eISSN: 2378-315X BBIJ

Biometrics & Biostatistics International Journal
Research Article
Volume 6 Issue 1 - 2017
A Quasi Shanker Distribution and its Applications
Rama Shanker* and Kamlesh Kumar Shukla
Department of Statistics, Eritrea Institute of Technology, Eritrea
Received: June 03, 2017 | Published: June 13, 2017
*Corresponding author: Rama Shanker, Department of Statistics, Eritrea Institute of Technology, Asmara, Eritrea, Email: ;
Citation: Shanker R, Shukla KK (2017) A Quasi Shanker Distribution and its Applications. Biom Biostat Int J 6(1): 00156. DOI: 10.15406/bbij.2017.06.00156

Abstract

In the present paper, a two-parameter quasi Shanker distribution (QSD) which includes one parameter Shanker distribution introduced by Shanker [1] as a special case has been proposed. Its statistical and mathematical properties including moments and moments based measures, hazard rate function, mean residual life function, stochastic ordering, mean deviations, Bonferroni and Lorenz curves and stress-strength reliability have also been discussed. The method of maximum likelihood estimation has been discussed for estimating the parameters of QSD. Finally, the goodness of fit of the QSD has been discussed with two real lifetime data and the fit is quite satisfactory over one parameter exponential, Lindley and Shanker distributions.

Keywords: Shanker distribution; Moments; Hazard rate function; Mean residual life function; stochastic ordering; Mean deviations; Stress-strength reliability; Estimation of parameters; Goodness of fit.

Introducton

Shanker [1] has introduced a one parameter lifetime distribution for modeling lifetime data from biomedical science and engineering having probability density function(pdf) and cumulative distribution function(cdf) given by

f 1 ( x;θ )= θ 2 θ 2 +1 ( θ+x ) e θx ;x>0,θ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadAgada WgaaqaaKqzadGaaGymaaqcfayabaWaaeWaaeaacaWG4bGaai4oaiab eI7aXbGaayjkaiaawMcaaiabg2da9maalaaabaGaeqiUde3aaWbaae qabaqcLbmacaaIYaaaaaqcfayaaiabeI7aXnaaCaaabeqaaKqzadGa aGOmaaaajuaGcqGHRaWkcaaIXaaaamaabmaabaGaeqiUdeNaey4kaS IaamiEaaGaayjkaiaawMcaaiaadwgadaahaaqabeaajugWaiabgkHi TiabeI7aXjaadIhaaaqcfaOaaGPaVlaaykW7caaMc8UaaGPaVlaacU dacaWG4bGaeyOpa4JaaGimaiaacYcacaaMc8UaaGPaVlabeI7aXjab g6da+iaaicdaaaa@6716@ …. (1.1)
F 1 ( x,θ )=1[ 1+ θx θ 2 +1 ] e θx ;x>0,θ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadAeada WgaaqaaKqzadGaaGymaaqcfayabaWaaeWaaeaacaWG4bGaaiilaiab eI7aXbGaayjkaiaawMcaaiabg2da9iaaigdacqGHsisldaWadaqaai aaigdacqGHRaWkdaWcaaqaaiabeI7aXjaadIhaaeaacqaH4oqCdaah aaqabeaajugWaiaaikdaaaqcfaOaey4kaSIaaGymaaaaaiaawUfaca GLDbaacaWGLbWaaWbaaeqabaqcLbmacqGHsislcqaH4oqCcaWG4baa aKqbakaaykW7caaMc8Uaai4oaiaadIhacqGH+aGpcaaIWaGaaiilai abeI7aXjabg6da+iaaicdaaaa@5F37@  (1.2)

Shanker [1] has shown that it gives better fit than both one parameter exponential and Lindley [2] distributions. This distribution is a mixture of exponential ( θ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacqaH4oqCaiaawIcacaGLPaaaaaa@39C3@ and gamma ( 2,θ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacaaIYaGaaiilaiabeI7aXbGaayjkaiaawMcaaaaa@3B2F@ distributions with their mixing proportion θ 2 θ 2 +1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacqaH4oqCdaahaaqabeaajugWaiaaikdaaaaajuaGbaGaeqiUde3a aWbaaeqabaqcLbmacaaIYaaaaKqbakabgUcaRiaaigdaaaaaaa@40D1@ and 1 θ 2 +1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacaaIXaaabaGaeqiUde3aaWbaaeqabaqcLbmacaaIYaaaaKqbakab gUcaRiaaigdaaaaaaa@3D3C@ respectively.
The first four moments about origin of Shanker distribution obtained by Shanker [1] are given as

μ 1 = θ 2 +2 θ( θ 2 +1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeY7aTn aaBaaabaqcLbmacaaIXaaajuaGbeaadaahaaqabeaacWaGGBOmGika aiabg2da9maalaaabaGaeqiUde3aaWbaaeqabaqcLbmacaaIYaaaaK qbakabgUcaRiaaikdaaeaacqaH4oqCdaqadaqaaiabeI7aXnaaCaaa beqaaKqzadGaaGOmaaaajuaGcqGHRaWkcaaIXaaacaGLOaGaayzkaa aaaaaa@4E5F@ , μ 2 = 2( θ 2 +3 ) θ 2 ( θ 2 +1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2cdaWgaaqcfayaaKqzadGaaGOmaaqcfayabaWcdaahaaqcfayabeaa jugWaiadacUHYaIOaaqcfaOaeyypa0ZaaSaaaeaacaaIYaWaaeWaae aacqaH4oqClmaaCaaajuaGbeqaaKqzadGaaGOmaaaajuaGcqGHRaWk caaIZaaacaGLOaGaayzkaaaabaGaeqiUde3cdaahaaqcfayabeaaju gWaiaaikdaaaqcfa4aaeWaaeaacqaH4oqClmaaCaaajuaGbeqaaKqz adGaaGOmaaaajuaGcqGHRaWkcaaIXaaacaGLOaGaayzkaaaaaaaa@57A6@ , μ 3 = 6( θ 2 +4 ) θ 3 ( θ 2 +1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaeaajugWaiaaiodaaKqbagqaamaaCaaabeqaaiadacUHYaIO aaGaeyypa0ZaaSaaaeaacaaI2aWaaeWaaeaacqaH4oqCdaahaaqabe aajugWaiaaikdaaaqcfaOaey4kaSIaaGinaaGaayjkaiaawMcaaaqa aiabeI7aXnaaCaaabeqaaKqzadGaaG4maaaajuaGdaqadaqaaiabeI 7aXnaaCaaabeqaaKqzadGaaGOmaaaajuaGcqGHRaWkcaaIXaaacaGL OaGaayzkaaaaaaaa@52F4@ , μ 4 = 24( θ 2 +5 ) θ 4 ( θ 2 +1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaeaajugWaiaaisdaaKqbagqaamaaCaaabeqaaiadacUHYaIO aaGaeyypa0ZaaSaaaeaacaaIYaGaaGinamaabmaabaGaeqiUde3aaW baaeqabaqcLbmacaaIYaaaaKqbakabgUcaRiaaiwdaaiaawIcacaGL PaaaaeaacqaH4oqCdaahaaqabeaajugWaiaaisdaaaqcfa4aaeWaae aacqaH4oqCdaahaaqabeaajugWaiaaikdaaaqcfaOaey4kaSIaaGym aaGaayjkaiaawMcaaaaaaaa@53B1@

The central moments of Shanker distribution obtained by Shanker [1] are

μ 2 = θ 4 +4 θ 2 +2 θ 2 ( θ 2 +1 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2cdaWgaaqcfayaaKqzadGaaGOmaaqcfayabaGaeyypa0ZaaSaaaeaa cqaH4oqCdaahaaqabeaajugWaiaaisdaaaqcfaOaey4kaSIaaGinai abeI7aXnaaCaaabeqaaKqzadGaaGOmaaaajuaGcqGHRaWkcaaIYaaa baGaeqiUde3cdaahaaqcfayabeaajugWaiaaikdaaaqcfa4aaeWaae aacqaH4oqCdaahaaqabeaajugWaiaaikdaaaqcfaOaey4kaSIaaGym aaGaayjkaiaawMcaaSWaaWbaaKqbagqabaqcLbmacaaIYaaaaaaaaa a@5766@
μ 3 = 2( θ 6 +6 θ 4 +6 θ 2 +2 ) θ 3 ( θ 2 +1 ) 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaeaajugWaiaaiodaaKqbagqaaiabg2da9maalaaabaGaaGOm amaabmaabaGaeqiUde3aaWbaaeqabaqcLbmacaaI2aaaaKqbakabgU caRiaaiAdacqaH4oqCdaahaaqabeaajugWaiaaisdaaaqcfaOaey4k aSIaaGOnaiabeI7aXnaaCaaabeqaaKqzadGaaGOmaaaajuaGcqGHRa WkcaaIYaaacaGLOaGaayzkaaaabaGaeqiUde3aaWbaaeqabaqcLbma caaIZaaaaKqbaoaabmaabaGaeqiUde3aaWbaaeqabaqcLbmacaaIYa aaaKqbakabgUcaRiaaigdaaiaawIcacaGLPaaadaahaaqabeaajugW aiaaiodaaaaaaaaa@5DDB@
μ 4 = 3( 3 θ 8 +24 θ 6 +44 θ 4 +32 θ 2 +8 ) θ 4 ( θ 2 +1 ) 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaeaajugWaiaaisdaaKqbagqaaiabg2da9maalaaabaGaaG4m amaabmaabaGaaG4maiabeI7aXnaaCaaabeqaaKqzadGaaGioaaaaju aGcqGHRaWkcaaIYaGaaGinaiabeI7aXnaaCaaabeqaaKqzadGaaGOn aaaajuaGcqGHRaWkcaaI0aGaaGinaiabeI7aXnaaCaaabeqaaKqzad GaaGinaaaajuaGcqGHRaWkcaaIZaGaaGOmaiabeI7aXnaaCaaabeqa aKqzadGaaGOmaaaajuaGcqGHRaWkcaaI4aaacaGLOaGaayzkaaaaba GaeqiUde3aaWbaaeqabaqcLbmacaaI0aaaaKqbaoaabmaabaGaeqiU de3aaWbaaeqabaqcLbmacaaIYaaaaKqbakabgUcaRiaaigdaaiaawI cacaGLPaaadaahaaqabeaajugWaiaaisdaaaaaaaaa@66C9@

Shanker [1] studied its important properties including coefficient of variation, skewness, kurtosis, Index of dispersion, hazard rate function, mean residual life function, stochastic ordering, mean deviations, order statistics, Bonferroni and Lorenz curves, Renyi entropy measure, and stress-strength reliability. The discrete Poisson - Shanker distribution, a Poisson mixture of Shanker distribution has also been studied by Shanker [3].

Recall that the Lindley distribution, introduced by Lindley [2] in the context of Bayesian analysis as a counter example of fiducial statistics, is defined by its pdf and cdf

f 2 ( x;θ )= θ 2 θ+1 ( 1+x ) e θx ;x>0,θ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aaBaaabaqcLbmacaaIYaaajuaGbeaadaqadaqaaiaadIhacaGG7aGa eqiUdehacaGLOaGaayzkaaGaeyypa0ZaaSaaaeaacqaH4oqCdaahaa qabeaajugWaiaaikdaaaaajuaGbaGaeqiUdeNaey4kaSIaaGymaaaa daqadaqaaiaaigdacqGHRaWkcaWG4baacaGLOaGaayzkaaGaamyzam aaCaaabeqaaKqzadGaeyOeI0IaeqiUdeNaamiEaaaajuaGcaaMc8Ua aGPaVlaaykW7caaMc8Uaai4oaiaadIhacqGH+aGpcaaIWaGaaiilai aaykW7caaMc8UaeqiUdeNaeyOpa4JaaGimaaaa@632F@  (1.3)
F 2 ( x;θ )=1[ 1+ θx θ+1 ] e θx ;x>0,θ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOram aaBaaabaqcLbmacaaIYaaajuaGbeaadaqadaqaaiaadIhacaGG7aGa eqiUdehacaGLOaGaayzkaaGaeyypa0JaaGymaiabgkHiTmaadmaaba GaaGymaiabgUcaRmaalaaabaGaeqiUdeNaamiEaaqaaiabeI7aXjab gUcaRiaaigdaaaaacaGLBbGaayzxaaGaamyzamaaCaaabeqaaKqzad GaeyOeI0IaeqiUdeNaamiEaaaajuaGcaaMc8UaaGPaVlaacUdacaWG 4bGaeyOpa4JaaGimaiaacYcacaaMc8UaeqiUdeNaeyOpa4JaaGimaa aa@5DE5@ (1.4)

In this paper, a two - parameter quasi Shanker distribution (QSD), of which one parameter Shanker distribution introduced by Shanker [1] is a particular case, has been proposed. Its raw moments and central moments have been obtained and coefficients of variation, skewness, kurtosis and index of dispersion have been discussed. Some of its important mathematical and statistical properties including hazard rate function, mean residual life function, stochastic ordering, mean deviations, Bonferroni and Lorenz curves and stress-strength reliability have also been discussed. The estimation of the parameters has been discussed using maximum likelihood estimation. The goodness of fit of QSD has been illustrated with two real lifetime data sets and the fit has been compared with one parameter exponential, Lindley and Shanker distributions.

