSqabaqcfa4aaeWaaOqaaKqz GeGaamiEaaGccaGLOaGaayzkaaaaaa@3EA4@  can be calculated using the following relationships

δ 1 ( x )= 0 μ ( μx ) f( x )dx+ μ ( xμ ) f( x )dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqiTdq wcfa4aaSbaaKqaGeaajugWaiaaigdaaSqabaqcfa4aaeWaaOqaaKqz GeGaamiEaaGccaGLOaGaayzkaaqcLbsacqGH9aqpjuaGdaWdXaGcba qcfa4aaeWaaOqaaKqzGeGaeqiVd0MaeyOeI0IaamiEaaGccaGLOaGa ayzkaaaajeaibaqcLbmacaaIWaaajeaibaqcLbmacqaH8oqBaKqzGe Gaey4kIipacaWGMbqcfa4aaeWaaOqaaKqzGeGaamiEaaGccaGLOaGa ayzkaaqcLbsacaWGKbGaamiEaiabgUcaRKqbaoaapedakeaajuaGda qadaGcbaqcLbsacaWG4bGaeyOeI0IaeqiVd0gakiaawIcacaGLPaaa aKqaGeaajugWaiabeY7aTbqcbasaaKqzadGaeyOhIukajugibiabgU IiYdGaamOzaKqbaoaabmaakeaajugibiaadIhaaOGaayjkaiaawMca aKqzGeGaamizaiaadIhaaaa@6D63@

=μF( μ ) 0 μ x f( x )dxμ[ 1F( μ ) ]+ μ x f( x )dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeyypa0 JaeqiVd0MaamOraKqbaoaabmaakeaajugibiabeY7aTbGccaGLOaGa ayzkaaqcLbsacqGHsisljuaGdaWdXaGcbaqcLbsacaWG4baajeaiba qcLbmacaaIWaaajeaibaqcLbmacqaH8oqBaKqzGeGaey4kIipacaWG Mbqcfa4aaeWaaOqaaKqzGeGaamiEaaGccaGLOaGaayzkaaqcLbsaca WGKbGaamiEaiabgkHiTiabeY7aTLqbaoaadmaakeaajugibiaaigda cqGHsislcaWGgbqcfa4aaeWaaOqaaKqzGeGaeqiVd0gakiaawIcaca GLPaaaaiaawUfacaGLDbaajugibiabgUcaRKqbaoaapedakeaajugi biaadIhaaKqaGeaajugWaiabeY7aTbqcbasaaKqzadGaeyOhIukaju gibiabgUIiYdGaamOzaKqbaoaabmaakeaajugibiaadIhaaOGaayjk aiaawMcaaKqzGeGaamizaiaadIhaaaa@70AA@

=2μF( μ )2μ+2 μ x f( x )dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeyypa0 JaaGOmaiabeY7aTjaadAeajuaGdaqadaGcbaqcLbsacqaH8oqBaOGa ayjkaiaawMcaaKqzGeGaeyOeI0IaaGOmaiabeY7aTjabgUcaRiaaik dajuaGdaWdXaGcbaqcLbsacaWG4baajeaibaqcLbmacqaH8oqBaKqa GeaajugWaiabg6HiLcqcLbsacqGHRiI8aiaadAgajuaGdaqadaGcba qcLbsacaWG4baakiaawIcacaGLPaaajugibiaadsgacaWG4baaaa@566B@

=2μF( μ )2 0 μ x f( x )dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeyypa0 JaaGOmaiabeY7aTjaadAeajuaGdaqadaGcbaqcLbsacqaH8oqBaOGa ayjkaiaawMcaaKqzGeGaeyOeI0IaaGOmaKqbaoaapedakeaajugibi aadIhaaKqaGeaajugWaiaaicdaaKqaGeaajugWaiabeY7aTbqcLbsa cqGHRiI8aiaadAgajuaGdaqadaGcbaqcLbsacaWG4baakiaawIcaca GLPaaajugibiaadsgacaWG4baaaa@5260@     (4.3.1)

and δ 2 ( x )= 0 M ( Mx )f( x )dx+ M ( xM ) f( x )dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqiTdq wcfa4aaSbaaKqaGeaajugWaiaaikdaaSqabaqcfa4aaeWaaOqaaKqz GeGaamiEaaGccaGLOaGaayzkaaqcLbsacqGH9aqpjuaGdaWdXaGcba qcfa4aaeWaaOqaaKqzGeGaamytaiabgkHiTiaadIhaaOGaayjkaiaa wMcaaKqzGeGaamOzaKqbaoaabmaakeaajugibiaadIhaaOGaayjkai aawMcaaKqzGeGaamizaiaadIhacqGHRaWkaKqaGeaajugWaiaaicda aKqaGeaajugWaiaad2eaaKqzGeGaey4kIipajuaGdaWdXbGcbaqcfa 4aaeWaaOqaaKqzGeGaamiEaiabgkHiTiaad2eaaOGaayjkaiaawMca aaqcbasaaKqzadGaamytaaqcbasaaKqzadGaeyOhIukajugibiabgU IiYdGaamOzaKqbaoaabmaakeaajugibiaadIhaaOGaayjkaiaawMca aKqzGeGaamizaiaadIhaaaa@6AA3@

=MF( M ) 0 M x f( x )dxM( 1F( M ) )+ M x f( x )dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeyypa0 JaamytaiaadAeajuaGdaqadaGcbaqcLbsacaWGnbaakiaawIcacaGL PaaajugibiabgkHiTKqbaoaapedakeaajugibiaadIhaaKqaGeaaju gWaiaaicdaaKqaGeaajugWaiaad2eaaKqzGeGaey4kIipacaWGMbqc fa4aaeWaaOqaaKqzGeGaamiEaaGccaGLOaGaayzkaaqcLbsacaWGKb GaamiEaiabgkHiTiaad2eajuaGdaqadaGcbaqcLbsacaaIXaGaeyOe I0IaamOraKqbaoaabmaakeaajugibiaad2eaaOGaayjkaiaawMcaaa GaayjkaiaawMcaaKqzGeGaey4kaSscfa4aa8qmaOqaaKqzGeGaamiE aaqcbauaaKqzGdGaamytaaqcbasaaKqzadGaeyOhIukajugibiabgU IiYdGaamOzaKqbaoaabmaakeaajugibiaadIhaaOGaayjkaiaawMca aKqzGeGaamizaiaadIhaaaa@6B29@

=μ+2 M x f( x )dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeyypa0 JaeqiVd0Maey4kaSIaaGOmaKqbaoaapedakeaajugibiaadIhaaKqa GeaajugWaiaad2eaaKqaGeaajugWaiabg6HiLcqcLbsacqGHRiI8ai aadAgajuaGdaqadaGcbaqcLbsacaWG4baakiaawIcacaGLPaaajugi biaadsgacaWG4baaaa@4BA2@

=μ2 0 M x f( x )dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeyypa0 JaeqiVd0MaeyOeI0IaaGOmaKqbaoaapedakeaajugibiaadIhaaKqa GeaajugWaiaaicdaaKazba4=baqcLbkacaWGnbaajugibiabgUIiYd GaamOzaKqbaoaabmaakeaajugibiaadIhaaOGaayjkaiaawMcaaKqz GeGaamizaiaadIhaaaa@4C99@    (4.3.2)

Using the pdf (2.1) and expression for the mean of TPSD, we get

0 μ x f 4 ( x;θ,α ) dx=μ [ θ 3 ( μ 3 + μ 2 +αμ )+ θ 2 ( 3 μ 2 +2μ+α )+2θ( 3μ+1 )+6 ] e θμ θ( α θ 2 +θ+2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaapedake aajugibiaadIhacaaMc8UaamOzaKqbaoaaBaaaleaajugibiaaisda aSqabaqcfa4aaeWaaOqaaKqzGeGaamiEaiaacUdacqaH4oqCcaGGSa GaeqySdegakiaawIcacaGLPaaaaKqaGeaajugWaiaaicdaaKqaGeaa jugWaiabeY7aTbqcLbsacqGHRiI8aiaaykW7caWGKbGaamiEaiabg2 da9iabeY7aTjabgkHiTKqbaoaalaaakeaajuaGdaWadaGcbaqcLbsa cqaH4oqCjuaGdaahaaWcbeqcbasaaKqzadGaaG4maaaajuaGdaqada GcbaqcLbsacqaH8oqBjuaGdaahaaWcbeqcbasaaKqzadGaaG4maaaa jugibiabgUcaRiabeY7aTLqbaoaaCaaaleqajeaibaqcLbmacaaIYa aaaKqzGeGaey4kaSIaeqySdeMaeqiVd0gakiaawIcacaGLPaaajugi biabgUcaRiabeI7aXLqbaoaaCaaaleqajeaibaqcLbmacaaIYaaaaK qbaoaabmaakeaajugibiaaiodacqaH8oqBjuaGdaahaaWcbeqcbasa aKqzadGaaGOmaaaajugibiabgUcaRiaaikdacqaH8oqBcqGHRaWkcq aHXoqyaOGaayjkaiaawMcaaKqzGeGaey4kaSIaaGOmaiabeI7aXLqb aoaabmaakeaajugibiaaiodacqaH8oqBcqGHRaWkcaaIXaaakiaawI cacaGLPaaajugibiabgUcaRiaaiAdaaOGaay5waiaaw2faaKqzGeGa amyzaKqbaoaaCaaaleqajeaibaqcLbmacqGHsislcqaH4oqCcqaH8o qBaaaakeaajugibiabeI7aXLqbaoaabmaakeaajugibiabeg7aHjab eI7aXLqbaoaaCaaaleqajeaibaqcLbmacaaIYaaaaKqzGeGaey4kaS IaeqiUdeNaey4kaSIaaGOmaaGccaGLOaGaayzkaaaaaaaa@A744@     (4.3.3)