A Quasi Shanker Distribution

A two - parameter quasi Shanker distribution (QSD) having parameters θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiUdehaaa@37AC@  and α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdegaaa@3795@  is defined by its pdf

f( x;θ,α )= θ 3 θ 3 +θ+2α ( θ+x+α x 2 ) e θx ;x>0,θ>0, θ 3 +θ+2α>0. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aabmaabaGaamiEaiaacUdacqaH4oqCcaGGSaGaeqySdegacaGLOaGa ayzkaaGaeyypa0ZaaSaaaeaacqaH4oqCdaahaaqabeaajugWaiaaio daaaaajuaGbaGaeqiUde3aaWbaaeqabaqcLbmacaaIZaaaaKqbakab gUcaRiabeI7aXjabgUcaRiaaikdacqaHXoqyaaWaaeWaaeaacqaH4o qCcqGHRaWkcaWG4bGaey4kaSIaeqySdeMaaGPaVlaadIhadaahaaqa beaajugWaiaaikdaaaaajuaGcaGLOaGaayzkaaGaaGPaVlaadwgada ahaaqabeaajugWaiabgkHiTiabeI7aXjaaykW7caWG4baaaKqbakaa ykW7caaMc8Uaai4oaiaadIhacqGH+aGpcaaIWaGaaiilaiaaykW7ca aMc8UaeqiUdeNaeyOpa4JaaGimaiaaykW7caGGSaGaaGPaVlabeI7a XnaaCaaabeqaaKqzadGaaG4maaaajuaGcqGHRaWkcqaH4oqCcqGHRa WkcaaIYaGaeqySdeMaeyOpa4JaaGimaOGaaiOlaaaa@82C1@ (2.1)
It can be easily verified that (2.1) reduces to the Shanker distribution (1.1) at α=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdeMaey ypa0JaaGimaaaa@3955@ . It can be easily verified that QSD is a three-component mixture of exponential ( θ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacqaH4oqCaiaawIcacaGLPaaaaaa@39C3@ , gamma ( 2,θ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacaaIYaGaaiilaiabeI7aXbGaayjkaiaawMcaaaaa@3B2F@ and gamma ( 3,θ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacaaIZaGaaiilaiabeI7aXbGaayjkaiaawMcaaaaa@3B30@ distributions. We have

f( x;θ,α )= p 1 f 1 ( x;θ )+ p 2 f 2 ( x;2,θ )+( 1 p 1 p 2 ) f 3 ( x;3,θ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aabmaabaGaamiEaiaacUdacqaH4oqCcaGGSaGaeqySdegacaGLOaGa ayzkaaGaeyypa0JaamiCamaaBaaabaqcLbmacaaIXaaajuaGbeaaca aMc8UaamOzamaaBaaabaqcLbmacaaIXaaajuaGbeaadaqadaqaaiaa dIhacaGG7aGaeqiUdehacaGLOaGaayzkaaGaey4kaSIaamiCamaaBa aabaqcLbmacaaIYaaajuaGbeaacaWGMbWaaSbaaeaajugWaiaaikda aKqbagqaamaabmaabaGaamiEaiaacUdacaaIYaGaaiilaiabeI7aXb GaayjkaiaawMcaaiabgUcaRmaabmaabaGaaGymaiabgkHiTiaadcha daWgaaqaaKqzadGaaGymaaqcfayabaGaeyOeI0IaamiCamaaBaaaba qcLbmacaaIYaaajuaGbeaaaiaawIcacaGLPaaacaWGMbWaaSbaaeaa jugWaiaaiodaaKqbagqaamaabmaabaGaamiEaiaacUdacaaIZaGaai ilaiabeI7aXbGaayjkaiaawMcaaaaa@71B8@ (2.2)

where

p 1 = θ 3 θ 3 +θ+2α , p 2 = θ θ 3 +θ+2α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiCam aaBaaabaqcLbmacaaIXaaajuaGbeaacqGH9aqpdaWcaaqaaiabeI7a XnaaCaaabeqaaKqzadGaaG4maaaaaKqbagaacqaH4oqCdaahaaqabe aajugWaiaaiodaaaqcfaOaey4kaSIaeqiUdeNaey4kaSIaaGOmaiab eg7aHbaacaaMc8UaaiilaiaaykW7caaMc8UaaGPaVlaadchadaWgaa qaaKqzadGaaGOmaaqcfayabaGaeyypa0ZaaSaaaeaacqaH4oqCaeaa cqaH4oqCdaahaaqabeaajugWaiaaiodaaaqcfaOaey4kaSIaeqiUde Naey4kaSIaaGOmaiabeg7aHbaaaaa@60FA@ ,
f 1 ( x;θ )=θ e θx ;x>0,θ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aaBaaabaqcLbmacaaIXaaajuaGbeaadaqadaqaaiaadIhacaGG7aGa eqiUdehacaGLOaGaayzkaaGaeyypa0JaeqiUdeNaaGPaVlaadwgada ahaaqabeaajugWaiabgkHiTiabeI7aXjaaykW7caWG4baaaKqbakaa cUdacaaMc8UaaGPaVlaaykW7caWG4bGaeyOpa4JaaGimaiaacYcaca aMc8UaaGPaVlabeI7aXjabg6da+iaaicdaaaa@5A99@
f 2 ( x;2,θ )= θ 2 Γ( 2 ) e θx x 21 ;x>0,θ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aaBaaabaqcLbmacaaIYaaajuaGbeaadaqadaqaaiaadIhacaGG7aGa aGOmaiaacYcacqaH4oqCaiaawIcacaGLPaaacqGH9aqpdaWcaaqaai abeI7aXnaaCaaabeqaaKqzadGaaGOmaaaaaKqbagaacqqHtoWrdaqa daqaaiaaikdaaiaawIcacaGLPaaaaaGaaGPaVlaadwgadaahaaqabe aajugWaiabgkHiTiabeI7aXjaaykW7caWG4baaaKqbakaadIhadaah aaqabeaajugWaiaaikdacqGHsislcaaIXaaaaKqbakaacUdacaaMc8 UaaGPaVlaaykW7caaMc8UaamiEaiabg6da+iaaicdacaGGSaGaaGPa VlaaykW7cqaH4oqCcqGH+aGpcaaIWaaaaa@6927@
f 3 ( x;3,θ )= θ 3 Γ( 3 ) e θx x 31 ;x>0,θ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aaBaaabaqcLbmacaaIZaaajuaGbeaadaqadaqaaiaadIhacaGG7aGa aG4maiaacYcacqaH4oqCaiaawIcacaGLPaaacqGH9aqpdaWcaaqaai abeI7aXnaaCaaabeqaaKqzadGaaG4maaaaaKqbagaacqqHtoWrdaqa daqaaiaaiodaaiaawIcacaGLPaaaaaGaaGPaVlaadwgadaahaaqabe aajugWaiabgkHiTiabeI7aXjaaykW7caWG4baaaKqbakaadIhadaah aaqabeaajugWaiaaiodacqGHsislcaaIXaaaaKqbakaacUdacaaMc8 UaaGPaVlaaykW7caaMc8UaamiEaiabg6da+iaaicdacaGGSaGaaGPa VlaaykW7cqaH4oqCcqGH+aGpcaaIWaaaaa@692C@

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

F( x;θ,α )=1[ 1+ α θ 2 x 2 +θx( θ+2α ) θ 3 +θ+2α ] e θx ;  x>0,θ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOram aabmaabaGaamiEaiaacUdacqaH4oqCcaGGSaGaeqySdegacaGLOaGa ayzkaaGaeyypa0JaaGymaiabgkHiTmaadmaabaGaaGymaiabgUcaRm aalaaabaGaeqySdeMaaGPaVlabeI7aXnaaCaaabeqaaKqzadGaaGOm aaaajuaGcaWG4bWaaWbaaeqabaqcLbmacaaIYaaaaKqbakabgUcaRi abeI7aXjaaykW7caWG4bWaaeWaaeaacqaH4oqCcqGHRaWkcaaIYaGa eqySdegacaGLOaGaayzkaaaabaGaeqiUde3aaWbaaeqabaqcLbmaca aIZaaaaKqbakabgUcaRiabeI7aXjabgUcaRiaaikdacqaHXoqyaaaa caGLBbGaayzxaaGaamyzamaaCaaabeqaaKqzadGaeyOeI0IaeqiUde NaaGPaVlaadIhaaaqcfaOaai4oaiaaykW7caaMc8oeaaaaaaaaa8qa caGGGcGaaiiOaiaadIhacqGH+aGpcaaIWaGaaiilaiabeI7aXjabg6 da+iaaicdaaaa@7AD0@ (2.3)

The nature and behavior of the pdf and the cdf of QSD for varying values of the parameters θandα MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde NaaGPaVlaaykW7caaMc8Uaaeyyaiaab6gacaqGKbGaaGPaVlaaykW7 caaMc8UaeqySdegaaa@45D7@ have been explained graphically and presented in Figures 1 & 2, respectively.


Figure 1: Graphs of the pdf of QSD for varying values of parameters θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde haaa@383A@  and α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqySde gaaa@3823@ .

Figure 2: Graphs of the cdf of QSD for varying values of parameters θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde haaa@383A@  and α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqySde gaaa@3823@ .

Statistical Constants

The r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCaaaa@36ED@ th moment about origin of QSD can be obtained as

μ r = r![ θ 3 +( r+1 )θ+( r+1 )( r+2 )α ] θ r ( θ 3 +θ+2α ) ;r=1,2,3,.. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaeaajugWaiaadkhaaKqbagqaamaaCaaabeqaaiadacUHYaIO aaGaeyypa0ZaaSaaaeaacaWGYbGaaiyiamaadmaabaGaeqiUde3aaW baaeqabaqcLbmacaaIZaaaaKqbakabgUcaRmaabmaabaGaamOCaiab gUcaRiaaigdaaiaawIcacaGLPaaacqaH4oqCcqGHRaWkdaqadaqaai aadkhacqGHRaWkcaaIXaaacaGLOaGaayzkaaWaaeWaaeaacaWGYbGa ey4kaSIaaGOmaaGaayjkaiaawMcaaiabeg7aHbGaay5waiaaw2faaa qaaiabeI7aXnaaCaaabeqaaKqzadGaamOCaaaajuaGdaqadaqaaiab eI7aXnaaCaaabeqaaKqzadGaaG4maaaajuaGcqGHRaWkcqaH4oqCcq GHRaWkcaaIYaGaeqySdegacaGLOaGaayzkaaaaaiaaykW7caaMc8Ua aGPaVlaacUdacaWGYbGaeyypa0JaaGymaiaacYcacaaIYaGaaiilai aaiodacaGGSaGaaiOlaiaac6caaaa@75BD@ (3.1)

Thus, the first four moments about origin of QSD are given by

μ 1 = θ 3 +2θ+6α θ( θ 3 +θ+2α ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaeaajugWaiaaigdaaKqbagqaamaaCaaabeqaaiadacUHYaIO aaGaeyypa0ZaaSaaaeaacqaH4oqCdaahaaqabeaajugWaiaaiodaaa qcfaOaey4kaSIaaGOmaiabeI7aXjabgUcaRiaaiAdacqaHXoqyaeaa cqaH4oqCdaqadaqaaiabeI7aXnaaCaaabeqaaKqzadGaaG4maaaaju aGcqGHRaWkcqaH4oqCcqGHRaWkcaaIYaGaeqySdegacaGLOaGaayzk aaaaaaaa@573D@  , μ 2 = 2( θ 3 +3θ+12α ) θ 2 ( θ 3 +θ+2α ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaeaajugWaiaaikdaaKqbagqaamaaCaaabeqaaiadacUHYaIO aaGaeyypa0ZaaSaaaeaacaaIYaWaaeWaaeaacqaH4oqCdaahaaqabe aajugWaiaaiodaaaqcfaOaey4kaSIaaG4maiabeI7aXjabgUcaRiaa igdacaaIYaGaeqySdegacaGLOaGaayzkaaaabaGaeqiUde3aaWbaae qabaqcLbmacaaIYaaaaKqbaoaabmaabaGaeqiUde3aaWbaaeqabaqc LbmacaaIZaaaaKqbakabgUcaRiabeI7aXjabgUcaRiaaikdacqaHXo qyaiaawIcacaGLPaaaaaaaaa@5CD5@
μ 3 = 6( θ 3 +4θ+20α ) θ 3 ( θ 3 +θ+2α ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaeaajugWaiaaiodaaKqbagqaamaaCaaabeqaaiadacUHYaIO aaGaeyypa0ZaaSaaaeaacaaI2aWaaeWaaeaacqaH4oqCdaahaaqabe aajugWaiaaiodaaaqcfaOaey4kaSIaaGinaiabeI7aXjabgUcaRiaa ikdacaaIWaGaeqySdegacaGLOaGaayzkaaaabaGaeqiUde3aaWbaae qabaqcLbmacaaIZaaaaKqbaoaabmaabaGaeqiUde3aaWbaaeqabaqc LbmacaaIZaaaaKqbakabgUcaRiabeI7aXjabgUcaRiaaikdacqaHXo qyaiaawIcacaGLPaaaaaaaaa@5CDB@  , μ 4 = 24( θ 3 +5θ+30α ) θ 4 ( θ 3 +θ+2α ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaeaajugWaiaaisdaaKqbagqaamaaCaaabeqaaiadacUHYaIO aaGaeyypa0ZaaSaaaeaacaaIYaGaaGinamaabmaabaGaeqiUde3aaW baaeqabaqcLbmacaaIZaaaaKqbakabgUcaRiaaiwdacqaH4oqCcqGH RaWkcaaIZaGaaGimaiabeg7aHbGaayjkaiaawMcaaaqaaiabeI7aXn aaCaaabeqaaKqzadGaaGinaaaajuaGdaqadaqaaiabeI7aXnaaCaaa beqaaKqzadGaaG4maaaajuaGcqGHRaWkcqaH4oqCcqGHRaWkcaaIYa GaeqySdegacaGLOaGaayzkaaaaaaaa@5D99@

Using relationship between central moments and moments about origin, the central moments of QSD (2.1) are thus obtained as