0 M x f 4 ( x;θ,α ) dx=μ [ θ 3 ( M 3 + M 2 +αM )+ θ 2 ( 3 M 2 +2M+α )+2θ( 3M+1 )+6 ] e θM θ( α θ 2 +θ+2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaapedake aajugibiaadIhacaaMc8UaamOzaKqbaoaaBaaaleaajugibiaaisda aSqabaqcfa4aaeWaaOqaaKqzGeGaamiEaiaacUdacqaH4oqCcaGGSa GaeqySdegakiaawIcacaGLPaaaaKqaGeaajugWaiaaicdaaKqaGeaa jugWaiaad2eaaKqzGeGaey4kIipacaaMc8UaaGPaVlaadsgacaWG4b Gaeyypa0JaeqiVd0MaeyOeI0scfa4aaSaaaOqaaKqbaoaadmaakeaa jugibiabeI7aXLqbaoaaCaaaleqajeaibaqcLbmacaaIZaaaaKqbao aabmaakeaajugibiaad2eajuaGdaahaaWcbeqcbasaaKqzadGaaG4m aaaajugibiabgUcaRiaad2eajuaGdaahaaWcbeqcbasaaKqzadGaaG OmaaaajugibiabgUcaRiabeg7aHjaad2eaaOGaayjkaiaawMcaaKqz GeGaey4kaSIaeqiUdexcfa4aaWbaaSqabKqaGeaajugWaiaaikdaaa qcfa4aaeWaaOqaaKqzGeGaaG4maiaad2eajuaGdaahaaWcbeqcbasa aKqzadGaaGOmaaaajugibiabgUcaRiaaikdacaWGnbGaey4kaSIaeq ySdegakiaawIcacaGLPaaajugibiabgUcaRiaaikdacqaH4oqCjuaG daqadaGcbaqcLbsacaaIZaGaamytaiabgUcaRiaaigdaaOGaayjkai aawMcaaKqzGeGaey4kaSIaaGOnaaGccaGLBbGaayzxaaqcLbsacaWG Lbqcfa4aaWbaaSqabKqaGeaajugWaiabgkHiTiabeI7aXjaad2eaaa aakeaajugibiabeI7aXLqbaoaabmaakeaajugibiabeg7aHjabeI7a XLqbaoaaCaaaleqajeaqbaqcLboacaaIYaaaaKqzGeGaey4kaSIaeq iUdeNaey4kaSIaaGOmaaGccaGLOaGaayzkaaaaaaaa@A1EF@    (4.3.4)

Using expressions from (4.3.1), (4.3.2), (4.3.3) and (4.3.4) and after some tedious algebraic simplifications, the mean deviation about the mean, δ 1 ( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqiTdq wcfa4aaSbaaKqaGeaajugWaiaaigdaaSqabaqcfa4aaeWaaOqaaKqz GeGaamiEaaGccaGLOaGaayzkaaaaaa@3EA3@ and the mean deviation about the median, δ 2 ( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqiTdq wcfa4aaSbaaKqaGeaajugWaiaaikdaaSqabaqcfa4aaeWaaOqaaKqz GeGaamiEaaGccaGLOaGaayzkaaaaaa@3EA4@  of TPSD are obtained as

δ 1 ( x )= 2[ θ 2 ( μ 2 +μ+α )+2θ( 2μ+1 )+6 ] e θμ θ( α θ 2 +θ+2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqiTdq wcfa4aaSbaaKqaGeaajugWaiaaigdaaSqabaqcfa4aaeWaaOqaaKqz GeGaamiEaaGccaGLOaGaayzkaaqcLbsacqGH9aqpjuaGdaWcaaGcba qcLbsacaaIYaqcfa4aamWaaOqaaKqzGeGaeqiUdexcfa4aaWbaaSqa bKqaGeaajugWaiaaikdaaaqcfa4aaeWaaOqaaKqzGeGaeqiVd0wcfa 4aaWbaaSqabKqaGeaajugWaiaaikdaaaqcLbsacqGHRaWkcqaH8oqB cqGHRaWkcqaHXoqyaOGaayjkaiaawMcaaKqzGeGaey4kaSIaaGOmai abeI7aXLqbaoaabmaakeaajugibiaaikdacqaH8oqBcqGHRaWkcaaI XaaakiaawIcacaGLPaaajugibiabgUcaRiaaiAdaaOGaay5waiaaw2 faaKqzGeGaamyzaKqbaoaaCaaaleqajeaibaqcLbmacqGHsislcqaH 4oqCcqaH8oqBaaaakeaajugibiabeI7aXLqbaoaabmaakeaajugibi abeg7aHjabeI7aXLqbaoaaCaaaleqajeaqbaqcLboacaaIYaaaaKqz GeGaey4kaSIaeqiUdeNaey4kaSIaaGOmaaGccaGLOaGaayzkaaaaaa aa@7BAE@    (4.3.5)

and δ 2 ( x )= 2[ θ 3 ( M 3 + M 2 +αM )+ θ 2 ( 3 M 2 +2M+α )+2θ( 3M+1 )+6 ] e θM θ( α θ 2 +θ+2 ) μ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqiTdq wcfa4aaSbaaKqaGeaajugWaiaaikdaaSqabaqcfa4aaeWaaOqaaKqz GeGaamiEaaGccaGLOaGaayzkaaqcLbsacqGH9aqpjuaGdaWcaaGcba qcLbsacaaIYaqcfa4aamWaaOqaaKqzGeGaeqiUdexcfa4aaWbaaSqa bKqaGeaajugWaiaaiodaaaqcfa4aaeWaaOqaaKqzGeGaamytaKqbao aaCaaaleqajeaibaqcLbmacaaIZaaaaKqzGeGaey4kaSIaamytaKqb aoaaCaaaleqajeaibaqcLbmacaaIYaaaaKqzGeGaey4kaSIaeqySde MaamytaaGccaGLOaGaayzkaaqcLbsacqGHRaWkcqaH4oqCjuaGdaah aaWcbeqcbasaaKqzadGaaGOmaaaajuaGdaqadaGcbaqcLbsacaaIZa GaamytaKqbaoaaCaaaleqajeaibaqcLbmacaaIYaaaaKqzGeGaey4k aSIaaGOmaiaad2eacqGHRaWkcqaHXoqyaOGaayjkaiaawMcaaKqzGe Gaey4kaSIaaGOmaiabeI7aXLqbaoaabmaakeaajugibiaaiodacaWG nbGaey4kaSIaaGymaaGccaGLOaGaayzkaaqcLbsacqGHRaWkcaaI2a aakiaawUfacaGLDbaajugibiaadwgajuaGdaahaaWcbeqcbasaaKqz adGaeyOeI0IaeqiUdeNaamytaaaaaOqaaKqzGeGaeqiUdexcfa4aae WaaOqaaKqzGeGaeqySdeMaeqiUdexcfa4aaWbaaSqabKqaafaajug4 aiaaikdaaaqcLbsacqGHRaWkcqaH4oqCcqGHRaWkcaaIYaaakiaawI cacaGLPaaaaaqcLbsacqGHsislcqaH8oqBaaa@9212@    (4.3.6)

Bonferroni and Lorenz curves and indices

The Bonferroni and Lorenz curves and Bonferroni9 and Gini indices have applications not only in economics to study income and poverty, but also in other fields like reliability, demography and medical science. The Bonferroni and Lorenz curves are defined as

B( p )= 1 pμ 0 q xf( x ) dx= 1 pμ [ 0 x f( x )dx q x f( x )dx ]= 1 pμ [ μ q x f( x )dx ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamOqaK qbaoaabmaakeaajugibiaadchaaOGaayjkaiaawMcaaKqzGeGaeyyp a0tcfa4aaSaaaOqaaKqzGeGaaGymaaGcbaqcLbsacaWGWbGaeqiVd0 gaaKqbaoaapehakeaajugibiaadIhacaWGMbqcfa4aaeWaaOqaaKqz GeGaamiEaaGccaGLOaGaayzkaaaajeaibaqcLbmacaaIWaaajeaiba qcLbmacaWGXbaajugibiabgUIiYdGaaGPaVlaaykW7caWGKbGaamiE aiabg2da9KqbaoaalaaakeaajugibiaaigdaaOqaaKqzGeGaamiCai abeY7aTbaajuaGdaWadaGcbaqcfa4aa8qCaOqaaKqzGeGaamiEaaqc basaaKqzadGaaGimaaqcbauaaKqzGdGaeyOhIukajugibiabgUIiYd GaamOzaKqbaoaabmaakeaajugibiaadIhaaOGaayjkaiaawMcaaKqz GeGaamizaiaadIhacqGHsisljuaGdaWdXbGcbaqcLbsacaWG4baaje aibaqcLbmacaWGXbaajeaibaqcLbmacqGHEisPaKqzGeGaey4kIipa caWGMbqcfa4aaeWaaOqaaKqzGeGaamiEaaGccaGLOaGaayzkaaqcLb sacaWGKbGaamiEaaGccaGLBbGaayzxaaqcLbsacqGH9aqpjuaGdaWc aaGcbaqcLbsacaaIXaaakeaajugibiaadchacqaH8oqBaaqcfa4aam WaaOqaaKqzGeGaeqiVd0MaeyOeI0scfa4aa8qCaOqaaKqzGeGaamiE aaqcbasaaKqzadGaamyCaaqcbasaaKqzadGaeyOhIukajugibiabgU IiYdGaamOzaKqbaoaabmaakeaajugibiaadIhaaOGaayjkaiaawMca aKqzGeGaamizaiaadIhaaOGaay5waiaaw2faaaaa@9EA5@   (4.4.1)

and L( p )= 1 μ 0 q x f( x )dx= 1 μ [ 0 x f( x )dx q x f( x )dx ]= 1 μ [ μ q x f( x )dx ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamitaK qbaoaabmaakeaajugibiaadchaaOGaayjkaiaawMcaaKqzGeGaeyyp a0tcfa4aaSaaaOqaaKqzGeGaaGymaaGcbaqcLbsacqaH8oqBaaqcfa 4aa8qCaOqaaKqzGeGaamiEaaqcbasaaKqzadGaaGimaaqcbasaaKqz adGaamyCaaqcLbsacqGHRiI8aiaadAgajuaGdaqadaGcbaqcLbsaca WG4baakiaawIcacaGLPaaajugibiaadsgacaWG4bGaeyypa0tcfa4a aSaaaOqaaKqzGeGaaGymaaGcbaqcLbsacqaH8oqBaaqcfa4aamWaaO qaaKqbaoaapehakeaajugibiaadIhaaKqaGeaajugWaiaaicdaaKqa GeaajugWaiabg6HiLcqcLbsacqGHRiI8aiaadAgajuaGdaqadaGcba qcLbsacaWG4baakiaawIcacaGLPaaajugibiaadsgacaWG4bGaeyOe I0scfa4aa8qCaOqaaKqzGeGaamiEaaqcbasaaKqzadGaamyCaaqcba saaKqzadGaeyOhIukajugibiabgUIiYdGaamOzaKqbaoaabmaakeaa jugibiaadIhaaOGaayjkaiaawMcaaKqzGeGaamizaiaadIhaaOGaay 5waiaaw2faaKqzGeGaeyypa0tcfa4aaSaaaOqaaKqzGeGaaGymaaGc baqcLbsacqaH8oqBaaqcfa4aamWaaOqaaKqzGeGaeqiVd0MaeyOeI0 scfa4aa8qCaOqaaKqzGeGaamiEaaqcbasaaKqzadGaamyCaaqcbasa aKqzadGaeyOhIukajugibiabgUIiYdGaamOzaKqbaoaabmaakeaaju gibiaadIhaaOGaayjkaiaawMcaaKqzGeGaamizaiaadIhaaOGaay5w aiaaw2faaaaa@9909@    (4.4.2)

respectively or equivalently.

B( p )= 1 pμ 0 p F 1 ( x ) dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamOqaK qbaoaabmaakeaajugibiaadchaaOGaayjkaiaawMcaaKqzGeGaeyyp a0tcfa4aaSaaaOqaaKqzGeGaaGymaaGcbaqcLbsacaWGWbGaeqiVd0 gaaKqbaoaapehakeaajugibiaadAeajuaGdaahaaWcbeqcbasaaKqz adGaeyOeI0IaaGymaaaajuaGdaqadaGcbaqcLbsacaWG4baakiaawI cacaGLPaaaaKqaGeaajugWaiaaicdaaKqaGeaajugWaiaadchaaKqz GeGaey4kIipacaaMc8UaaGPaVlaadsgacaWG4baaaa@575A@    (4.4.3)

 and L( p )= 1 μ 0 p F 1 ( x ) dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamitaK qbaoaabmaakeaajugibiaadchaaOGaayjkaiaawMcaaKqzGeGaeyyp a0tcfa4aaSaaaOqaaKqzGeGaaGymaaGcbaqcLbsacqaH8oqBaaqcfa 4aa8qCaOqaaKqzGeGaamOraKqbaoaaCaaaleqajeaibaqcLbmacqGH sislcaaIXaaaaKqbaoaabmaakeaajugibiaadIhaaOGaayjkaiaawM caaaqcbauaaKqzGdGaaGimaaqcbasaaKqzadGaamiCaaqcLbsacqGH RiI8aiaaykW7caaMc8UaamizaiaadIhaaaa@56AF@     (4.4.4)

respectively, where μ=E( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqiVd0 Maeyypa0JaamyraKqbaoaabmaakeaajugibiaadIhaaOGaayjkaiaa wMcaaaaa@3DB7@  and q= F 1 ( p ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamyCai abg2da9iaadAeajuaGdaahaaWcbeqcbasaaKqzadGaeyOeI0IaaGym aaaajuaGdaqadaGcbaqcLbsacaWGWbaakiaawIcacaGLPaaaaaa@40AB@ .

The Bonferroni and Gini indices are thus defined as

B=1 0 1 B( p ) dp MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamOqai abg2da9iaaigdacqGHsisljuaGdaWdXbGcbaqcLbsacaWGcbqcfa4a aeWaaOqaaKqzGeGaamiCaaGccaGLOaGaayzkaaaajeaibaqcLbmaca aIWaaajeaibaqcLbmacaaIXaaajugibiabgUIiYdGaaGPaVlaaykW7 caWGKbGaamiCaaaa@4B85@    (4.4.5)

and G=12 0 1 L( p ) dp MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaam4rai abg2da9iaaigdacqGHsislcaaIYaqcfa4aa8qCaOqaaKqzGeGaamit aKqbaoaabmaakeaajugibiaadchaaOGaayjkaiaawMcaaaqcbasaaK qzadGaaGimaaqcbasaaKqzadGaaGymaaqcLbsacqGHRiI8aiaaykW7 caaMc8Uaamizaiaadchaaaa@4C50@    (4.4.6)

respectively.

Using pdf of TPSD (2.1), we get

q x f( x )dx= { θ 3 ( q 3 + q 2 +αq )+ θ 2 ( 3 q 2 +2q+α )+2θ( 3q+1 )+6} e θq θ( α θ 2 +2θ+6 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaapehake aajugibiaadIhaaKqaafaajug4aiaadghaaKqaGeaajugWaiabg6Hi LcqcLbsacqGHRiI8aiaadAgajuaGdaqadaGcbaqcLbsacaWG4baaki aawIcacaGLPaaajugibiaadsgacaWG4bGaeyypa0tcfa4aaSaaaOqa aKqzGeGaai4EaiabeI7aXLqbaoaaCaaaleqajeaibaqcLbmacaaIZa aaaKqbaoaabmaakeaajugibiaadghajuaGdaahaaWcbeqcbasaaKqz adGaaG4maaaajugibiabgUcaRiaadghajuaGdaahaaWcbeqcbasaaK qzadGaaGOmaaaajugibiabgUcaRiabeg7aHjaadghaaOGaayjkaiaa wMcaaKqzGeGaey4kaSIaeqiUdexcfa4aaWbaaSqabKqaafaajug4ai aaikdaaaqcfa4aaeWaaOqaaKqzGeGaaG4maiaadghajuaGdaahaaWc beqcbasaaKqzadGaaGOmaaaajugibiabgUcaRiaaikdacaWGXbGaey 4kaSIaeqySdegakiaawIcacaGLPaaajugibiabgUcaRiaaikdacqaH 4oqCjuaGdaqadaGcbaqcLbsacaaIZaGaamyCaiabgUcaRiaaigdaaO GaayjkaiaawMcaaKqzGeGaey4kaSIaaGOnaiaac2hacaWGLbqcfa4a aWbaaSqabKqaGeaajugWaiabgkHiTiabeI7aXjaadghaaaaakeaaju gibiabeI7aXLqbaoaabmaakeaajugibiabeg7aHjabeI7aXLqbaoaa CaaaleqajeaibaqcLbmacaaIYaaaaKqzGeGaey4kaSIaaGOmaiabeI 7aXjabgUcaRiaaiAdaaOGaayjkaiaawMcaaaaaaaa@9658@    (4.4.7)