                                                                   μ 2 = θ 6 +4 θ 4 +16 θ 3 α+2 θ 2 +12θα+12 α 2 θ 2 ( θ 3 +θ+2α ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaeaajugWaiaaikdaaKqbagqaaiabg2da9maalaaabaGaeqiU de3aaWbaaeqabaqcLbmacaaI2aaaaKqbakabgUcaRiaaisdacqaH4o qCdaahaaqabeaajugWaiaaisdaaaqcfaOaey4kaSIaaGymaiaaiAda caaMc8UaeqiUde3aaWbaaeqabaqcLbmacaaIZaaaaKqbakabeg7aHj abgUcaRiaaikdacqaH4oqCdaahaaqabeaajugWaiaaikdaaaqcfaOa ey4kaSIaaGymaiaaikdacaaMc8UaeqiUdeNaaGPaVlabeg7aHjabgU caRiaaigdacaaIYaGaeqySde2aaWbaaeqabaqcLbmacaaIYaaaaaqc fayaaiabeI7aXnaaCaaabeqaaKqzadGaaGOmaaaajuaGdaqadaqaai abeI7aXnaaCaaabeqaaKqzadGaaG4maaaajuaGcqGHRaWkcqaH4oqC cqGHRaWkcaaIYaGaeqySdegacaGLOaGaayzkaaWaaWbaaeqabaqcLb macaaIYaaaaaaaaaa@7756@
μ 3 = 2{ θ 9 +6 θ 7 +30 θ 6 α+6 θ 5 +42 θ 4 α+( 36 α 2 +2 ) θ 3 +18 θ 2 α+36θ α 2 +24 α 3 } θ 3 ( θ 3 +θ+2α ) 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaeaajugWaiaaiodaaKqbagqaaiabg2da9maalaaabaGaaGOm amaacmaabaGaeqiUde3aaWbaaeqabaqcLbmacaaI5aaaaKqbakabgU caRiaaiAdacqaH4oqCdaahaaqabeaajugWaiaaiEdaaaqcfaOaey4k aSIaaG4maiaaicdacqaH4oqCdaahaaqabeaajugWaiaaiAdaaaqcfa OaeqySdeMaey4kaSIaaGOnaiaaykW7cqaH4oqCdaahaaqabeaajugW aiaaiwdaaaqcfaOaey4kaSIaaGinaiaaikdacqaH4oqCdaahaaqabe aajugWaiaaisdaaaqcfaOaeqySdeMaey4kaSYaaeWaaeaacaaIZaGa aGOnaiabeg7aHnaaCaaabeqaaKqzadGaaGOmaaaajuaGcqGHRaWkca aIYaaacaGLOaGaayzkaaGaeqiUde3aaWbaaeqabaqcLbmacaaIZaaa aKqbakabgUcaRiaaigdacaaI4aGaeqiUde3aaWbaaeqabaqcLbmaca aIYaaaaKqbakabeg7aHjabgUcaRiaaiodacaaI2aGaeqiUdeNaaGPa Vlabeg7aHnaaCaaabeqaaKqzadGaaGOmaaaajuaGcqGHRaWkcaaIYa GaaGinaiabeg7aHnaaCaaabeqaaKqzadGaaG4maaaaaKqbakaawUha caGL9baaaeaacqaH4oqCdaahaaqabeaajugWaiaaiodaaaqcfa4aae WaaeaacqaH4oqCdaahaaqabeaajugWaiaaiodaaaqcfaOaey4kaSIa eqiUdeNaey4kaSIaaGOmaiabeg7aHbGaayjkaiaawMcaamaaCaaabe qaaKqzadGaaG4maaaaaaaaaa@9A18@
μ 4 = 3{ 3 θ 12 +24 θ 10 +128 θ 9 α+44 θ 8 +344 θ 7 α+( 408 α 2 +32 ) θ 6 +320 θ 5 α +( 768 α 2 +8 ) θ 4 +( 576 α 3 +96α ) θ 3 +336 θ 2 α 2 +480θ α 3 +240 α 4 } θ 4 ( θ 3 +θ+2α ) 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaeaajugWaiaaisdaaKqbagqaaiabg2da9maalaaabaGaaG4m amaacmaaeaqabeaacaaIZaGaeqiUde3aaWbaaeqabaqcLbmacaaIXa GaaGOmaaaajuaGcqGHRaWkcaaIYaGaaGinaiabeI7aXnaaCaaabeqa aKqzadGaaGymaiaaicdaaaqcfaOaey4kaSIaaGymaiaaikdacaaI4a GaeqiUde3aaWbaaeqabaqcLbmacaaI5aaaaKqbakabeg7aHjabgUca RiaaisdacaaI0aGaeqiUde3aaWbaaeqabaqcLbmacaaI4aaaaKqbak abgUcaRiaaiodacaaI0aGaaGinaiabeI7aXnaaCaaabeqaaKqzadGa aG4naaaajuaGcqaHXoqycqGHRaWkdaqadaqaaiaaisdacaaIWaGaaG ioaiabeg7aHnaaCaaabeqaaKqzadGaaGOmaaaajuaGcqGHRaWkcaaI ZaGaaGOmaaGaayjkaiaawMcaaiabeI7aXnaaCaaabeqaaKqzadGaaG OnaaaajuaGcqGHRaWkcaaIZaGaaGOmaiaaicdacaaMc8UaeqiUde3a aWbaaeqabaqcLbmacaaI1aaaaKqbakabeg7aHbqaaiabgUcaRmaabm aabaGaaG4naiaaiAdacaaI4aGaeqySde2aaWbaaeqabaqcLbmacaaI YaaaaKqbakabgUcaRiaaiIdaaiaawIcacaGLPaaacqaH4oqCdaahaa qabeaajugWaiaaisdaaaqcfaOaey4kaSYaaeWaaeaacaaI1aGaaG4n aiaaiAdacqaHXoqydaahaaqabeaajugWaiaaiodaaaqcfaOaey4kaS IaaGyoaiaaiAdacqaHXoqyaiaawIcacaGLPaaacqaH4oqCdaahaaqa beaajugWaiaaiodaaaqcfaOaey4kaSIaaG4maiaaiodacaaI2aGaeq iUde3aaWbaaeqabaqcLbmacaaIYaaaaKqbakabeg7aHnaaCaaabeqa aKqzadGaaGOmaaaajuaGcqGHRaWkcaaI0aGaaGioaiaaicdacaaMc8 UaeqiUdeNaeqySde2aaWbaaeqabaqcLbmacaaIZaaaaKqbakabgUca RiaaikdacaaI0aGaaGimaiabeg7aHnaaCaaabeqaaKqzadGaaGinaa aaaaqcfaOaay5Eaiaaw2haaaqaaiabeI7aXnaaCaaabeqaaKqzadGa aGinaaaajuaGdaqadaqaaiabeI7aXnaaCaaabeqaaKqzadGaaG4maa aajuaGcqGHRaWkcqaH4oqCcqGHRaWkcaaIYaGaeqySdegacaGLOaGa ayzkaaWaaWbaaeqabaqcLbmacaaI0aaaaaaaaaa@CEB3@
The coefficient of variation ( C.V ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacaWGdbGaaiOlaiaadAfaaiaawIcacaGLPaaaaaa@3A62@ , coefficient of skewness ( β 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aadaGcaaqaaiabek7aInaaBaaabaqcLbmacaaIXaaajuaGbeaaaeqa aaGaayjkaiaawMcaaaaa@3C56@ , coefficient of kurtosis ( β 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacqaHYoGydaWgaaqaaKqzadGaaGOmaaqcfayabaaacaGLOaGaayzk aaaaaa@3C47@ and index of dispersion ( γ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacqaHZoWzaiaawIcacaGLPaaaaaa@39B4@ of QSD are obtained as

C.V= σ μ 1 = θ 6 +4 θ 4 +16 θ 3 α+2 θ 2 +12θα+12 α 2 θ 3 +2θ+6α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4qai aac6cacaWGwbGaeyypa0ZaaSaaaeaacqaHdpWCaeaacqaH8oqBdaWg aaqaaKqzadGaaGymaaqcfayabaWaaWbaaeqabaGamai4gkdiIcaaaa Gaeyypa0ZaaSaaaeaadaGcaaqaaiabeI7aXnaaCaaabeqaaKqzadGa aGOnaaaajuaGcqGHRaWkcaaI0aGaeqiUde3aaWbaaeqabaqcLbmaca aI0aaaaKqbakabgUcaRiaaigdacaaI2aGaaGPaVlabeI7aXnaaCaaa beqaaKqzadGaaG4maaaajuaGcqaHXoqycqGHRaWkcaaIYaGaeqiUde 3aaWbaaeqabaqcLbmacaaIYaaaaKqbakabgUcaRiaaigdacaaIYaGa aGPaVlabeI7aXjaaykW7cqaHXoqycqGHRaWkcaaIXaGaaGOmaiabeg 7aHnaaCaaabeqaaKqzadGaaGOmaaaaaKqbagqaaaqaaiabeI7aXnaa CaaabeqaaKqzadGaaG4maaaajuaGcqGHRaWkcaaIYaGaeqiUdeNaey 4kaSIaaGOnaiabeg7aHbaaaaa@7878@

β 1 = μ 3 μ 2 3/2 = 2{ θ 9 +6 θ 7 +30 θ 6 α+6 θ 5 +42 θ 4 α+( 36 α 2 +2 ) θ 3 +18 θ 2 α+36θ α 2 +24 α 3 } ( θ 6 +4 θ 4 +16 θ 3 α+2 θ 2 +12θα+12 α 2 ) 3/2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaOaaae aacqaHYoGydaWgaaqaaKqzadGaaGymaaqcfayabaaabeaacqGH9aqp daWcaaqaaiabeY7aTnaaBaaabaqcLbmacaaIZaaajuaGbeaaaeaacq aH8oqBdaWgaaqaaKqzadGaaGOmaaqcfayabaWaaWbaaeqabaqcLbma caaIZaGaai4laiaaikdaaaaaaKqbakabg2da9maalaaabaGaaGOmam aacmaabaGaeqiUde3aaWbaaeqabaqcLbmacaaI5aaaaKqbakabgUca RiaaiAdacqaH4oqCdaahaaqabeaajugWaiaaiEdaaaqcfaOaey4kaS IaaG4maiaaicdacqaH4oqCdaahaaqabeaajugWaiaaiAdaaaqcfaOa eqySdeMaey4kaSIaaGOnaiaaykW7cqaH4oqCdaahaaqabeaajugWai aaiwdaaaqcfaOaey4kaSIaaGinaiaaikdacqaH4oqCdaahaaqabeaa jugWaiaaisdaaaqcfaOaeqySdeMaey4kaSYaaeWaaeaacaaIZaGaaG Onaiabeg7aHnaaCaaabeqaaKqzadGaaGOmaaaajuaGcqGHRaWkcaaI YaaacaGLOaGaayzkaaGaeqiUde3aaWbaaeqabaqcLbmacaaIZaaaaK qbakabgUcaRiaaigdacaaI4aGaeqiUde3aaWbaaeqabaqcLbmacaaI YaaaaKqbakabeg7aHjabgUcaRiaaiodacaaI2aGaeqiUdeNaaGPaVl abeg7aHnaaCaaabeqaaKqzadGaaGOmaaaajuaGcqGHRaWkcaaIYaGa aGinaiabeg7aHnaaCaaabeqaaKqzadGaaG4maaaaaKqbakaawUhaca GL9baaaeaadaqadaqaaiabeI7aXnaaCaaabeqaaKqzadGaaGOnaaaa juaGcqGHRaWkcaaI0aGaeqiUde3aaWbaaeqabaqcLbmacaaI0aaaaK qbakabgUcaRiaaigdacaaI2aGaaGPaVlabeI7aXnaaCaaabeqaaKqz adGaaG4maaaajuaGcqaHXoqycqGHRaWkcaaIYaGaeqiUde3aaWbaae qabaqcLbmacaaIYaaaaKqbakabgUcaRiaaigdacaaIYaGaaGPaVlab eI7aXjaaykW7cqaHXoqycqGHRaWkcaaIXaGaaGOmaiabeg7aHnaaCa aabeqaaKqzadGaaGOmaaaaaKqbakaawIcacaGLPaaadaahaaqabeaa jugWaiaaiodacaGGVaGaaGOmaaaaaaaaaa@C42A@

β 2 = μ 4 μ 2 2 = 3{ 3 θ 12 +24 θ 10 +128 θ 9 α+44 θ 8 +344 θ 7 α+( 408 α 2 +32 ) θ 6 +320 θ 5 α +( 768 α 2 +8 ) θ 4 +( 576 α 3 +96α ) θ 3 +336 θ 2 α 2 +480θ α 3 +240 α 4 } ( θ 6 +4 θ 4 +16 θ 3 α+2 θ 2 +12θα+12 α 2 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOSdi 2aaSbaaeaajugWaiaaikdaaKqbagqaaiabg2da9maalaaabaGaeqiV d02aaSbaaeaajugWaiaaisdaaKqbagqaaaqaaiabeY7aTnaaBaaaba qcLbmacaaIYaaajuaGbeaadaahaaqabeaajugWaiaaikdaaaaaaKqb akabg2da9maalaaabaGaaG4mamaacmaaeaqabeaacaaIZaGaeqiUde 3aaWbaaeqabaqcLbmacaaIXaGaaGOmaaaajuaGcqGHRaWkcaaIYaGa aGinaiabeI7aXnaaCaaabeqaaKqzadGaaGymaiaaicdaaaqcfaOaey 4kaSIaaGymaiaaikdacaaI4aGaeqiUde3aaWbaaeqabaqcLbmacaaI 5aaaaKqbakabeg7aHjabgUcaRiaaisdacaaI0aGaeqiUde3aaWbaae qabaqcLbmacaaI4aaaaKqbakabgUcaRiaaiodacaaI0aGaaGinaiab eI7aXnaaCaaabeqaaKqzadGaaG4naaaajuaGcqaHXoqycqGHRaWkda qadaqaaiaaisdacaaIWaGaaGioaiabeg7aHnaaCaaabeqaaKqzadGa aGOmaaaajuaGcqGHRaWkcaaIZaGaaGOmaaGaayjkaiaawMcaaiabeI 7aXnaaCaaabeqaaKqzadGaaGOnaaaajuaGcqGHRaWkcaaIZaGaaGOm aiaaicdacaaMc8UaeqiUde3aaWbaaeqabaqcLbmacaaI1aaaaKqbak abeg7aHbqaaiabgUcaRmaabmaabaGaaG4naiaaiAdacaaI4aGaeqyS de2aaWbaaeqabaqcLbmacaaIYaaaaKqbakabgUcaRiaaiIdaaiaawI cacaGLPaaacqaH4oqCdaahaaqabeaajugWaiaaisdaaaqcfaOaey4k aSYaaeWaaeaacaaI1aGaaG4naiaaiAdacqaHXoqydaahaaqabeaaju gWaiaaiodaaaqcfaOaey4kaSIaaGyoaiaaiAdacqaHXoqyaiaawIca caGLPaaacqaH4oqCdaahaaqabeaajugWaiaaiodaaaqcfaOaey4kaS IaaG4maiaaiodacaaI2aGaeqiUde3aaWbaaeqabaqcLbmacaaIYaaa aKqbakabeg7aHnaaCaaabeqaaKqzadGaaGOmaaaajuaGcqGHRaWkca aI0aGaaGioaiaaicdacaaMc8UaeqiUdeNaeqySde2aaWbaaeqabaqc LbmacaaIZaaaaKqbakabgUcaRiaaikdacaaI0aGaaGimaiabeg7aHn aaCaaabeqaaKqzadGaaGinaaaaaaqcfaOaay5Eaiaaw2haaaqaamaa bmaabaGaeqiUde3aaWbaaeqabaGaaGOnaaaacqGHRaWkcaaI0aGaeq iUde3aaWbaaeqabaGaaGinaaaacqGHRaWkcaaIXaGaaGOnaiaaykW7 cqaH4oqCdaahaaqabeaacaaIZaaaaiabeg7aHjabgUcaRiaaikdacq aH4oqCdaahaaqabeaacaaIYaaaaiabgUcaRiaaigdacaaIYaGaaGPa VlabeI7aXjaaykW7cqaHXoqycqGHRaWkcaaIXaGaaGOmaiabeg7aHn aaCaaabeqaaiaaikdaaaaacaGLOaGaayzkaaWaaWbaaeqabaGaaGOm aaaaaaaaaa@EBFA@