Now using equation (4.4.7), (4.4.1) and (4.4.2), we get

B( p )= 1 p [ 1 { θ 3 ( q 3 + q 2 +αq )+ θ 2 ( 3 q 2 +2q+α )+2θ( 3q+1 )+6} e θq α θ 2 +2θ+6 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamOqaK qbaoaabmaakeaajugibiaadchaaOGaayjkaiaawMcaaKqzGeGaeyyp a0tcfa4aaSaaaOqaaKqzGeGaaGymaaGcbaqcLbsacaWGWbaaaKqbao aadmaakeaajugibiaaigdacqGHsisljuaGdaWcaaGcbaqcLbsacaGG 7bGaeqiUdexcfa4aaWbaaSqabKqaGeaajugWaiaaiodaaaqcfa4aae WaaOqaaKqzGeGaamyCaKqbaoaaCaaaleqajeaibaqcLbmacaaIZaaa aKqzGeGaey4kaSIaamyCaKqbaoaaCaaaleqajeaibaqcLbmacaaIYa aaaKqzGeGaey4kaSIaeqySdeMaamyCaaGccaGLOaGaayzkaaqcLbsa cqGHRaWkcqaH4oqCjuaGdaahaaWcbeqcbasaaKqzadGaaGOmaaaaju aGdaqadaGcbaqcLbsacaaIZaGaamyCaKqbaoaaCaaaleqajeaibaqc LbmacaaIYaaaaKqzGeGaey4kaSIaaGOmaiaadghacqGHRaWkcqaHXo qyaOGaayjkaiaawMcaaKqzGeGaey4kaSIaaGOmaiabeI7aXLqbaoaa bmaakeaajugibiaaiodacaWGXbGaey4kaSIaaGymaaGccaGLOaGaay zkaaqcLbsacqGHRaWkcaaI2aGaaiyFaiaadwgajuaGdaahaaWcbeqc basaaKqzadGaeyOeI0IaeqiUdeNaamyCaaaaaOqaaKqzGeGaeqySde MaeqiUdexcfa4aaWbaaSqabKqaGeaajugWaiaaikdaaaqcLbsacqGH RaWkcaaIYaGaeqiUdeNaey4kaSIaaGOnaaaaaOGaay5waiaaw2faaa aa@8E15@    (4.4.8)

and L( p )=1 { θ 3 ( q 3 + q 2 +αq )+ θ 2 ( 3 q 2 +2q+α )+2θ( 3q+1 )+6} e θq α θ 2 +2θ+6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamitaK qbaoaabmaakeaajugibiaadchaaOGaayjkaiaawMcaaKqzGeGaeyyp a0JaaGymaiabgkHiTKqbaoaalaaakeaajugibiaacUhacqaH4oqCju aGdaahaaWcbeqcbasaaKqzadGaaG4maaaajuaGdaqadaGcbaqcLbsa caWGXbqcfa4aaWbaaSqabKqaGeaajugWaiaaiodaaaqcLbsacqGHRa WkcaWGXbqcfa4aaWbaaSqabKqaGeaajugWaiaaikdaaaqcLbsacqGH RaWkcqaHXoqycaWGXbaakiaawIcacaGLPaaajugibiabgUcaRiabeI 7aXLqbaoaaCaaaleqajeaibaqcLbmacaaIYaaaaKqbaoaabmaakeaa jugibiaaiodacaWGXbqcfa4aaWbaaSqabKqaGeaajugWaiaaikdaaa qcLbsacqGHRaWkcaaIYaGaamyCaiabgUcaRiabeg7aHbGccaGLOaGa ayzkaaqcLbsacqGHRaWkcaaIYaGaeqiUdexcfa4aaeWaaOqaaKqzGe GaaG4maiaadghacqGHRaWkcaaIXaaakiaawIcacaGLPaaajugibiab gUcaRiaaiAdacaGG9bGaamyzaKqbaoaaCaaaleqajeaibaqcLbmacq GHsislcqaH4oqCcaWGXbaaaaGcbaqcLbsacqaHXoqycqaH4oqCjuaG daahaaWcbeqcbasaaKqzadGaaGOmaaaajugibiabgUcaRiaaikdacq aH4oqCcqGHRaWkcaaI2aaaaaaa@877C@  (4.4.9)

Now using the equations (4.4.8) and (4.4.9) in (4.4.5) and (4.4.6), the Bonferroni and Gini indices of TPSD (2.1) are obtained as

B=1 { θ 3 ( q 3 + q 2 +αq )+ θ 2 ( 3 q 2 +2q+α )+2θ( 3q+1 )+6} e θq α θ 2 +2θ+6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamOqai abg2da9iaaigdacqGHsisljuaGdaWcaaGcbaqcLbsacaGG7bGaeqiU dexcfa4aaWbaaSqabKqaGeaajugWaiaaiodaaaqcfa4aaeWaaOqaaK qzGeGaamyCaKqbaoaaCaaaleqajeaibaqcLbmacaaIZaaaaKqzGeGa ey4kaSIaamyCaKqbaoaaCaaaleqajeaibaqcLbmacaaIYaaaaKqzGe Gaey4kaSIaeqySdeMaamyCaaGccaGLOaGaayzkaaqcLbsacqGHRaWk cqaH4oqCjuaGdaahaaWcbeqcbasaaKqzadGaaGOmaaaajuaGdaqada GcbaqcLbsacaaIZaGaamyCaKqbaoaaCaaaleqajeaibaqcLbmacaaI YaaaaKqzGeGaey4kaSIaaGOmaiaadghacqGHRaWkcqaHXoqyaOGaay jkaiaawMcaaKqzGeGaey4kaSIaaGOmaiabeI7aXLqbaoaabmaakeaa jugibiaaiodacaWGXbGaey4kaSIaaGymaaGccaGLOaGaayzkaaqcLb sacqGHRaWkcaaI2aGaaiyFaiaadwgajuaGdaahaaWcbeqcbasaaKqz adGaeyOeI0IaeqiUdeNaamyCaaaaaOqaaKqzGeGaeqySdeMaeqiUde xcfa4aaWbaaSqabKqaGeaajugWaiaaikdaaaqcLbsacqGHRaWkcaaI YaGaeqiUdeNaey4kaSIaaGOnaaaaaaa@8334@  (4.4.10)

G=1+ 2{ θ 3 ( q 3 + q 2 +αq )+ θ 2 ( 3 q 2 +2q+α )+2θ( 3q+1 )+6} e θq α θ 2 +2θ+6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaam4rai abg2da9iabgkHiTiaaigdacqGHRaWkjuaGdaWcaaGcbaqcLbsacaaI YaGaai4EaiabeI7aXLqbaoaaCaaaleqajeaibaqcLbmacaaIZaaaaK qbaoaabmaakeaajugibiaadghajuaGdaahaaWcbeqcbasaaKqzadGa aG4maaaajugibiabgUcaRiaadghajuaGdaahaaWcbeqcbasaaKqzad GaaGOmaaaajugibiabgUcaRiabeg7aHjaadghaaOGaayjkaiaawMca aKqzGeGaey4kaSIaeqiUdexcfa4aaWbaaSqabKqaGeaajugWaiaaik daaaqcfa4aaeWaaOqaaKqzGeGaaG4maiaadghajuaGdaahaaWcbeqc basaaKqzadGaaGOmaaaajugibiabgUcaRiaaikdacaWGXbGaey4kaS IaeqySdegakiaawIcacaGLPaaajugibiabgUcaRiaaikdacqaH4oqC juaGdaqadaGcbaqcLbsacaaIZaGaamyCaiabgUcaRiaaigdaaOGaay jkaiaawMcaaKqzGeGaey4kaSIaaGOnaiaac2hacaWGLbqcfa4aaWba aSqabKqaGeaajugWaiabgkHiTiabeI7aXjaadghaaaaakeaajugibi abeg7aHjabeI7aXLqbaoaaCaaaleqajeaqbaqcLboacaaIYaaaaKqz GeGaey4kaSIaaGOmaiabeI7aXjabgUcaRiaaiAdaaaaaaa@8517@  (4.4.11)

Stress–strength reliability

The stress–strength reliability of a component illustrates the life of the component which has random strength that is subjected to random stress. When the stress of the component Y applied to it exceeds the strength of the component X, the component fails instantly and the component will function satisfactorily till X > Y . Therefore, R = P (Y < X ) is a measure of the component reliability and is known as stress–strength reliability in statistical literature. It has extensive application in almost all areas of knowledge especially in engineering such as structure, deterioration of rocket motor, static fatigue of ceramic component, aging of concrete pressure vessels etc.