γ= σ 2 μ 1 = θ 6 +4 θ 4 +16 θ 3 α+2 θ 2 +12θα+12 α 2 θ( θ 3 +θ+2α )( θ 3 +2θ+6α ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4SdC Maeyypa0ZaaSaaaeaacqaHdpWCdaahaaqabeaajugWaiaaikdaaaaa juaGbaGaeqiVd02aaSbaaeaajugWaiaaigdaaKqbagqaamaaCaaabe qaaiadacUHYaIOaaaaaiabg2da9maalaaabaGaeqiUde3aaWbaaeqa baqcLbmacaaI2aaaaKqbakabgUcaRiaaisdacqaH4oqCdaahaaqabe aajugWaiaaisdaaaqcfaOaey4kaSIaaGymaiaaiAdacaaMc8UaeqiU de3aaWbaaeqabaqcLbmacaaIZaaaaKqbakabeg7aHjabgUcaRiaaik dacqaH4oqCdaahaaqabeaajugWaiaaikdaaaqcfaOaey4kaSIaaGym aiaaikdacaaMc8UaeqiUdeNaaGPaVlabeg7aHjabgUcaRiaaigdaca aIYaGaeqySde2aaWbaaeqabaqcLbmacaaIYaaaaaqcfayaaiabeI7a XnaabmaabaGaeqiUde3aaWbaaeqabaqcLbmacaaIZaaaaKqbakabgU caRiabeI7aXjabgUcaRiaaikdacqaHXoqyaiaawIcacaGLPaaadaqa daqaaiabeI7aXnaaCaaabeqaaKqzadGaaG4maaaajuaGcqGHRaWkca aIYaGaeqiUdeNaey4kaSIaaGOnaiabeg7aHbGaayjkaiaawMcaaaaa aaa@8942@

Graphs of C.V, β 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaOaaae aacqaHYoGydaWgaaqaaiaaykW7jugWaiaaigdaaKqbagqaaaqabaaa aa@3C58@ , β 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOSdi 2aaSbaaeaajugWaiaaykW7caaIYaaajuaGbeaaaaa@3C49@ and γ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4SdC gaaa@382B@ of QSD for varying values of the parameters θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde haaa@383A@ and α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqySde gaaa@3823@ have been presented in Figure 3.


Figure 3: Graphs of C.V, β 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaOaaae aacqaHYoGylmaaBaaajuaGbaqcLbmacaaMc8UaaGymaaqcfayabaaa beaaaaa@3CF1@ , β 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOSdi 2aaSbaaeaacaaMc8EcLbmacaaIYaaajuaGbeaaaaa@3C49@ and γ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4SdC gaaa@382B@  of QSD for varying values of the parameter θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde haaa@383A@  and α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqySde gaaa@3823@ .

Hazard Rate Function And Mean Residual Life Function

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

h( x )= lim Δx0 P( X<x+Δx|X>x ) Δx = f( x ) 1F( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiAam aabmaabaGaamiEaaGaayjkaiaawMcaaiabg2da9maaxababaGaciiB aiaacMgacaGGTbaabaGaeyiLdqKaamiEaiabgkziUkaaicdaaeqaam aalaaabaGaamiuamaabmaabaWaaqGaaeaacaWGybGaeyipaWJaamiE aiabgUcaRiabgs5aejaadIhacaaMc8oacaGLiWoacaaMc8Uaamiwai abg6da+iaadIhaaiaawIcacaGLPaaaaeaacqGHuoarcaWG4baaaiab g2da9maalaaabaGaamOzamaabmaabaGaamiEaaGaayjkaiaawMcaaa qaaiaaigdacqGHsislcaWGgbWaaeWaaeaacaWG4baacaGLOaGaayzk aaaaaaaa@5F0A@   (4.1)
And  m( x )=E[ Xx|X>x ]= 1 1F( x ) x [ 1F( t ) ] dt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaaeiiai aad2gadaqadaqaaiaadIhaaiaawIcacaGLPaaacqGH9aqpcaWGfbWa amWaaeaadaabcaqaaiaadIfacqGHsislcaWG4baacaGLiWoacaWGyb GaeyOpa4JaamiEaaGaay5waiaaw2faaiaaysW7cqGH9aqpcaaMe8+a aSaaaeaacaaIXaaabaGaaGymaiabgkHiTiaadAeadaqadaqaaiaadI haaiaawIcacaGLPaaaaaWaa8qmaeaadaWadaqaaiaaigdacqGHsisl caWGgbWaaeWaaeaacaWG0baacaGLOaGaayzkaaaacaGLBbGaayzxaa aabaGaamiEaaqaaiabg6HiLcGaey4kIipacaaMe8UaaGPaVlaadsga caWG0baaaa@5FE4@   (4.2)
The corresponding hazard rate function h( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiAam aabmaabaGaamiEaaGaayjkaiaawMcaaaaa@39F7@ , and the mean residual life function m( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyBam aabmaabaGaamiEaaGaayjkaiaawMcaaaaa@39FC@ of QSD are thus obtained as

h( x )= θ 3 ( θ+x+α x 2 ) α θ 2 x 2 +θ( θ+2α )x+( θ 3 +θ+2α ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiAam aabmaabaGaamiEaaGaayjkaiaawMcaaiabg2da9maalaaabaGaeqiU de3aaWbaaeqabaqcLbmacaaIZaaaaKqbaoaabmaabaGaeqiUdeNaey 4kaSIaamiEaiabgUcaRiabeg7aHjaadIhadaahaaqabeaajugWaiaa ikdaaaaajuaGcaGLOaGaayzkaaaabaGaeqySdeMaaGPaVlabeI7aXn aaCaaabeqaaKqzadGaaGOmaaaajuaGcaWG4bWaaWbaaeqabaqcLbma caaIYaaaaKqbakabgUcaRiabeI7aXnaabmaabaGaeqiUdeNaey4kaS IaaGOmaiabeg7aHbGaayjkaiaawMcaaiaadIhacqGHRaWkdaqadaqa aiabeI7aXnaaCaaabeqaaKqzadGaaG4maaaajuaGcqGHRaWkcqaH4o qCcqGHRaWkcaaIYaGaeqySdegacaGLOaGaayzkaaaaaaaa@6C47@  (4.3)
and m( x )= 1 [ α θ 2 x 2 +θ( θ+2α )x+( θ 3 +θ+2α ) ] e θx x [ α θ 2 t 2 +θ( θ+2α )t +( θ 3 +θ+2α ) ] e θt dt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyBam aabmaabaGaamiEaaGaayjkaiaawMcaaiabg2da9maalaaabaGaaGym aaqaamaadmaabaGaeqySdeMaaGPaVlabeI7aXnaaCaaabeqaaKqzad GaaGOmaaaajuaGcaWG4bWaaWbaaeqabaqcLbmacaaIYaaaaKqbakab gUcaRiabeI7aXnaabmaabaGaeqiUdeNaey4kaSIaaGOmaiabeg7aHb GaayjkaiaawMcaaiaadIhacqGHRaWkdaqadaqaaiabeI7aXnaaCaaa beqaaKqzadGaaG4maaaajuaGcqGHRaWkcqaH4oqCcqGHRaWkcaaIYa GaeqySdegacaGLOaGaayzkaaaacaGLBbGaayzxaaGaamyzamaaCaaa beqaaKqzadGaeyOeI0IaeqiUdeNaamiEaaaaaaqcfa4aa8qCaeaada WadaabaeqabaGaeqySdeMaaGPaVlabeI7aXnaaCaaabeqaaKqzadGa aGOmaaaajuaGcaWG0bWaaWbaaeqabaqcLbmacaaIYaaaaKqbakabgU caRiabeI7aXnaabmaabaGaeqiUdeNaey4kaSIaaGOmaiabeg7aHbGa ayjkaiaawMcaaiaadshaaeaacqGHRaWkdaqadaqaaiabeI7aXnaaCa aabeqaaKqzadGaaG4maaaajuaGcqGHRaWkcqaH4oqCcqGHRaWkcaaI YaGaeqySdegacaGLOaGaayzkaaaaaiaawUfacaGLDbaacaWGLbWaaW baaeqabaqcLbmacqGHsislcqaH4oqCcaaMc8UaamiDaaaajuaGcaWG KbGaamiDaaqaaiaadIhaaeaacqGHEisPaiabgUIiYdaaaa@980E@

= α θ 2 x 2 +θ( θ+4α )x+( θ 3 +2θ+6α ) θ[ α θ 2 x 2 +θ( θ+2α )x+( θ 3 +θ+2α ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyypa0 ZaaSaaaeaacqaHXoqycaaMc8UaeqiUde3aaWbaaeqabaqcLbmacaaI YaaaaKqbakaadIhadaahaaqabeaajugWaiaaikdaaaqcfaOaey4kaS IaeqiUde3aaeWaaeaacqaH4oqCcqGHRaWkcaaI0aGaeqySdegacaGL OaGaayzkaaGaamiEaiabgUcaRmaabmaabaGaeqiUde3aaWbaaeqaba qcLbmacaaIZaaaaKqbakabgUcaRiaaikdacqaH4oqCcqGHRaWkcaaI 2aGaeqySdegacaGLOaGaayzkaaaabaGaeqiUde3aamWaaeaacqaHXo qycaaMc8UaeqiUde3aaWbaaeqabaqcLbmacaaIYaaaaKqbakaadIha daahaaqabeaajugWaiaaikdaaaqcfaOaey4kaSIaeqiUde3aaeWaae aacqaH4oqCcqGHRaWkcaaIYaGaeqySdegacaGLOaGaayzkaaGaamiE aiabgUcaRmaabmaabaGaeqiUde3aaWbaaeqabaqcLbmacaaIZaaaaK qbakabgUcaRiabeI7aXjabgUcaRiaaikdacqaHXoqyaiaawIcacaGL PaaaaiaawUfacaGLDbaaaaaaaa@7F6A@  (4.4)
It can be easily verified that h( 0 )= θ 4 θ 3 +θ+2α =f( 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiAam aabmaabaGaaGimaaGaayjkaiaawMcaaiabg2da9maalaaabaGaeqiU de3aaWbaaeqabaqcLbmacaaI0aaaaaqcfayaaiabeI7aXnaaCaaabe qaaKqzadGaaG4maaaajuaGcqGHRaWkcqaH4oqCcqGHRaWkcaaIYaGa eqySdegaaiabg2da9iaadAgadaqadaqaaiaaicdaaiaawIcacaGLPa aaaaa@4D76@  and m( 0 )= θ 3 +2θ+6α θ( θ 3 +θ+2α ) = μ 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyBam aabmaabaGaaGimaaGaayjkaiaawMcaaiabg2da9maalaaabaGaeqiU de3aaWbaaeqabaqcLbmacaaIZaaaaKqbakabgUcaRiaaikdacqaH4o qCcqGHRaWkcaaI2aGaeqySdegabaGaeqiUde3aaeWaaeaacqaH4oqC daahaaqabeaajugWaiaaiodaaaqcfaOaey4kaSIaeqiUdeNaey4kaS IaaGOmaiabeg7aHbGaayjkaiaawMcaaaaacqGH9aqpcqaH8oqBdaWg aaqaaKqzadGaaGymaaqcfayabaWaaWbaaeqabaGamai4gkdiIcaaaa a@5B78@
The nature and behavior of h( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiAam aabmaabaGaamiEaaGaayjkaiaawMcaaaaa@39F7@ and m( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyBam aabmaabaGaamiEaaGaayjkaiaawMcaaaaa@39FC@ of QSD for varying values of parameters θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde haaa@383A@ and α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqySde gaaa@3823@ have been shown graphically in Figures 4 & 5. It is obvious that h( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiAam aabmaabaGaamiEaaGaayjkaiaawMcaaaaa@39F7@ of QSD is monotonically increasing whereas h( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiAam aabmaabaGaamiEaaGaayjkaiaawMcaaaaa@39F7@ is monotonically decreasing

Figure 4: Graphs of h( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiAam aabmaabaGaamiEaaGaayjkaiaawMcaaaaa@39F7@  of QSD for varying values of parameters θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde haaa@383A@ and α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqySde gaaa@3823@ .

Figure 5: Graphs of m( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyBam aabmaabaGaamiEaaGaayjkaiaawMcaaaaa@39FC@  of QSD for varying values of parameters θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde haaa@383A@  and α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqySde gaaa@3823@ .