Let X and Y be independent strength and stress random variables having TPSD (2.1) with parameter ( θ 1 , α 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaake aajugibiabeI7aXLqbaoaaBaaajqwaG9FaaKqzadGaaGymaaWcbeaa jugibiaacYcacqaHXoqyjuaGdaWgaaqcKfay=haajugWaiaaigdaaS qabaaakiaawIcacaGLPaaaaaa@4619@  and ( θ 2 , α 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaake aajugibiabeI7aXLqbaoaaBaaajeaibaqcLbmacaaIYaaaleqaaKqz GeGaaiilaiabeg7aHLqbaoaaBaaajeaibaqcLbmacaaIYaaaleqaaa GccaGLOaGaayzkaaaaaa@42D5@  respectively. Then the stress–strength reliability R of TPSD can be obtained as

R=P( Y<X )= 0 P( Y<X|X=x ) f x ( x )dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaae+qcLbsaca WGsbGaeyypa0JaamiuaKqbaoaabmaakeaajugibiaadMfacqGH8aap caWGybaakiaawIcacaGLPaaajugibiabg2da9Kqbaoaapehakeaaju gibiaadcfajuaGdaqadaGcbaqcLbsacaWGzbGaeyipaWJaamiwaiaa cYhacaWGybGaeyypa0JaamiEaaGccaGLOaGaayzkaaaajeaibaqcLb macaaIWaaajeaibaqcLbmacqGHEisPaKqzGeGaey4kIipacaWGMbqc fa4aaSbaaKqaGeaajugWaiaadIhaaSqabaqcfa4aaeWaaOqaaKqzGe GaamiEaaGccaGLOaGaayzkaaqcLbsacaWGKbGaamiEaaaa@5E06@

= 0 f( x; θ 1, α 1 ) F( x; θ 2, α 2 )dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeyypa0 tcfa4aa8qCaOqaaKqzGeGaamOzaKqbaoaabmaakeaajugibiaadIha caGG7aGaeqiUdexcfa4aaSbaaKqaGeaajugWaiaaigdacaGGSaaale qaaKqzGeGaeqySdewcfa4aaSbaaKqaGeaajugWaiaaigdaaSqabaaa kiaawIcacaGLPaaaaKqaGeaajugWaiaaicdaaKqaGeaajugWaiabg6 HiLcqcLbsacqGHRiI8aiaadAeajuaGdaqadaGcbaqcLbsacaWG4bGa ai4oaiabeI7aXLqbaoaaBaaajeaibaqcLbmacaaIYaGaaiilaaWcbe aajugibiabeg7aHLqbaoaaBaaajeaibaqcLbmacaaIYaaaleqaaaGc caGLOaGaayzkaaqcLbsacaWGKbGaamiEaaaa@61D7@

=1 θ 1 3 [ ( α 1 α 2 ) θ 2 6 +( 2 α 1 + α 2 +4 α 1 α 2 θ 1 ) θ 2 5 +( 7 α 1 θ 1 +3 α 2 θ 1 +6 α 1 +2 α 2 +6 α 1 α 2 θ 1 2 +3 ) θ 2 4 +( 9 α 1 θ 1 2 +3 α 2 θ 1 2 +18 α 1 θ 1 +4 α 2 θ 1 +7 θ 1 +4 α 1 α 2 θ 1 3 +20 ) θ 2 3 +( 5 α 1 θ 1 3 + α 2 θ 1 3 +20 α 1 θ 1 2 +2 α 2 θ 1 2 +5 θ 1 2 +30 θ 1 + α 1 α 2 θ 1 4 +40 ) θ 2 2 +( α 1 θ 1 3 +10 α 1 θ 1 2 + θ 1 2 +12 θ 1 +20 ) θ 1 θ 2 +2( α 1 θ 1 2 + θ 1 +2 ) θ 1 2 ] ( α 1 θ 1 2 + θ 1 +2 )( α 2 θ 2 2 + θ 2 +2 ) ( θ 1 + θ 2 ) 5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeyypa0 JaaGymaiabgkHiTKqbaoaalaaakeaajugibiabeI7aXLqbaoaaDaaa jeaibaqcLbmacaaIXaaajeaibaqcLbmacaaIZaaaaKqbaoaadmaaju gibqaabeGcbaqcLbsacaGGOaGaeqySdewcfa4aaSbaaKqaGeaajugW aiaaigdaaSqabaqcLbsacqaHXoqyjuaGdaWgaaqcbasaaKqzadGaaG OmaaWcbeaajugibiaacMcacqaH4oqCjuaGdaqhaaqcbasaaKqzadGa aGOmaaqcbasaaKqzadGaaGOnaaaajugibiabgUcaRKqbaoaabmaake aajugibiaaikdacqaHXoqyjuaGdaWgaaqcbasaaKqzadGaaGymaaWc beaajugibiabgUcaRiabeg7aHLqbaoaaBaaajeaibaqcLbmacaaIYa aaleqaaKqzGeGaey4kaSIaaGinaiabeg7aHLqbaoaaBaaajeaibaqc LbmacaaIXaaaleqaaKqzGeGaeqySdewcfa4aaSbaaKqaGeaajugWai aaikdaaSqabaqcLbsacqaH4oqCjuaGdaWgaaqcKfaG=haajugOaiaa igdaaSqabaaakiaawIcacaGLPaaajugibiabeI7aXLqbaoaaDaaaje aibaqcLbmacaaIYaaajeaibaqcLbmacaaI1aaaaKqzGeGaey4kaSsc fa4aaeWaaOqaaKqzGeGaaG4naiabeg7aHLqbaoaaBaaajeaibaqcLb macaaIXaaaleqaaKqzGeGaeqiUdexcfa4aaSbaaKqaGeaajugWaiaa igdaaSqabaqcLbsacqGHRaWkcaaIZaGaeqySdewcfa4aaSbaaKqaaf aajug4aiaaikdaaSqabaqcLbsacqaH4oqCjuaGdaWgaaqcbasaaKqz adGaaGymaaWcbeaajugibiabgUcaRiaaiAdacqaHXoqyjuaGdaWgaa qcbasaaKqzadGaaGymaaWcbeaajugibiabgUcaRiaaikdacqaHXoqy juaGdaWgaaqcbasaaKqzadGaaGOmaaWcbeaajugibiabgUcaRiaaiA dacqaHXoqyjuaGdaWgaaqcbasaaKqzadGaaGymaaWcbeaajugibiab eg7aHLqbaoaaBaaajeaibaqcLbmacaaIYaaaleqaaKqzGeGaeqiUde 3cdaqhaaqcbasaaKqzadGaaGymaaqcbasaaKqzadGaaGOmaaaajugi biabgUcaRiaaiodaaOGaayjkaiaawMcaaKqzGeGaeqiUde3cdaqhaa qcbasaaKqzadGaaGOmaaqcbasaaKqzadGaaGinaaaaaOqaaKqzGeGa ey4kaSscfa4aaeWaaOqaaKqzGeGaaGyoaiabeg7aHLqbaoaaBaaaje aibaqcLbmacaaIXaaaleqaaKqzGeGaeqiUde3cdaqhaaqcbasaaKqz adGaaGymaaqcbasaaKqzadGaaGOmaaaajugibiabgUcaRiaaiodacq aHXoqyjuaGdaWgaaWcbaqcLbsacaaIYaaaleqaaKqzGeGaeqiUde3c daqhaaqcbasaaKqzadGaaGymaaqcbasaaKqzadGaaGOmaaaajugibi abgUcaRiaaigdacaaI4aGaeqySdewcfa4aaSbaaKqaGeaajugWaiaa igdaaSqabaqcLbsacqaH4oqCjuaGdaWgaaqcbasaaKqzadGaaGymaa WcbeaajugibiabgUcaRiaaisdacqaHXoqyjuaGdaWgaaqcbasaaKqz adGaaGOmaaWcbeaajugibiabeI7aXLqbaoaaBaaajeaqbaqcLboaca aIXaaaleqaaKqzGeGaey4kaSIaaG4naiabeI7aXLqbaoaaBaaajeai baqcLbmacaaIXaaaleqaaKqzGeGaey4kaSIaaGinaiabeg7aHLqbao aaBaaajeaibaqcLbmacaaIXaaaleqaaKqzGeGaeqySdewcfa4aaSba aKqaGeaajugWaiaaikdaaSqabaqcLbsacqaH4oqClmaaDaaajeaiba qcLbmacaaIXaaajeaibaqcLbmacaaIZaaaaKqzGeGaey4kaSIaaGOm aiaaicdaaOGaayjkaiaawMcaaKqzGeGaeqiUdexcfa4aa0baaKqaGe 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aibaqcLbmacaaIYaaajeaibaqcLbmacaaIYaaaaaGcbaqcLbsacqGH RaWkjuaGdaqadaGcbaqcLbsacqaHXoqyjuaGdaWgaaqcbasaaKqzad GaaGymaaWcbeaajugibiabeI7aXLqbaoaaDaaajeaibaqcLbmacaaI XaaajeaibaqcLbmacaaIZaaaaKqzGeGaey4kaSIaaGymaiaaicdacq aHXoqyjuaGdaWgaaqcbauaaKqzGdGaaGymaaWcbeaajugibiabeI7a XLqbaoaaDaaajeaibaqcLbmacaaIXaaajeaibaqcLbmacaaIYaaaaK qzGeGaey4kaSIaeqiUdexcfa4aa0baaKqaGeaajugWaiaaigdaaKqa GeaajugWaiaaikdaaaqcLbsacqGHRaWkcaaIXaGaaGOmaiabeI7aXL qbaoaaBaaajeaibaqcLbmacaaIXaaaleqaaKqzGeGaey4kaSIaaGOm aiaaicdaaOGaayjkaiaawMcaaKqzGeGaeqiUdexcfa4aaSbaaKqaGe aajugWaiaaigdaaSqabaqcLbsacqaH4oqCjuaGdaWgaaqcbasaaKqz adGaaGOmaaWcbeaajugibiabgUcaRiaaikdajuaGdaqadaGcbaqcLb sacqaHXoqyjuaGdaWgaaqcbasaaKqzadGaaGymaaWcbeaajugibiab eI7aXLqbaoaaDaaajqwaa+FaaKqzGcGaaGymaaqcKfaG=haajugOai aaikdaaaqcLbsacqGHRaWkcqaH4oqCjuaGdaWgaaqcbasaaKqzadGa aGymaaWcbeaajugibiabgUcaRiaaikdaaOGaayjkaiaawMcaaKqzGe GaeqiUdexcfa4aa0baaKqaGeaajugWaiaaigdaaKqaGeaajugWaiaa ikdaaaaaaOGaay5waiaaw2faaaqaaKqbaoaabmaakeaajugibiabeg 7aHLqbaoaaBaaajeaqbaqcLboacaaIXaaaleqaaKqzGeGaeqiUdexc fa4aa0baaKqaGeaajugWaiaaigdaaKqaGeaajugWaiaaikdaaaqcLb sacqGHRaWkcqaH4oqCjuaGdaWgaaqcbasaaKqzadGaaGymaaWcbeaa jugibiabgUcaRiaaikdaaOGaayjkaiaawMcaaKqbaoaabmaakeaaju gibiabeg7aHLqbaoaaBaaajeaqbaqcLboacaaIYaaaleqaaKqzGeGa eqiUdexcfa4aa0baaKqaGeaajugWaiaaikdaaKqaGeaajugWaiaaik daaaqcLbsacqGHRaWkcqaH4oqCjuaGdaWgaaqcbasaaKqzadGaaGOm aaWcbeaajugibiabgUcaRiaaikdaaOGaayjkaiaawMcaaKqbaoaabm aakeaajugibiabeI7aXLqbaoaaBaaajeaibaqcLbmacaaIXaaaleqa aKqzGeGaey4kaSIaeqiUdexcfa4aaSbaaSqaaKqzGeGaaGOmaaWcbe aaaOGaayjkaiaawMcaaKqbaoaaCaaaleqajeaibaqcLbmacaaI1aaa aaaaaaa@19CD@