Stochastic Orderings

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

  1. stochastic order ( X st Y ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacaWGybGaeyizIm6aaSbaaeaajugWaiaadohacaWG0baajuaGbeaa caWGzbaacaGLOaGaayzkaaaaaa@3F4B@ if F X ( x ) F Y ( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOram aaBaaabaqcLbmacaWGybaajuaGbeaadaqadaqaaiaadIhaaiaawIca caGLPaaacqGHLjYScaWGgbWaaSbaaeaajugWaiaadMfaaKqbagqaam aabmaabaGaamiEaaGaayjkaiaawMcaaaaa@4461@ for all x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEaa aa@3781@
  2. hazard rate order ( X hr Y ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacaWGybGaeyizIm6aaSbaaeaajugWaiaadIgacaWGYbaajuaGbeaa caWGzbaacaGLOaGaayzkaaaaaa@3F3E@  if h X ( x ) h Y ( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiAam aaBaaabaqcLbmacaWGybaajuaGbeaadaqadaqaaiaadIhaaiaawIca caGLPaaacqGHLjYScaWGObWaaSbaaeaajugWaiaadMfaaKqbagqaam aabmaabaGaamiEaaGaayjkaiaawMcaaaaa@44A5@ for all x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEaa aa@3781@
  3. mean residual life order ( X mrl Y ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacaWGybGaeyizIm6aaSbaaeaajugWaiaad2gacaWGYbGaamiBaaqc fayabaGaamywaaGaayjkaiaawMcaaaaa@4034@ if m X ( x ) m Y ( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyBam aaBaaabaqcLbmacaWGybaajuaGbeaadaqadaqaaiaadIhaaiaawIca caGLPaaacqGHKjYOcaWGTbWaaSbaaeaajugWaiaadMfaaKqbagqaam aabmaabaGaamiEaaGaayjkaiaawMcaaaaa@449E@ for all x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEaa aa@3781@
  4. likelihood ratio order ( X lr Y ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacaWGybGaeyizIm6aaSbaaeaajugWaiaadYgacaWGYbaajuaGbeaa caWGzbaacaGLOaGaayzkaaaaaa@3F42@ if f X ( x ) f Y ( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacaWGMbWaaSbaaeaajugWaiaadIfaaKqbagqaamaabmaabaGaamiE aaGaayjkaiaawMcaaaqaaiaadAgadaWgaaqaaKqzadGaamywaaqcfa yabaWaaeWaaeaacaWG4baacaGLOaGaayzkaaaaaaaa@42EB@ decreases in x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEaa aa@3781@ .

The following results due to Shaked and Shanthikumar [4] are well known for establishing stochastic ordering of distributions

X lr YX hr YX mrl Y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiwai abgsMiJoaaBaaabaqcLbmacaWGSbGaamOCaaqcfayabaGaamywaiab gkDiElaadIfacqGHKjYOdaWgaaqaaKqzadGaamiAaiaadkhaaKqbag qaaiaadMfacqGHshI3caWGybGaeyizIm6aaSbaaeaajugWaiaad2ga caWGYbGaamiBaaqcfayabaGaamywaaaa@51CB@
X st Y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaCbeae aacqGHthY3aeaacaWGybGaeyizIm6aaSbaaeaajugWaiaadohacaWG 0baajuaGbeaacaWGzbaabeaaaaa@404F@

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

Theorem: Let X MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiwaa aa@3761@ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcfaOae8 hpI4haaa@37EF@ QSD ( θ 1 , α 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacqaH4oqCdaWgaaqaaKqzadGaaGymaaqcfayabaGaaiilaiabeg7a HnaaBaaabaqcLbmacaaIXaaajuaGbeaaaiaawIcacaGLPaaaaaa@4142@ and Y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamywaa aa@3762@ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaGae8hpI4 haaa@3761@ QSD ( θ 2 , α 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacqaH4oqCdaWgaaqaaKqzadGaaGOmaaqcfayabaGaaiilaiabeg7a HnaaBaaabaqcLbmacaaIYaaajuaGbeaaaiaawIcacaGLPaaaaaa@4144@ . If α 1 = α 2 and θ 1 > θ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqySde 2aaSbaaeaajugWaiaaigdaaKqbagqaaiabg2da9iabeg7aHnaaBaaa baqcLbmacaaIYaaajuaGbeaacaaMc8Uaaeyyaiaab6gacaqGKbGaaG PaVlabeI7aXnaaBaaabaqcLbmacaaIXaaajuaGbeaacqGH+aGpcqaH 4oqCdaWgaaqaaKqzadGaaGOmaaqcfayabaaaaa@4F70@ (or θ 1 = θ 2 and α 1 < α 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde 3aaSbaaeaajugWaiaaigdaaKqbagqaaiabg2da9iabeI7aXnaaBaaa baqcLbmacaaIYaaajuaGbeaacaaMc8UaaGPaVlaabggacaqGUbGaae izaiaaykW7caaMc8UaeqySde2aaSbaaeaajugWaiaaigdaaKqbagqa aiabgYda8iabeg7aHnaaBaaabaqcLbmacaaIYaaajuaGbeaaaaa@5282@ ), then X lr Y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiwai abgsMiJoaaBaaabaqcLbmacaWGSbGaamOCaaqcfayabaGaamywaaaa @3DB9@ and hence X hr Y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiwai abgsMiJoaaBaaabaqcLbmacaWGObGaamOCaaqcfayabaGaamywaaaa @3DB5@ , X mrl Y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiwai abgsMiJoaaBaaabaqcLbmacaWGTbGaamOCaiaadYgaaKqbagqaaiaa dMfaaaa@3EAB@ and X st Y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiwai abgsMiJoaaBaaabaqcLbmacaWGZbGaamiDaaqcfayabaGaamywaaaa @3DC2@ .
Proof: We have

f X ( x; θ 1 , α 1 ) f Y ( x; θ 2 , α 2 ) = θ 1 3 ( θ 2 3 + θ 2 +2 α 2 ) θ 2 3 ( θ 1 3 + θ 1 +2 α 1 ) ( θ 1 +x+ α 1 x 2 θ 2 +x+ α 2 x 2 ) e ( θ 1 θ 2 )x ;x>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSqaae aacaWGMbWaaSbaaeaajugWaiaadIfaaKqbagqaamaabmaabaGaamiE aiaacUdacqaH4oqCdaWgaaqaaKqzadGaaGymaaqcfayabaGaaiilai abeg7aHnaaBaaabaqcLbmacaaIXaaajuaGbeaaaiaawIcacaGLPaaa aeaacaWGMbWaaSbaaeaajugWaiaadMfaaKqbagqaamaabmaabaGaam iEaiaacUdacqaH4oqCdaWgaaqaaKqzadGaaGOmaaqcfayabaGaaiil aiabeg7aHnaaBaaabaqcLbmacaaIYaaajuaGbeaaaiaawIcacaGLPa aaaaGaeyypa0ZaaSaaaeaacqaH4oqCdaWgaaqaaKqzadGaaGymaaqc fayabaWaaWbaaeqabaqcLbmacaaIZaaaaKqbaoaabmaabaGaeqiUde 3aaSbaaeaajugWaiaaikdaaKqbagqaamaaCaaabeqaaKqzadGaaG4m aaaajuaGcqGHRaWkcaaMc8UaeqiUde3aaSbaaeaajugWaiaaikdaaK qbagqaaiabgUcaRiaaikdacqaHXoqydaWgaaqaaKqzadGaaGOmaaqc fayabaaacaGLOaGaayzkaaaabaGaeqiUde3aaSbaaeaajugWaiaaik daaKqbagqaamaaCaaabeqaaKqzadGaaG4maaaajuaGdaqadaqaaiab eI7aXnaaBaaabaqcLbmacaaIXaaajuaGbeaadaahaaqabeaajugWai aaiodaaaqcfaOaey4kaSIaaGPaVlabeI7aXnaaBaaabaqcLbmacaaI XaaajuaGbeaacqGHRaWkcaaIYaGaeqySde2aaSbaaeaajugWaiaaig daaKqbagqaaaGaayjkaiaawMcaaaaacaaMc8+aaeWaaeaadaWcaaqa aiabeI7aXnaaBaaabaqcLbmacaaIXaaajuaGbeaacqGHRaWkcaWG4b Gaey4kaSIaeqySde2aaSbaaeaajugWaiaaigdaaKqbagqaaiaadIha daahaaqabeaajugWaiaaikdaaaaajuaGbaGaeqiUde3aaSbaaeaaju gWaiaaikdaaKqbagqaaiabgUcaRiaadIhacqGHRaWkcqaHXoqydaWg aaqaaKqzadGaaGOmaaqcfayabaGaamiEamaaCaaabeqaaKqzadGaaG OmaaaaaaaajuaGcaGLOaGaayzkaaGaaGPaVlaadwgadaahaaqabeaa jugWaiabgkHiTSWaaeWaaKqbagaajugWaiabeI7aXTWaaSbaaKqbag aajugWaiaaigdaaKqbagqaaKqzadGaeyOeI0IaeqiUde3cdaWgaaqc fayaaKqzadGaaGOmaaqcfayabaaacaGLOaGaayzkaaqcLbmacaaMc8 UaamiEaaaajuaGcaaMc8UaaGPaVlaacUdacaWG4bGaeyOpa4JaaGim aaaa@CF6A@

Now

ln f X ( x; θ 1 , α 1 ) f Y ( x; θ 2 , α 2 ) =log[ θ 1 3 ( θ 2 3 + θ 2 +2 α 2 ) θ 2 3 ( θ 1 3 + θ 1 +2 α 1 ) ]+ln( θ 1 +x+ α 1 x 2 θ 2 +x+ α 2 x 2 )( θ 1 θ 2 )x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaciiBai aac6gadaWcbaqaaiaadAgadaWgaaqaaiaadIfaaeqaamaabmaabaGa amiEaiaacUdacqaH4oqCdaWgaaqaaKqzadGaaGymaaqcfayabaGaai ilaiabeg7aHnaaBaaabaqcLbmacaaIXaaajuaGbeaaaiaawIcacaGL PaaaaeaacaWGMbWaaSbaaeaacaWGzbaabeaadaqadaqaaiaadIhaca GG7aGaeqiUde3aaSbaaeaajugWaiaaikdaaKqbagqaaiaacYcacqaH XoqydaWgaaqaaKqzadGaaGOmaaqcfayabaaacaGLOaGaayzkaaaaai abg2da9iGacYgacaGGVbGaai4zamaadmaabaWaaSaaaeaacqaH4oqC daWgaaqaaKqzadGaaGymaaqcfayabaWaaWbaaeqabaqcLbmacaaIZa aaaKqbaoaabmaabaGaeqiUde3aaSbaaeaajugWaiaaikdaaKqbagqa amaaCaaabeqaaKqzadGaaG4maaaajuaGcqGHRaWkcaaMc8UaeqiUde 3aaSbaaeaajugWaiaaikdaaKqbagqaaiabgUcaRiaaikdacqaHXoqy daWgaaqaaKqzadGaaGOmaaqcfayabaaacaGLOaGaayzkaaaabaGaeq iUde3aaSbaaeaajugWaiaaikdaaKqbagqaamaaCaaabeqaaKqzadGa aG4maaaajuaGdaqadaqaaiabeI7aXnaaBaaabaqcLbmacaaIXaaaju aGbeaadaahaaqabeaajugWaiaaiodaaaqcfaOaey4kaSIaaGPaVlab eI7aXnaaBaaabaqcLbmacaaIXaaajuaGbeaacqGHRaWkcaaIYaGaeq ySde2aaSbaaeaajugWaiaaigdaaKqbagqaaaGaayjkaiaawMcaaaaa aiaawUfacaGLDbaacqGHRaWkciGGSbGaaiOBamaabmaabaWaaSaaae aacqaH4oqCdaWgaaqaaKqzadGaaGymaaqcfayabaGaey4kaSIaamiE aiabgUcaRiabeg7aHnaaBaaabaqcLbmacaaIXaaajuaGbeaacaWG4b WaaWbaaeqabaqcLbmacaaIYaaaaaqcfayaaiabeI7aXnaaBaaabaqc LbmacaaIYaaajuaGbeaacqGHRaWkcaWG4bGaey4kaSIaeqySde2aaS baaeaajugWaiaaikdaaKqbagqaaiaadIhadaahaaqabeaajugWaiaa ikdaaaaaaaqcfaOaayjkaiaawMcaaiabgkHiTmaabmaabaGaeqiUde 3aaSbaaeaajugWaiaaigdaaKqbagqaaiabgkHiTiabeI7aXnaaBaaa baqcLbmacaaIYaaajuaGbeaaaiaawIcacaGLPaaacaWG4baaaa@C20C@

This gives

d dx { ln f X ( x; θ 1 , α 1 ) f Y ( x; θ 2 , α 2 ) }= ( θ 2 θ 1 )+( α 2 α 1 )+2( α 1 θ 2 α 2 θ 1 )x+2( α 1 α 2 ) x 2 ( θ 1 +x+ α 1 x 2 )( θ 2 +x+ α 2 x 2 ) ( θ 1 θ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacaWGKbaabaGaamizaiaadIhaaaWaaiWaaeaaciGGSbGaaiOBamaa leaabaGaamOzamaaBaaabaqcLbmacaWGybaajuaGbeaadaqadaqaai aadIhacaGG7aGaeqiUde3aaSbaaeaajugWaiaaigdaaKqbagqaaiaa cYcacqaHXoqydaWgaaqaaiaaigdaaeqaaaGaayjkaiaawMcaaaqaai aadAgadaWgaaqaaKqzadGaamywaaqcfayabaWaaeWaaeaacaWG4bGa ai4oaiabeI7aXnaaBaaabaqcLbmacaaIYaaajuaGbeaacaGGSaGaeq ySde2aaSbaaeaajugWaiaaikdaaKqbagqaaaGaayjkaiaawMcaaaaa aiaawUhacaGL9baacqGH9aqpdaWcaaqaamaabmaabaGaeqiUde3aaS baaeaajugWaiaaikdaaKqbagqaaiabgkHiTiabeI7aXnaaBaaabaqc LbmacaaIXaaajuaGbeaaaiaawIcacaGLPaaacqGHRaWkdaqadaqaai abeg7aHnaaBaaabaqcLbmacaaIYaaajuaGbeaacqGHsislcqaHXoqy daWgaaqaaKqzadGaaGymaaqcfayabaaacaGLOaGaayzkaaGaey4kaS IaaGOmamaabmaabaGaeqySde2aaSbaaeaajugWaiaaigdaaKqbagqa aiabeI7aXnaaBaaabaqcLbmacaaIYaaajuaGbeaacqGHsislcqaHXo qydaWgaaqaaKqzadGaaGOmaaqcfayabaGaeqiUde3aaSbaaeaajugW aiaaigdaaKqbagqaaaGaayjkaiaawMcaaiaadIhacqGHRaWkcaaIYa WaaeWaaeaacqaHXoqydaWgaaqaaKqzadGaaGymaaqcfayabaGaeyOe I0IaeqySde2aaSbaaeaajugWaiaaikdaaKqbagqaaaGaayjkaiaawM caaiaadIhadaahaaqabeaajugWaiaaikdaaaaajuaGbaWaaeWaaeaa cqaH4oqCdaWgaaqaaKqzadGaaGymaaqcfayabaGaey4kaSIaamiEai abgUcaRiabeg7aHnaaBaaabaqcLbmacaaIXaaajuaGbeaacaWG4bWa aWbaaeqabaqcLbmacaaIYaaaaaqcfaOaayjkaiaawMcaamaabmaaba GaeqiUde3aaSbaaeaajugWaiaaikdaaKqbagqaaiabgUcaRiaadIha cqGHRaWkcqaHXoqydaWgaaqaaKqzadGaaGOmaaqcfayabaGaamiEam aaCaaabeqaaKqzadGaaGOmaaaaaKqbakaawIcacaGLPaaaaaGaeyOe I0YaaeWaaeaacqaH4oqCdaWgaaqaaKqzadGaaGymaaqcfayabaGaey OeI0IaeqiUde3aaSbaaeaajugWaiaaikdaaKqbagqaaaGaayjkaiaa wMcaaaaa@C703@

Thus if α 1 = α 2 and θ 1 > θ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqySde 2aaSbaaeaajugWaiaaigdaaKqbagqaaiabg2da9iabeg7aHnaaBaaa baqcLbmacaaIYaaajuaGbeaacaaMc8Uaaeyyaiaab6gacaqGKbGaaG PaVlabeI7aXnaaBaaabaqcLbmacaaIXaaajuaGbeaacqGH+aGpcqaH 4oqCdaWgaaqaaKqzadGaaGOmaaqcfayabaaaaa@4F70@  or θ 1 = θ 2 and α 1 < α 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde 3aaSbaaeaajugWaiaaigdaaKqbagqaaiabg2da9iabeI7aXnaaBaaa baqcLbmacaaIYaaajuaGbeaacaaMc8UaaGPaVlaabggacaqGUbGaae izaiaaykW7caaMc8UaeqySde2aaSbaaeaajugWaiaaigdaaKqbagqa aiabgYda8iabeg7aHnaaBaaabaqcLbmacaaIYaaajuaGbeaaaaa@5282@ , d dx ln f X ( x; θ 1 , α 1 ) f Y ( x; θ 2 , α 2 ) <0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacaWGKbaabaGaamizaiaadIhaaaGaciiBaiaac6gadaWcbaqaaiaa dAgadaWgaaqaaKqzadGaamiwaaqcfayabaWaaeWaaeaacaWG4bGaai 4oaiabeI7aXnaaBaaabaqcLbmacaaIXaaajuaGbeaacaGGSaGaeqyS de2aaSbaaeaajugWaiaaigdaaKqbagqaaaGaayjkaiaawMcaaaqaai aadAgadaWgaaqaaKqzadGaamywaaqcfayabaWaaeWaaeaacaWG4bGa ai4oaiabeI7aXnaaBaaabaqcLbmacaaIYaaajuaGbeaacaGGSaGaeq ySde2aaSbaaeaajugWaiaaikdaaKqbagqaaaGaayjkaiaawMcaaaaa cqGH8aapcaaIWaaaaa@5D57@ . This means that X lr Y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaWNajuaGca WGybGaeyizIm6aaSbaaeaajugWaiaadYgacaWGYbaajuaGbeaacaWG zbaaaa@3E3A@ and hence X hr Y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiwai abgsMiJoaaBaaabaqcLbmacaWGObGaamOCaaqcfayabaGaamywaaaa @3DB5@ , X mrl Y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiwai abgsMiJoaaBaaabaGaamyBaiaadkhacaWGSbaabeaacaWGzbaaaa@3CEF@ and X st Y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiwai abgsMiJoaaBaaabaqcLbmacaWGZbGaamiDaaqcfayabaGaamywaaaa @3DC2@ .

Mean Deviations From the Mean And the Median

The amount of scatter in a population is measured to some extent by the totality of deviations usually from mean and median. These are known as the mean deviation about the mean and the mean deviation about the median defined by
δ 1 ( X )= 0 | xμ | f( x )dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiTdq 2aaSbaaeaajugWaiaaigdaaKqbagqaamaabmaabaGaamiwaaGaayjk aiaawMcaaiabg2da9maapehabaWaaqWaaeaacaWG4bGaeyOeI0Iaeq iVd0gacaGLhWUaayjcSdaabaqcLbmacaaIWaaajuaGbaqcLbmacqGH EisPaKqbakabgUIiYdGaaGPaVlaadAgadaqadaqaaiaadIhaaiaawI cacaGLPaaacaWGKbGaamiEaaaa@53C5@  and δ 2 ( X )= 0 | xM | f( x )dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiTdq 2aaSbaaeaajugWaiaaikdaaKqbagqaamaabmaabaGaamiwaaGaayjk aiaawMcaaiabg2da9maapehabaWaaqWaaeaacaWG4bGaeyOeI0Iaam ytaaGaay5bSlaawIa7aaqaaKqzadGaaGimaaqcfayaaKqzadGaeyOh IukajuaGcqGHRiI8aiaaykW7caWGMbWaaeWaaeaacaWG4baacaGLOa GaayzkaaGaamizaiaadIhaaaa@52E2@ , respectively, where μ=E( X ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 Maeyypa0JaamyramaabmaabaGaamiwaaGaayjkaiaawMcaaaaa@3C70@  and M=Median ( X ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamytai abg2da9iaab2eacaqGLbGaaeizaiaabMgacaqGHbGaaeOBaiaabcca daqadaqaaiaadIfaaiaawIcacaGLPaaaaaa@40C5@ . The measures δ 1 ( X ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiTdq 2aaSbaaeaajugWaiaaigdaaKqbagqaamaabmaabaGaamiwaaGaayjk aiaawMcaaaaa@3D27@  and δ 2 ( X ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiTdq 2aaSbaaeGabaaIeKqzadGaaGOmaaqcfayabaWaaeWaaeaacaWGybaa caGLOaGaayzkaaaaaa@3DB7@ can be calculated using the following simplified relationships

δ 1 ( X )= 0 μ ( μx ) f( x )dx+ μ ( xμ ) f( x )dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiTdq 2aaSbaaeaajugWaiaaigdaaKqbagqaamaabmaabaGaamiwaaGaayjk aiaawMcaaiabg2da9maapehabaWaaeWaaeaacqaH8oqBcqGHsislca WG4baacaGLOaGaayzkaaaabaqcLbmacaaIWaaajuaGbaqcLbmacqaH 8oqBaKqbakabgUIiYdGaamOzamaabmaabaGaamiEaaGaayjkaiaawM caaiaadsgacaWG4bGaey4kaSYaa8qCaeaadaqadaqaaiaadIhacqGH sislcqaH8oqBaiaawIcacaGLPaaaaeaajugWaiabeY7aTbqcfayaaK qzadGaeyOhIukajuaGcqGHRiI8aiaadAgadaqadaqaaiaadIhaaiaa wIcacaGLPaaacaWGKbGaamiEaaaa@6538@
=μF( μ ) 0 μ xf( x )dx μ[ 1F( μ ) ]+ μ x f( x )dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyypa0 JaeqiVd0wcLbmacaWGgbqcfa4aaeWaaeaacqaH8oqBaiaawIcacaGL PaaacqGHsisldaWdXbqaaiaadIhacaaMc8UaamOzamaabmaabaGaam iEaaGaayjkaiaawMcaaiaadsgacaWG4baabaqcLbmacaaIWaaajuaG baqcLbmacqaH8oqBaKqbakabgUIiYdGaeyOeI0IaeqiVd02aamWaae aacaaIXaGaeyOeI0IaamOramaabmaabaGaeqiVd0gacaGLOaGaayzk aaaacaGLBbGaayzxaaGaey4kaSYaa8qCaeaacaWG4bGaaGPaVdqaaK qzadGaeqiVd0gajuaGbaqcLbmacqGHEisPaKqbakabgUIiYdGaamOz amaabmaabaGaamiEaaGaayjkaiaawMcaaiaadsgacaWG4baaaa@6C03@
=2μF( μ )2μ+2 μ x f( x )dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyypa0 JaaGOmaiabeY7aTLqzadGaamOraKqbaoaabmaabaGaeqiVd0gacaGL OaGaayzkaaGaeyOeI0IaaGOmaiabeY7aTjabgUcaRiaaikdadaWdXb qaaiaadIhacaaMc8oabaqcLbmacqaH8oqBaKqbagaajugWaiabg6Hi LcqcfaOaey4kIipacaWGMbWaaeWaaeaacaWG4baacaGLOaGaayzkaa GaamizaiaadIhaaaa@558E@
=2μF( μ )2 0 μ x f( x )dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyypa0 JaaGOmaiabeY7aTLqzadGaamOraKqbaoaabmaabaGaeqiVd0gacaGL OaGaayzkaaGaeyOeI0IaaGOmamaapehabaGaamiEaiaaykW7aeaaju gWaiaaicdaaKqbagaajugWaiabeY7aTbqcfaOaey4kIipacaWGMbWa aeWaaeaacaWG4baacaGLOaGaayzkaaGaamizaiaadIhaaaa@5183@ (6.1)

and

δ 2 ( X )= 0 M ( Mx ) f( x )dx+ M ( xM ) f( x )dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiTdq 2aaSbaaeaajugWaiaaikdaaKqbagqaamaabmaabaGaamiwaaGaayjk aiaawMcaaiabg2da9maapehabaWaaeWaaeaacaWGnbGaeyOeI0Iaam iEaaGaayjkaiaawMcaaaqaaKqzadGaaGimaaqcfayaaKqzadGaamyt aaqcfaOaey4kIipacaWGMbWaaeWaaeaacaWG4baacaGLOaGaayzkaa GaamizaiaadIhacqGHRaWkdaWdXbqaamaabmaabaGaamiEaiabgkHi Tiaad2eaaiaawIcacaGLPaaaaeaajugWaiaad2eaaKqbagaajugWai abg6HiLcqcfaOaey4kIipacaWGMbWaaeWaaeaacaWG4baacaGLOaGa ayzkaaGaamizaiaadIhaaaa@61A9@
=MF( M ) 0 M xf( x )dx M[ 1F( M ) ]+ M x f( x )dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyypa0 JaamytaiaaykW7caWGgbWaaeWaaeaacaWGnbaacaGLOaGaayzkaaGa eyOeI0Yaa8qCaeaacaWG4bGaaGPaVlaadAgadaqadaqaaiaadIhaai aawIcacaGLPaaacaWGKbGaamiEaaqaaKqzadGaaGimaaqcfayaaKqz adGaamytaaqcfaOaey4kIipacqGHsislcaWGnbWaamWaaeaacaaIXa GaeyOeI0IaamOramaabmaabaGaamytaaGaayjkaiaawMcaaaGaay5w aiaaw2faaiabgUcaRmaapehabaGaamiEaiaaykW7aeaajugWaiaad2 eaaKqbagaajugWaiabg6HiLcqcfaOaey4kIipacaWGMbWaaeWaaeaa caWG4baacaGLOaGaayzkaaGaamizaiaadIhaaaa@667A@
=μ+2 M x f( x )dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyypa0 JaeyOeI0IaeqiVd0Maey4kaSIaaGOmamaapehabaGaamiEaiaaykW7 aeaajugWaiaad2eaaKqbagaajugWaiabg6HiLcqcfaOaey4kIipaca WGMbWaaeWaaeaacaWG4baacaGLOaGaayzkaaGaamizaiaadIhaaaa@4BB6@
=μ2 0 M x f( x )dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyypa0 JaeqiVd0MaeyOeI0IaaGOmamaapehabaGaamiEaiaaykW7aeaajugW aiaaicdaaKqbagaajugWaiaad2eaaKqbakabgUIiYdGaamOzamaabm aabaGaamiEaaGaayjkaiaawMcaaiaadsgacaWG4baaaa@4A1D@  (6.2)
Using p.d.f. (2.1) and expression for the mean of QSD, we get

0 μ x f( x )dx=μ { α θ 3 μ 3 + θ 2 ( θ+3α ) μ 2 +θ( θ 3 +2θ+6α )μ+( θ 3 +2θ+6α ) } e θμ θ( θ 3 +θ+2α ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aa8qCae aacaWG4bGaaGPaVdqaaKqzadGaaGimaaqcfayaaKqzadGaeqiVd0ga juaGcqGHRiI8aiaadAgadaqadaqaaiaadIhaaiaawIcacaGLPaaaca WGKbGaamiEaiabg2da9iabeY7aTjabgkHiTmaalaaabaWaaiWaaeaa cqaHXoqycaaMc8UaeqiUde3aaWbaaeqabaqcLbmacaaIZaaaaKqbak abeY7aTnaaCaaabeqaaKqzadGaaG4maaaajuaGcqGHRaWkcqaH4oqC daahaaqabeaajugWaiaaikdaaaqcfa4aaeWaaeaacqaH4oqCcqGHRa WkcaaIZaGaeqySdegacaGLOaGaayzkaaGaeqiVd02aaWbaaeqabaqc LbmacaaIYaaaaKqbakabgUcaRiabeI7aXnaabmaabaGaeqiUde3aaW baaeqabaqcLbmacaaIZaaaaKqbakabgUcaRiaaikdacqaH4oqCcqGH RaWkcaaI2aGaeqySdegacaGLOaGaayzkaaGaeqiVd0Maey4kaSYaae WaaeaacqaH4oqCdaahaaqabeaajugWaiaaiodaaaqcfaOaey4kaSIa aGOmaiabeI7aXjabgUcaRiaaiAdacqaHXoqyaiaawIcacaGLPaaaai aawUhacaGL9baacaWGLbWaaWbaaeqabaqcLbmacqGHsislcqaH4oqC caaMc8UaeqiVd0gaaaqcfayaaiabeI7aXnaabmaabaGaeqiUde3aaW baaeqabaqcLbmacaaIZaaaaKqbakabgUcaRiabeI7aXjabgUcaRiaa ikdacqaHXoqyaiaawIcacaGLPaaaaaaaaa@9C70@                                                                                                                                               (6.3)
0 M x f( x )dx=μ { α θ 3 M 3 + θ 2 ( θ+3α ) M 2 +θ( θ 3 +2θ+6α )M+( θ 3 +2θ+6α ) } e θM θ( θ 3 +θ+2α ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aa8qCae aacaWG4bGaaGPaVdqaaKqzadGaaGimaaqcfayaaKqzadGaamytaaqc faOaey4kIipacaWGMbWaaeWaaeaacaWG4baacaGLOaGaayzkaaGaam izaiaadIhacqGH9aqpcqaH8oqBcqGHsisldaWcaaqaamaacmaabaGa eqySdeMaaGPaVlabeI7aXnaaCaaabeqaaKqzadGaaG4maaaajuaGca WGnbWaaWbaaeqabaqcLbmacaaIZaaaaKqbakabgUcaRiabeI7aXnaa CaaabeqaaKqzadGaaGOmaaaajuaGdaqadaqaaiabeI7aXjabgUcaRi aaiodacqaHXoqyaiaawIcacaGLPaaacaWGnbWaaWbaaeqabaqcLbma caaIYaaaaKqbakabgUcaRiabeI7aXnaabmaabaGaeqiUde3aaWbaae qabaqcLbmacaaIZaaaaKqbakabgUcaRiaaikdacqaH4oqCcqGHRaWk caaI2aGaeqySdegacaGLOaGaayzkaaGaamytaiabgUcaRmaabmaaba GaeqiUde3aaWbaaeqabaqcLbmacaaIZaaaaKqbakabgUcaRiaaikda cqaH4oqCcqGHRaWkcaaI2aGaeqySdegacaGLOaGaayzkaaaacaGL7b GaayzFaaGaamyzamaaCaaabeqaaKqzadGaeyOeI0IaeqiUdeNaaGPa Vlaad2eaaaaajuaGbaGaeqiUde3aaeWaaeaacqaH4oqCdaahaaqabe aajugWaiaaiodaaaqcfaOaey4kaSIaeqiUdeNaey4kaSIaaGOmaiab eg7aHbGaayjkaiaawMcaaaaaaaa@97FC@                       (6.4)
Using expressions from (6.1), (6.2), (6.3), and (6.4), the mean deviation about mean, δ 1 ( X ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiTdq 2aaSbaaeaajugWaiaaigdaaKqbagqaamaabmaabaGaamiwaaGaayjk aiaawMcaaaaa@3D27@ and the mean deviation about median, δ 2 ( X ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiTdq 2aaSbaaeaajugWaiaaikdaaKqbagqaamaabmaabaGaamiwaaGaayjk aiaawMcaaaaa@3D28@ of QSD are finally obtained as