It can be verified that the stress–strength reliability of Sujatha distribution is a particular case of stress–strength reliability of TPSD at α 1 = α 2 =1. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqySde wcfa4aaSbaaKqaGeaajugWaiaaigdaaSqabaqcLbsacqGH9aqpcqaH XoqyjuaGdaWgaaqcbasaaKqzadGaaGOmaaWcbeaajugibiabg2da9i aaigdacaaMc8UaaiOlaaaa@4575@

Estimation of parameters

In this section, the estimations of parameters of TPSD using method of moments and method of maximum likelihood have been discussed.

Method of moment estimates (MOME)

Since TPSD (2.1) has two parameters to be estimated, the first two moments about the origin are required to estimate its parameters using method of moments. Equating the population mean to the sample mean, we have

x ¯ = α θ 2 +2θ+6 θ( α θ 2 +θ+2 ) = α θ 2 +θ+2 θ( α θ 2 +θ+2 ) + θ+4 θ( α θ 2 +θ+2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGabmiEay aaraGaeyypa0tcfa4aaSaaaOqaaKqzGeGaeqySdeMaeqiUdexcfa4a aWbaaSqabKqaGeaajugWaiaaikdaaaqcLbsacqGHRaWkcaaIYaGaeq iUdeNaey4kaSIaaGOnaaGcbaqcLbsacqaH4oqCjuaGdaqadaGcbaqc LbsacqaHXoqycqaH4oqCjuaGdaahaaWcbeqcbasaaKqzadGaaGOmaa aajugibiabgUcaRiabeI7aXjabgUcaRiaaikdaaOGaayjkaiaawMca aaaajugibiabg2da9Kqbaoaalaaakeaajugibiabeg7aHjabeI7aXL qbaoaaCaaaleqajeaibaqcLbmacaaIYaaaaKqzGeGaey4kaSIaeqiU deNaey4kaSIaaGOmaaGcbaqcLbsacqaH4oqCjuaGdaqadaGcbaqcLb sacqaHXoqycqaH4oqCjuaGdaahaaWcbeqcbasaaKqzadGaaGOmaaaa jugibiabgUcaRiabeI7aXjabgUcaRiaaikdaaOGaayjkaiaawMcaaa aajugibiabgUcaRKqbaoaalaaakeaajugibiabeI7aXjabgUcaRiaa isdaaOqaaKqzGeGaeqiUdexcfa4aaeWaaOqaaKqzGeGaeqySdeMaeq iUdexcfa4aaWbaaSqabKqaGeaajugWaiaaikdaaaqcLbsacqGHRaWk cqaH4oqCcqGHRaWkcaaIYaaakiaawIcacaGLPaaaaaaaaa@88FE@

x ¯ = 1 θ + θ+4 θ( α θ 2 +θ+2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGabmiEay aaraGaeyypa0tcfa4aaSaaaOqaaKqzGeGaaGymaaGcbaqcLbsacqaH 4oqCaaGaey4kaSscfa4aaSaaaOqaaKqzGeGaeqiUdeNaey4kaSIaaG inaaGcbaqcLbsacqaH4oqCjuaGdaqadaGcbaqcLbsacqaHXoqycqaH 4oqCjuaGdaahaaWcbeqcbasaaKqzadGaaGOmaaaajugibiabgUcaRi abeI7aXjabgUcaRiaaikdaaOGaayjkaiaawMcaaaaaaaa@5237@

( α θ 2 +θ+2 )= θ+4 θ x ¯ 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaake aajugibiabeg7aHjabeI7aXLqbaoaaCaaaleqajeaibaqcLbmacaaI YaaaaKqzGeGaey4kaSIaeqiUdeNaey4kaSIaaGOmaaGccaGLOaGaay zkaaqcLbsacqGH9aqpjuaGdaWcaaGcbaqcLbsacqaH4oqCcqGHRaWk caaI0aaakeaajugibiabeI7aXjqadIhagaqeaiabgkHiTiaaigdaaa aaaa@4EBC@    (5.1.1)

Again equating the second population moment with the corresponding sample moment, we have

m 2 ' = 2( α θ 2 +3θ+12 ) θ 2 ( α θ 2 +θ+2 ) = 2( α θ 2 +θ+2 ) θ 2 ( α θ 2 +θ+2 ) + 4( θ+5 ) θ 2 ( α θ 2 +θ+2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamyBaS Waa0baaKqaGeaajugWaiaaikdaaKqaGeaajugWaiaacEcaaaqcLbsa cqGH9aqpjuaGdaWcaaGcbaqcLbsacaaIYaqcfa4aaeWaaOqaaKqzGe GaeqySdeMaeqiUdexcfa4aaWbaaSqabKqaGeaajugWaiaaikdaaaqc LbsacqGHRaWkcaaIZaGaeqiUdeNaey4kaSIaaGymaiaaikdaaOGaay jkaiaawMcaaaqaaKqzGeGaeqiUdexcfa4aaWbaaSqabeaajugibiaa ikdaaaqcfa4aaeWaaOqaaKqzGeGaeqySdeMaeqiUdexcfa4aaWbaaS qabKqaGeaajugWaiaaikdaaaqcLbsacqGHRaWkcqaH4oqCcqGHRaWk caaIYaaakiaawIcacaGLPaaaaaqcLbsacqGH9aqpjuaGdaWcaaGcba qcLbsacaaIYaqcfa4aaeWaaOqaaKqzGeGaeqySdeMaeqiUdexcfa4a aWbaaSqabKqaGeaajugWaiaaikdaaaqcLbsacqGHRaWkcqaH4oqCcq GHRaWkcaaIYaaakiaawIcacaGLPaaaaeaajugibiabeI7aXLqbaoaa CaaaleqabaqcLbsacaaIYaaaaKqbaoaabmaakeaajugibiabeg7aHj abeI7aXLqbaoaaCaaaleqajeaibaqcLbmacaaIYaaaaKqzGeGaey4k aSIaeqiUdeNaey4kaSIaaGOmaaGccaGLOaGaayzkaaaaaKqzGeGaey 4kaSscfa4aaSaaaOqaaKqzGeGaaGinaKqbaoaabmaakeaajugibiab eI7aXjabgUcaRiaaiwdaaOGaayjkaiaawMcaaaqaaKqzGeGaeqiUde xcfa4aaWbaaSqabKqaGeaajugWaiaaikdaaaqcfa4aaeWaaOqaaKqz GeGaeqySdeMaeqiUdexcfa4aaWbaaSqabKqaafaajug4aiaaikdaaa qcLbsacqGHRaWkcqaH4oqCcqGHRaWkcaaIYaaakiaawIcacaGLPaaa aaaaaa@9FC8@

m 2 ' = 2( α θ 2 +3θ+12 ) θ 2 ( α θ 2 +θ+2 ) = 2 θ 2 + 4( θ+5 ) θ 2 ( α θ 2 +θ+2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamyBaS Waa0baaKqaGeaajugWaiaaikdaaKqaGeaajugWaiaacEcaaaqcLbsa cqGH9aqpjuaGdaWcaaGcbaqcLbsacaaIYaqcfa4aaeWaaOqaaKqzGe GaeqySdeMaeqiUdexcfa4aaWbaaSqabKqaGeaajugWaiaaikdaaaqc LbsacqGHRaWkcaaIZaGaeqiUdeNaey4kaSIaaGymaiaaikdaaOGaay jkaiaawMcaaaqaaKqzGeGaeqiUdexcfa4aaWbaaSqabKqaafaajug4 aiaaikdaaaqcfa4aaeWaaOqaaKqzGeGaeqySdeMaeqiUdexcfa4aaW baaSqabKqaafaajug4aiaaikdaaaqcLbsacqGHRaWkcqaH4oqCcqGH RaWkcaaIYaaakiaawIcacaGLPaaaaaqcLbsacqGH9aqpjuaGdaWcaa GcbaqcLbsacaaIYaaakeaajugibiabeI7aXLqbaoaaCaaaleqajeai baqcLbmacaaIYaaaaaaajugibiabgUcaRKqbaoaalaaakeaajugibi aaisdajuaGdaqadaGcbaqcLbsacqaH4oqCcqGHRaWkcaaI1aaakiaa wIcacaGLPaaaaeaajugibiabeI7aXLqbaoaaCaaaleqajeaibaqcLb macaaIYaaaaKqbaoaabmaakeaajugibiabeg7aHjabeI7aXLqbaoaa CaaaleqajeaqbaqcLboacaaIYaaaaKqzGeGaey4kaSIaeqiUdeNaey 4kaSIaaGOmaaGccaGLOaGaayzkaaaaaaaa@869E@

α θ 2 +θ+2= 4( θ+5 ) m 2 ' θ 2 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqySde MaeqiUdexcfa4aaWbaaSqabKqaGeaajugWaiaaikdaaaqcLbsacqGH RaWkcqaH4oqCcqGHRaWkcaaIYaGaeyypa0tcfa4aaSaaaOqaaKqzGe GaaGinaKqbaoaabmaakeaajugibiabeI7aXjabgUcaRiaaiwdaaOGa ayjkaiaawMcaaaqaaKqzGeGaamyBaSWaa0baaKqaGeaajugWaiaaik daaKqaGeaajugWaiaacEcaaaqcLbsacqaH4oqCjuaGdaahaaWcbeqc basaaKqzadGaaGOmaaaajugibiabgkHiTiaaikdaaaaaaa@5780@     (5.1.2)

Equations (5.1.1) and (5.1.2) give the following cubic equation in θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqiUde haaa@3830@  

m 2 ' θ 3 +4( m 2 ' X ¯ ) θ 2 2( 10 X ¯ 1 )θ+12=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamyBaS Waa0baaKqaGeaajugWaiaaikdaaKqaGeaajugWaiaacEcaaaqcLbsa cqaH4oqCjuaGdaahaaWcbeqcbasaaKqzadGaaG4maaaajugibiabgU caRiaaisdajuaGdaqadaGcbaqcLbsacaWGTbWcdaqhaaqcbasaaKqz adGaaGOmaaqcbasaaKqzadGaai4jaaaajugibiabgkHiTiqadIfaga qeaaGccaGLOaGaayzkaaqcLbsacqaH4oqCjuaGdaahaaWcbeqcbasa aKqzadGaaGOmaaaajugibiabgkHiTiaaikdajuaGdaqadaGcbaqcLb sacaaIXaGaaGimaiqadIfagaqeaiabgkHiTiaaigdaaOGaayjkaiaa wMcaaKqzGeGaeqiUdeNaey4kaSIaaGymaiaaikdacqGH9aqpcaaIWa aaaa@61CB@    (5.1.3)

Solving equation (5.1.3) using any iterative method such as Newton–Raphson method, Regula–Falsi method or Bisection method, method of moment estimation (MOME) θ ˜ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGafqiUde NbaGaaaaa@383F@ of θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqiUde haaa@3830@  can be obtained and substituting the value of θ ˜ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGafqiUde NbaGaaaaa@383F@  in equation (5.1.1), MOME α ˜ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGafqySde MbaGaaaaa@3828@ of α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqySde gaaa@3819@ can be obtained as

α ˜ = x ¯ θ ˜ 2 2( x ¯ 1) θ ˜ +6 θ ˜ 2 ( θ ˜ x ¯ 1) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGafqySde MbaGaacqGH9aqpjuaGdaWcaaGcbaqcLbsacqGHsislceWG4bGbaeba cuaH4oqCgaacaKqbaoaaCaaaleqajeaibaqcLbmacaaIYaaaaKqzGe GaeyOeI0IaaGOmaiaacIcaceWG4bGbaebacqGHsislcaaIXaGaaiyk aiqbeI7aXzaaiaGaey4kaSIaaGOnaaGcbaqcLbsacuaH4oqCgaacaK qbaoaaCaaaleqajeaibaqcLbmacaaIYaaaaKqzGeGaaiikaiqbeI7a XzaaiaGabmiEayaaraGaeyOeI0IaaGymaiaacMcaaaaaaa@5647@    (5.1.4)

Maximum likelihood estimates (MLE)

Let ( x 1 , x 2 , x 3 ,... x n ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaake aajugibiaadIhajuaGdaWgaaqcbasaaKqzadGaaGymaaWcbeaajugi biaacYcacaWG4bqcfa4aaSbaaKqaGeaajugWaiaaikdaaSqabaqcLb sacaGGSaGaamiEaKqbaoaaBaaajeaibaqcLbmacaaIZaaaleqaaKqz GeGaaiilaiaac6cacaGGUaGaaiOlaiaadIhajuaGdaWgaaqcbasaaK qzadGaamOBaaWcbeaaaOGaayjkaiaawMcaaaaa@4DDB@  be random sample from TPSD (2.1). The likelihood function L is given by

L= ( θ 3 α θ 2 +θ+2 ) n i=1 n ( α+ x i + x i 2 ) e nθ x ¯ , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamitai abg2da9KqbaoaabmaakeaajuaGdaWcaaGcbaqcLbsacqaH4oqCjuaG daahaaWcbeqcbasaaKqzadGaaG4maaaaaOqaaKqzGeGaeqySdeMaeq iUdexcfa4aaWbaaSqabKqaGeaajugWaiaaikdaaaqcLbsacqGHRaWk cqaH4oqCcqGHRaWkcaaIYaaaaaGccaGLOaGaayzkaaqcfa4aaWbaaS qabeaajugibiaad6gaaaqcfa4aaebCaOqaaKqbaoaabmaakeaajugi biabeg7aHjabgUcaRiaadIhajuaGdaWgaaqcbasaaKqzadGaamyAaa WcbeaajugibiabgUcaRiaadIhalmaaDaaajeaibaqcLbmacaWGPbaa jeaibaqcLbmacaaIYaaaaaGccaGLOaGaayzkaaqcLbsacaWGLbqcfa 4aaWbaaSqabKqaGeaajugWaiabgkHiTiaad6gacqaH4oqCceWG4bGb aebaaaaajeaibaqcLbmacaWGPbGaeyypa0JaaGymaaqcbasaaKqzad GaamOBaaqcLbsacqGHpis1aiaaykW7caGGSaaaaa@7229@

where x ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGabmiEay aaraaaaa@378F@  is the sample mean.