δ 1 ( X )= 2{ α θ 2 μ 2 +θ( θ+4α )μ+( θ 3 +2θ+6α ) } e θμ θ( θ 3 +θ+2α ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiTdq 2aaSbaaeaajugWaiaaigdaaKqbagqaamaabmaabaGaamiwaaGaayjk aiaawMcaaiabg2da9maalaaabaGaaGOmamaacmaabaGaeqySdeMaaG PaVlabeI7aXnaaCaaabeqaaKqzadGaaGOmaaaajuaGcqaH8oqBdaah aaqabeaajugWaiaaikdaaaqcfaOaey4kaSIaeqiUde3aaeWaaeaacq aH4oqCcqGHRaWkcaaI0aGaaGPaVlabeg7aHbGaayjkaiaawMcaaiaa ykW7cqaH8oqBcqGHRaWkdaqadaqaaiabeI7aXnaaCaaabeqaaKqzad GaaG4maaaajuaGcqGHRaWkcaaIYaGaeqiUdeNaey4kaSIaaGOnaiab eg7aHbGaayjkaiaawMcaaaGaay5Eaiaaw2haaiaadwgadaahaaqabe aajugWaiabgkHiTiabeI7aXjaaykW7cqaH8oqBaaaajuaGbaGaeqiU de3aaeWaaeaacqaH4oqCdaahaaqabeaajugWaiaaiodaaaqcfaOaey 4kaSIaeqiUdeNaey4kaSIaaGOmaiabeg7aHbGaayjkaiaawMcaaaaa aaa@7E38@ (6.5)
δ 2 ( X )= 2{ α θ 3 M 3 + θ 2 ( θ+3α ) M 2 +θ( θ 3 +2θ+6α )M+( θ 3 +2θ+6α ) } e θM θ( θ 3 +θ+2α ) μ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiTdq 2aaSbaaeaajugWaiaaikdaaKqbagqaamaabmaabaGaamiwaaGaayjk aiaawMcaaiabg2da9maalaaabaGaaGOmamaacmaabaGaeqySdeMaaG PaVlabeI7aXnaaCaaabeqaaKqzadGaaG4maaaajuaGcaWGnbWaaWba aeqabaqcLbmacaaIZaaaaKqbakabgUcaRiabeI7aXnaaCaaabeqaaK qzadGaaGOmaaaajuaGdaqadaqaaiabeI7aXjabgUcaRiaaiodacqaH XoqyaiaawIcacaGLPaaacaWGnbWaaWbaaeqabaqcLbmacaaIYaaaaK qbakabgUcaRiabeI7aXnaabmaabaGaeqiUde3aaWbaaeqabaqcLbma caaIZaaaaKqbakabgUcaRiaaikdacqaH4oqCcqGHRaWkcaaI2aGaeq ySdegacaGLOaGaayzkaaGaamytaiabgUcaRmaabmaabaGaeqiUde3a aWbaaeqabaqcLbmacaaIZaaaaKqbakabgUcaRiaaikdacqaH4oqCcq GHRaWkcaaI2aGaeqySdegacaGLOaGaayzkaaaacaGL7bGaayzFaaGa amyzamaaCaaabeqaaKqzadGaeyOeI0IaeqiUdeNaaGPaVlaad2eaaa aajuaGbaGaeqiUde3aaeWaaeaacqaH4oqCdaahaaqabeaajugWaiaa iodaaaqcfaOaey4kaSIaeqiUdeNaey4kaSIaaGOmaiabeg7aHbGaay jkaiaawMcaaaaacqGHsislcqaH8oqBaaa@9028@  (6.6)

Bonferroni And Lorenz Curves

The Bonferroni and Lorenz curves [5] and Bonferroni and Gini indices have applications not only in economics to study income and poverty, but also in other fields like reliability, demography, insurance and medicine. The Bonferroni and Lorenz curves are defined as

B( p )= 1 pμ 0 q xf( x ) dx= 1 pμ [ 0 xf( x )dx q xf( x ) dx ]= 1 pμ [ μ q xf( x ) dx ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOqam aabmaabaGaamiCaaGaayjkaiaawMcaaiabg2da9maalaaabaGaaGym aaqaaiaadchacqaH8oqBaaWaa8qCaeaacaWG4bGaaGPaVlaadAgada qadaqaaiaadIhaaiaawIcacaGLPaaacaaMc8oabaqcLbmacaaIWaaa juaGbaqcLbmacaWGXbaajuaGcqGHRiI8aiaadsgacaWG4bGaeyypa0 ZaaSaaaeaacaaIXaaabaGaamiCaiabeY7aTbaadaWadaqaamaapeha baGaamiEaiaaykW7caWGMbWaaeWaaeaacaWG4baacaGLOaGaayzkaa GaamizaiaadIhacqGHsislaeaajugWaiaaicdaaKqbagaajugWaiab g6HiLcqcfaOaey4kIipadaWdXbqaaiaadIhacaaMc8UaamOzamaabm aabaGaamiEaaGaayjkaiaawMcaaaqaaKqzadGaamyCaaqcfayaaKqz adGaeyOhIukajuaGcqGHRiI8aiaaykW7caWGKbGaamiEaaGaay5wai aaw2faaiabg2da9maalaaabaGaaGymaaqaaiaadchacqaH8oqBaaWa amWaaeaacqaH8oqBcqGHsisldaWdXbqaaiaadIhacaaMc8UaamOzam aabmaabaGaamiEaaGaayjkaiaawMcaaaqaaKqzadGaamyCaaqcfaya aKqzadGaeyOhIukajuaGcqGHRiI8aiaaykW7caWGKbGaamiEaaGaay 5waiaaw2faaaaa@929B@ (7.1)
and L( p )= 1 μ 0 q xf( x ) dx= 1 μ [ 0 xf( x )dx q xf( x ) dx ]= 1 μ [ μ q xf( x ) dx ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamitam aabmaabaGaamiCaaGaayjkaiaawMcaaiabg2da9maalaaabaGaaGym aaqaaiabeY7aTbaadaWdXbqaaiaadIhacaaMc8UaamOzamaabmaaba GaamiEaaGaayjkaiaawMcaaaqaaKqzadGaaGimaaqcfayaaKqzadGa amyCaaqcfaOaey4kIipacaaMc8UaamizaiaadIhacqGH9aqpdaWcaa qaaiaaigdaaeaacqaH8oqBaaWaamWaaeaadaWdXbqaaiaadIhacaaM c8UaamOzamaabmaabaGaamiEaaGaayjkaiaawMcaaiaadsgacaWG4b GaeyOeI0cabaqcLbmacaaIWaaajuaGbaqcLbmacqGHEisPaKqbakab gUIiYdWaa8qCaeaacaWG4bGaaGPaVlaadAgadaqadaqaaiaadIhaai aawIcacaGLPaaacaaMc8oabaqcLbmacaWGXbaajuaGbaqcLbmacqGH EisPaKqbakabgUIiYdGaamizaiaadIhaaiaawUfacaGLDbaacqGH9a qpdaWcaaqaaiaaigdaaeaacqaH8oqBaaWaamWaaeaacqaH8oqBcqGH sisldaWdXbqaaiaadIhacaaMc8UaamOzamaabmaabaGaamiEaaGaay jkaiaawMcaaaqaaKqzadGaamyCaaqcfayaaKqzadGaeyOhIukajuaG cqGHRiI8aiaaykW7caWGKbGaamiEaaGaay5waiaaw2faaaaa@8FC6@ (7.2)

Respectively or equivalently

B( p )= 1 pμ 0 p F 1 ( x ) dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOqam aabmaabaGaamiCaaGaayjkaiaawMcaaiabg2da9maalaaabaGaaGym aaqaaiaadchacqaH8oqBaaWaa8qCaeaacaWGgbWaaWbaaeqabaqcLb macqGHsislcaaIXaaaaKqbaoaabmaabaGaamiEaaGaayjkaiaawMca aaqaaKqzadGaaGimaaqcfayaaKqzadGaamiCaaqcfaOaey4kIipaca aMc8UaamizaiaadIhaaaa@5005@  (7.3)
and L( p )= 1 μ 0 p F 1 ( x ) dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamitam aabmaabaGaamiCaaGaayjkaiaawMcaaiabg2da9maalaaabaGaaGym aaqaaiabeY7aTbaadaWdXbqaaiaadAeadaahaaqabeaajugWaiabgk HiTiaaigdaaaqcfa4aaeWaaeaacaWG4baacaGLOaGaayzkaaaabaqc LbmacaaIWaaajuaGbaqcLbmacaWGWbaajuaGcqGHRiI8aiaaykW7ca WGKbGaamiEaaaa@4F1A@ (7.4)

Respectively, where μ=E( X ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 Maeyypa0JaamyramaabmaabaGaamiwaaGaayjkaiaawMcaaaaa@3C70@ and q= F 1 ( p ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyCai abg2da9iaadAeadaahaaqabeaajugWaiabgkHiTiaaigdaaaqcfa4a aeWaaeaacaWGWbaacaGLOaGaayzkaaaaaa@3F4F@ .
The Bonferroni and Gini indices are thus defined as

B=1 0 1 B( p ) dp MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOqai abg2da9iaaigdacqGHsisldaWdXbqaaiaadkeadaqadaqaaiaadcha aiaawIcacaGLPaaaaeaajugWaiaaicdaaKqbagaajugWaiaaigdaaK qbakabgUIiYdGaaGPaVlaadsgacaWGWbaaaa@47E5@  (7.5)
and G=12 0 1 L( p ) dp MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4rai abg2da9iaaigdacqGHsislcaaIYaWaa8qCaeaacaWGmbWaaeWaaeaa caWGWbaacaGLOaGaayzkaaGaaGPaVdqaaKqzadGaaGimaaqcfayaaK qzadGaaGymaaqcfaOaey4kIipacaWGKbGaamiCaaaa@48B0@  (7.6) respectively.

Using p.d.f. of QSD (2.1), we get

q xf( x ) dx= { α θ 3 q 3 + θ 2 ( θ+3α ) q 2 +θ( θ 3 +2θ+6α )q+( θ 3 +2θ+6α ) } e θq θ( θ 3 +θ+2α ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aa8qCae aacaWG4bGaaGPaVlaadAgadaqadaqaaiaadIhaaiaawIcacaGLPaaa aeaajugWaiaadghaaKqbagaajugWaiabg6HiLcqcfaOaey4kIipaca aMc8UaamizaiaadIhacqGH9aqpdaWcaaqaamaacmaabaGaeqySdeMa aGPaVlabeI7aXnaaCaaabeqaaKqzadGaaG4maaaajuaGcaWGXbWaaW baaeqabaqcLbmacaaIZaaaaKqbakabgUcaRiabeI7aXnaaCaaabeqa aKqzadGaaGOmaaaajuaGdaqadaqaaiabeI7aXjabgUcaRiaaiodacq aHXoqyaiaawIcacaGLPaaacaWGXbWaaWbaaeqabaqcLbmacaaIYaaa aKqbakabgUcaRiabeI7aXnaabmaabaGaeqiUde3aaWbaaeqabaqcLb macaaIZaaaaKqbakabgUcaRiaaikdacqaH4oqCcqGHRaWkcaaI2aGa eqySdegacaGLOaGaayzkaaGaamyCaiabgUcaRmaabmaabaGaeqiUde 3aaWbaaeqabaqcLbmacaaIZaaaaKqbakabgUcaRiaaikdacqaH4oqC cqGHRaWkcaaI2aGaeqySdegacaGLOaGaayzkaaaacaGL7bGaayzFaa GaamyzamaaCaaabeqaaKqzadGaeyOeI0IaeqiUdeNaaGPaVlaadgha aaaajuaGbaGaeqiUde3aaeWaaeaacqaH4oqCdaahaaqabeaajugWai aaiodaaaqcfaOaey4kaSIaeqiUdeNaey4kaSIaaGOmaiabeg7aHbGa ayjkaiaawMcaaaaaaaa@984F@  (7.7)
Now using equation (7.7) in (7.1) and (7.2), we get
B( p )= 1 p [ 1 { α θ 3 q 3 + θ 2 ( θ+3α ) q 2 +θ( θ 3 +2θ+6α )q+( θ 3 +2θ+6α ) } e θq θ 3 +2θ+6α ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOqam aabmaabaGaamiCaaGaayjkaiaawMcaaiabg2da9maalaaabaGaaGym aaqaaiaadchaaaWaamWaaeaacaaIXaGaeyOeI0YaaSaaaeaadaGada qaaiabeg7aHjaaykW7cqaH4oqCdaahaaqabeaajugWaiaaiodaaaqc faOaamyCamaaCaaabeqaaKqzadGaaG4maaaajuaGcqGHRaWkcqaH4o qCdaahaaqabeaajugWaiaaikdaaaqcfa4aaeWaaeaacqaH4oqCcqGH RaWkcaaIZaGaeqySdegacaGLOaGaayzkaaGaamyCamaaCaaabeqaaK qzadGaaGOmaaaajuaGcqGHRaWkcqaH4oqCdaqadaqaaiabeI7aXnaa CaaabeqaaKqzadGaaG4maaaajuaGcqGHRaWkcaaIYaGaeqiUdeNaey 4kaSIaaGOnaiabeg7aHbGaayjkaiaawMcaaiaadghacqGHRaWkdaqa daqaaiabeI7aXnaaCaaabeqaaKqzadGaaG4maaaajuaGcqGHRaWkca aIYaGaeqiUdeNaey4kaSIaaGOnaiabeg7aHbGaayjkaiaawMcaaaGa ay5Eaiaaw2haaiaadwgadaahaaqabeaajugWaiabgkHiTiabeI7aXj aaykW7caWGXbaaaaqcfayaaiabeI7aXnaaCaaabeqaaKqzadGaaG4m aaaajuaGcqGHRaWkcaaIYaGaeqiUdeNaey4kaSIaaGOnaiabeg7aHb aaaiaawUfacaGLDbaaaaa@8CD5@ (7.8)