The natural log likelihood function is thus obtained as

lnL=n[ 3lnθln( α θ 2 +θ+2 ) ]+ i=1 n ln( α+ x i + x i 2 )nθ x ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaciiBai aac6gacaWGmbGaeyypa0JaamOBaKqbaoaadmaakeaajugibiaaioda ciGGSbGaaiOBaiabeI7aXjabgkHiTiGacYgacaGGUbqcfa4aaeWaaO qaaKqzGeGaeqySdeMaeqiUdexcfa4aaWbaaSqabKqaGeaajugWaiaa ikdaaaqcLbsacqGHRaWkcqaH4oqCcqGHRaWkcaaIYaaakiaawIcaca GLPaaaaiaawUfacaGLDbaajugibiabgUcaRKqbaoaaqahakeaajugi biGacYgacaGGUbqcfa4aaeWaaOqaaKqzGeGaeqySdeMaey4kaSIaam iEaKqbaoaaBaaajeaibaqcLbmacaWGPbaaleqaaKqzGeGaey4kaSIa amiEaKqbaoaaDaaajeaibaqcLbmacaWGPbaajeaibaqcLbmacaaIYa aaaaGccaGLOaGaayzkaaqcLbsacqGHsislcaWGUbGaaGPaVlabeI7a XjaaykW7ceWG4bGbaebaaKqaGeaajugWaiaadMgacqGH9aqpcaaIXa aajeaqbaqcLboacaWGUbaajugibiabggHiLdaaaa@79D3@

The maximum likelihood estimate (MLE’s) ( θ ^ , α ^ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaacIcacuaH4o qCgaqcaiaacYcacuaHXoqygaqcaiaacMcaaaa@3B69@  of (θ,α) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaacIcacqaH4o qCcaGGSaGaeqySdeMaaiykaaaa@3B49@  are then the solutions of the following non–linear equations

lnL θ = 3n θ n( 2αθ+1 ) α θ 2 +θ+2 n x ¯ =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaake aajugibiabgkGi2kGacYgacaGGUbGaamitaaGcbaqcLbsacqGHciIT cqaH4oqCaaGaeyypa0tcfa4aaSaaaOqaaKqzGeGaaG4maiaad6gaaO qaaKqzGeGaeqiUdehaaiabgkHiTKqbaoaalaaakeaajugibiaad6ga juaGdaqadaGcbaqcLbsacaaIYaGaeqySdeMaeqiUdeNaey4kaSIaaG ymaaGccaGLOaGaayzkaaaabaqcLbsacqaHXoqycqaH4oqCjuaGdaah aaWcbeqcbasaaKqzadGaaGOmaaaajugibiabgUcaRiabeI7aXjabgU caRiaaikdaaaGaeyOeI0IaamOBaiaaykW7ceWG4bGbaebacqGH9aqp caaIWaaaaa@6265@     (5.2.1)

lnL α = n θ 2 α θ 2 +θ+2 + i=1 n 1 α+ x i + x i 2 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaake aajugibiabgkGi2kGacYgacaGGUbGaamitaaGcbaqcLbsacqGHciIT cqaHXoqyaaGaeyypa0JaeyOeI0scfa4aaSaaaOqaaKqzGeGaamOBai abeI7aXLqbaoaaCaaaleqajeaibaqcLbmacaaIYaaaaaGcbaqcLbsa cqaHXoqycqaH4oqCjuaGdaahaaWcbeqcbasaaKqzadGaaGOmaaaaju gibiabgUcaRiabeI7aXjabgUcaRiaaikdaaaGaey4kaSscfa4aaabC aOqaaKqbaoaalaaakeaajugibiaaigdaaOqaaKqzGeGaeqySdeMaey 4kaSIaamiEaKqbaoaaBaaajeaibaqcLbmacaWGPbaaleqaaKqzGeGa ey4kaSIaamiEaKqbaoaaDaaajeaibaqcLbmacaWGPbaajqwaa+FaaK qzGcGaaGOmaaaaaaaajeaibaqcLbmacaWGPbGaeyypa0JaaGymaaqc basaaKqzadGaamOBaaqcLbsacqGHris5aiabg2da9iaaicdaaaa@7168@      (5.2.2)

These two natural log likelihood equations do not seem to be solved directly, because they cannot be expressed in closed forms. The (MLE’s) ( θ ^ , α ^ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaaiikai qbeI7aXzaajaGaaiilaiqbeg7aHzaajaGaaiykaaaa@3BF8@  of (θ,α) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaaiikai abeI7aXjaacYcacqaHXoqycaGGPaaaaa@3BD8@  can be computed directly by solving the natural log likelihood equations using Newton–Raphson iteration method using R–software till sufficiently close values of θ ^ and α ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGafqiUde NbaKaacaaMc8Uaaeyyaiaab6gacaqGKbGaaGPaVlqbeg7aHzaajaaa aa@3FC1@  are obtained. The initial values of parameters  and α are the MOME ( θ ˜ , α ˜ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaaiikai qbeI7aXzaaiaGaaiilaiqbeg7aHzaaiaGaaiykaaaa@3BF6@  of the parameters (θ,α). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaaiikai abeI7aXjaacYcacqaHXoqycaGGPaGaaGPaVlaac6caaaa@3E15@

A numerical example

In this section an application of TPSD using maximum likelihood estimates has been discussed with a real lifetime data set. The data set regarding vinyl chloride obtained from clean up gradient monitoring wells in mg/l, available in Bhaumik et al.10 has been considered. The data set is

5.1

1.2

1.3

0.6

0.5

2.4

0.5

1.1

8

0.8

0.4

0.6

0.9

0.4

2

0.5

5.3

3.2

2.7

2.9

2.5

2.3

1

0.2

0.1

0.1

1.8

0.9

2.4

6.8

1.2

0.4

0.2

In order to compare lifetime distributions, values of 2lnL MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGHsi slcaaIYaGaciiBaiaac6gacaWGmbaaaa@3AE3@ , AIC (Akaike Information Criterion), BIC (Bayesian Information Criterion) and K–S Statistic (Kolmogorov–Smirnov Statistic) for the above data set has been computed. The formulae for computing AIC, BIC, and K–S Statistics are as follows:

AIC=2lnL+2k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGbb GaamysaiaadoeacqGH9aqpcqGHsislcaaIYaGaciiBaiaac6gacaWG mbGaey4kaSIaaGOmaiaadUgaaaa@40D3@ , BIC=2lnL+klnn MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGcb GaamysaiaadoeacqGH9aqpcqGHsislcaaIYaGaciiBaiaac6gacaWG mbGaey4kaSIaam4AaiGacYgacaGGUbGaaGPaVlaad6gaaaa@447A@  and D= Sup x | F n ( x ) F 0 ( x ) | MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGeb Gaeyypa0tcfa4aaCbeaOqaaKqzGeGaae4uaiaabwhacaqGWbaajeai baqcLbmacaWG4baaleqaaKqbaoaaemaakeaajugibiaadAeajuaGda WgaaqcbasaaKqzadGaamOBaaWcbeaajuaGdaqadaGcbaqcLbsacaWG 4baakiaawIcacaGLPaaajugibiabgkHiTiaadAeajuaGdaWgaaqcba saaKqzadGaaGimaaWcbeaajuaGdaqadaGcbaqcLbsacaWG4baakiaa wIcacaGLPaaaaiaawEa7caGLiWoaaaa@5364@ , where k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGRb aaaa@3775@ = the number of parameters, n= MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGUb Gaeyypa0daaa@387E@  the sample size, and the F n ( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGgb qcfa4aaSbaaKqaGeaajugWaiaad6gaaSqabaqcfa4aaeWaaOqaaKqz GeGaamiEaaGccaGLOaGaayzkaaaaaa@3E0C@ = empirical distribution function. The best distribution is the distribution which corresponds to lower values of 2lnL MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGHsi slcaaIYaGaciiBaiaac6gacaWGmbaaaa@3AE3@ , AIC, and K–S statistic and higher p–value. The MLE ( θ ^ , α ^ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeGafqiUdeNbaKaacaGGSaGafqySdeMbaKaaaOGaayjkaiaa wMcaaaaa@3CD5@  along with their standard errors, 2lnL MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGHsi slcaaIYaGaciiBaiaac6gacaWGmbaaaa@3AE3@ , AIC, BIC, K–S Statistic and p–value of the fitted distributions are presented in the Table 5.

It is obvious that TPSD gives much closer fit than Sujatha and Lindley distributions. Therefore, TPSD can be considered as an important two–parameter lifetime distribution. In order to see the closeness of the fit given by Lindley, Sujatha and TPSD, the fitted pdf plots of these distributions for the given dataset have been shown in Figure 6. It is also obvious from the fitted plots of the distribution along with the histogram of the original dataset that TPSD gives much closer fit than Lindley and Sujatha distributions.

Figure 6Fitted pdf plots of distributions for the given dataset.

Conclusion

A two parameter Sujatha distribution (TPSD) has been introduced which includes size–biased Lindley distribution and Sujatha distribution, proposed by Shanker (2016a) as particular cases. Moments about origin and moments about mean have been obtained and nature of coefficient of variation, coefficient of skewness, coefficient of kurtosis and index of dispersion of TPSD have been studied with varying values of the parameters. The nature of probability density function, cumulative distribution function, hazard rate function and mean residual life function have been discussed with varying values of the parameters. The stochastic ordering, mean deviations, Bonferroni and Lorenz curves, and stress–strength reliability have also been discussed. The method of moments and method of maximum likelihood have been discussed for estimating parameters. A numerical example of real lifetime data have been presented to show the application of TPSD and the goodness of fit of TPSD gives much closer fit over Sujatha and Lindley distributions.

Acknowledgement

Authors are grateful to the editor–in–chief of the journal and the anonymous reviewer for constructive comments on the paper.

Conflict of interest

Authors declare that there is no conflict of interest.

References

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©2018 Tesfay, et al . This is an open access article distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and build upon your work non-commercially.