and

L( p )=1 { α θ 3 q 3 + θ 2 ( θ+3α ) q 2 +θ( θ 3 +2θ+6α )q+( θ 3 +2θ+6α ) } e θq θ 3 +2θ+6α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamitam aabmaabaGaamiCaaGaayjkaiaawMcaaiabg2da9iaaigdacqGHsisl daWcaaqaamaacmaabaGaeqySdeMaaGPaVlabeI7aXnaaCaaabeqaaK qzadGaaG4maaaajuaGcaWGXbWaaWbaaeqabaqcLbmacaaIZaaaaKqb akabgUcaRiabeI7aXnaaCaaabeqaaKqzadGaaGOmaaaajuaGdaqada qaaiabeI7aXjabgUcaRiaaiodacqaHXoqyaiaawIcacaGLPaaacaWG XbWaaWbaaeqabaqcLbmacaaIYaaaaKqbakabgUcaRiabeI7aXnaabm aabaGaeqiUde3aaWbaaeqabaqcLbmacaaIZaaaaKqbakabgUcaRiaa ikdacqaH4oqCcqGHRaWkcaaI2aGaeqySdegacaGLOaGaayzkaaGaam yCaiabgUcaRmaabmaabaGaeqiUde3aaWbaaeqabaqcLbmacaaIZaaa aKqbakabgUcaRiaaikdacqaH4oqCcqGHRaWkcaaI2aGaeqySdegaca GLOaGaayzkaaaacaGL7bGaayzFaaGaamyzamaaCaaabeqaaKqzadGa eyOeI0IaeqiUdeNaaGPaVlaadghaaaaajuaGbaGaeqiUde3aaWbaae qabaqcLbmacaaIZaaaaKqbakabgUcaRiaaikdacqaH4oqCcqGHRaWk caaI2aGaeqySdegaaaaa@892D@  (7.9)

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

B=1 { α θ 3 q 3 + θ 2 ( θ+3α ) q 2 +θ( θ 3 +2θ+6α )q+( θ 3 +2θ+6α ) } e θq θ 3 +2θ+6α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOqai abg2da9iaaigdacqGHsisldaWcaaqaamaacmaabaGaeqySdeMaaGPa VlabeI7aXnaaCaaabeqaaKqzadGaaG4maaaajuaGcaWGXbWaaWbaae qabaqcLbmacaaIZaaaaKqbakabgUcaRiabeI7aXnaaCaaabeqaaKqz adGaaGOmaaaajuaGdaqadaqaaiabeI7aXjabgUcaRiaaiodacqaHXo qyaiaawIcacaGLPaaacaWGXbWaaWbaaeqabaqcLbmacaaIYaaaaKqb akabgUcaRiabeI7aXnaabmaabaGaeqiUde3aaWbaaeqabaqcLbmaca aIZaaaaKqbakabgUcaRiaaikdacqaH4oqCcqGHRaWkcaaI2aGaeqyS degacaGLOaGaayzkaaGaamyCaiabgUcaRmaabmaabaGaeqiUde3aaW baaeqabaqcLbmacaaIZaaaaKqbakabgUcaRiaaikdacqaH4oqCcqGH RaWkcaaI2aGaeqySdegacaGLOaGaayzkaaaacaGL7bGaayzFaaGaam yzamaaCaaabeqaaKqzadGaeyOeI0IaeqiUdeNaaGPaVlaadghaaaaa juaGbaGaeqiUde3aaWbaaeqabaqcLbmacaaIZaaaaKqbakabgUcaRi aaikdacqaH4oqCcqGHRaWkcaaI2aGaeqySdegaaaaa@86A5@   (7.10)
G= 2{ α θ 3 q 3 + θ 2 ( θ+3α ) q 2 +θ( θ 3 +2θ+6α )q+( θ 3 +2θ+6α ) } e θq θ 3 +2θ+6α 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4rai abg2da9maalaaabaGaaGOmamaacmaabaGaeqySdeMaaGPaVlabeI7a XnaaCaaabeqaaKqzadGaaG4maaaajuaGcaWGXbWaaWbaaeqabaqcLb macaaIZaaaaKqbakabgUcaRiabeI7aXnaaCaaabeqaaKqzadGaaGOm aaaajuaGdaqadaqaaiabeI7aXjabgUcaRiaaiodacqaHXoqyaiaawI cacaGLPaaacaWGXbWaaWbaaeqabaqcLbmacaaIYaaaaKqbakabgUca RiabeI7aXnaabmaabaGaeqiUde3aaWbaaeqabaqcLbmacaaIZaaaaK qbakabgUcaRiaaikdacqaH4oqCcqGHRaWkcaaI2aGaeqySdegacaGL OaGaayzkaaGaamyCaiabgUcaRmaabmaabaGaeqiUde3aaWbaaeqaba qcLbmacaaIZaaaaKqbakabgUcaRiaaikdacqaH4oqCcqGHRaWkcaaI 2aGaeqySdegacaGLOaGaayzkaaaacaGL7bGaayzFaaGaamyzamaaCa aabeqaaKqzadGaeyOeI0IaeqiUdeNaaGPaVlaadghaaaaajuaGbaGa eqiUde3aaWbaaeqabaqcLbmacaaIZaaaaKqbakabgUcaRiaaikdacq aH4oqCcqGHRaWkcaaI2aGaeqySdegaaiabgkHiTiaaigdaaaa@8766@  (7.11)

Stress-Strength Reliability

The stress- strength reliability describes the life of a component which has random strength X MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiwaa aa@3761@ that is subjected to a random stress Y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamywaa aa@3762@ . When the stress applied to it exceeds the strength, the component fails instantly and the component will function satisfactorily till X>Y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiwai abg6da+iaadMfaaaa@3947@ . Therefore, R=P( Y<X ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOuai abg2da9iaadcfadaqadaqaaiaadMfacqGH8aapcaWGybaacaGLOaGa ayzkaaaaaa@3D7E@  is a measure of component reliability and in statistical literature it is known as stress-strength parameter. It has wide applications in almost all areas of knowledge especially in engineering such as structures, deterioration of rocket motors, static fatigue of ceramic components, aging of concrete pressure vessels etc. Let X MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiwaa aa@3761@ and Y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamywaa aa@3762@ be independent strength and stress random variables having QSD (2.1) with parameter ( θ 1 , α 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacqaH4oqCdaWgaaqaaKqzadGaaGymaaqcfayabaGaaiilaiabeg7a HnaaBaaabaqcLbmacaaIXaaajuaGbeaaaiaawIcacaGLPaaaaaa@4142@  and ( θ 2 , α 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacqaH4oqCdaWgaaqaaKqzadGaaGOmaaqcfayabaGaaiilaiabeg7a HnaaBaaabaqcLbmacaaIYaaajuaGbeaaaiaawIcacaGLPaaaaaa@4144@  respectively. Then the stress-strength reliability R MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOuaa aa@375B@ of QSD (2.1) can be obtained as

R=P( Y<X )= 0 P( Y<X|X=x ) f X ( x )dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOuai abg2da9iaadcfadaqadaqaaiaadMfacqGH8aapcaWGybaacaGLOaGa ayzkaaGaeyypa0Zaa8qCaeaacaWGqbWaaeWaaeaacaWGzbGaeyipaW JaamiwaiaacYhacaWGybGaeyypa0JaamiEaaGaayjkaiaawMcaaaqa aKqzadGaaGimaaqcfayaaKqzadGaeyOhIukajuaGcqGHRiI8aiaadA gadaWgaaqaaiaadIfaaeqaamaabmaabaGaamiEaaGaayjkaiaawMca aiaadsgacaWG4baaaa@55CA@
= 0 f( x; θ 1 , α 1 ) F( x; θ 2 , α 2 )dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyypa0 Zaa8qCaeaacaWGMbWaaeWaaeaacaWG4bGaai4oaiabeI7aXnaaBaaa baqcLbmacaaIXaaajuaGbeaacaGGSaGaeqySde2aaSbaaeaajugWai aaigdaaKqbagqaaaGaayjkaiaawMcaaaqaaKqzadGaaGimaaqcfaya aKqzadGaeyOhIukajuaGcqGHRiI8aiaaykW7caaMc8UaamOramaabm aabaGaamiEaiaacUdacqaH4oqCdaWgaaqaaKqzadGaaGOmaaqcfaya baGaaiilaiabeg7aHnaaBaaabaqcLbmacaaIYaaajuaGbeaaaiaawI cacaGLPaaacaWGKbGaamiEaaaa@5F26@
=1 θ 1 3 [ θ 1 θ 2 7 +( 4 θ 1 2 +1 ) θ 2 6 +( 6 θ 1 3 +5 θ 1 +2 α 1 ) θ 2 5 +( 4 θ 1 4 +10 θ 1 2 +4 α 1 θ 1 +4 α 2 θ 1 +3 ) θ 2 4 +( θ 1 5 +10 θ 1 3 +2 α 1 θ 1 2 +14 α 2 θ 1 2 +8 α 1 +7 θ 1 +2 α 2 θ 1 +6 α 2 ) θ 2 3 +( 5 θ 1 4 +18 α 2 θ 1 3 +4 α 2 θ 1 2 +5 θ 1 2 +16 α 1 α 2 +10 α 1 θ 1 +14 α 2 θ 2 +6 α 2 ) θ 2 2 +( θ 1 5 +10 α 2 θ 1 4 +2 α 2 θ 1 3 + θ 1 3 +10 α 2 θ 1 2 +2 α 1 θ 1 2 +20 α 1 α 2 θ 1 +24 α 1 α 2 +6 α 2 θ 1 ) θ 2 +2( α 2 θ 1 5 +2 α 1 α 2 θ 1 2 +2 α 2 θ 1 3 ) ] ( θ 1 3 + θ 1 +2 α 1 )( θ 2 3 + θ 2 +2 α 2 ) ( θ 1 + θ 2 ) 5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyypa0 JaaGymaiabgkHiTmaalaaabaGaeqiUde3aaSbaaeaajugWaiaaigda aKqbagqaamaaCaaabeqaaKqzadGaaG4maaaajuaGdaWadaabaeqaba GaeqiUde3aaSbaaeaajugWaiaaigdaaKqbagqaaiaaykW7cqaH4oqC daWgaaqaaKqzadGaaGOmaaqcfayabaWaaWbaaeqabaqcLbmacaaI3a aaaKqbakabgUcaRmaabmaabaGaaGinaiabeI7aXnaaBaaabaqcLbma caaIXaaajuaGbeaadaahaaqabeaajugWaiaaikdaaaqcfaOaey4kaS IaaGymaaGaayjkaiaawMcaaiabeI7aXnaaBaaabaqcLbmacaaIYaaa juaGbeaadaahaaqabeaajugWaiaaiAdaaaqcfaOaey4kaSYaaeWaae aacaaI2aGaaGPaVlabeI7aXnaaBaaabaqcLbmacaaIXaaajuaGbeaa daahaaqabeaajugWaiaaiodaaaqcfaOaey4kaSIaaGynaiaaykW7cq aH4oqCdaWgaaqaaKqzadGaaGymaaqcfayabaGaey4kaSIaaGOmaiab eg7aHnaaBaaabaqcLbmacaaIXaaajuaGbeaaaiaawIcacaGLPaaacq aH4oqCdaWgaaqaaKqzadGaaGOmaaqcfayabaWaaWbaaeqabaqcLbma caaI1aaaaKqbakabgUcaRmaabmaabaGaaGinaiabeI7aXnaaBaaaba qcLbmacaaIXaaajuaGbeaadaahaaqabeaajugWaiaaisdaaaqcfaOa ey4kaSIaaGymaiaaicdacqaH4oqCdaWgaaqaaKqzadGaaGymaaqcfa yabaWaaWbaaeqabaqcLbmacaaIYaaaaKqbakabgUcaRiaaisdacqaH XoqydaWgaaqaaKqzadGaaGymaaqcfayabaGaeqiUde3aaSbaaeaaju gWaiaaigdaaKqbagqaaiabgUcaRiaaisdacqaHXoqydaWgaaqaaKqz adGaaGOmaaqcfayabaGaeqiUde3aaSbaaeaajugWaiaaigdaaKqbag qaaiabgUcaRiaaiodaaiaawIcacaGLPaaacqaH4oqCdaWgaaqaaKqz 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adGaaGOmaaqcfayabaGaeqiUde3aaSbaaeaajugWaiaaigdaaKqbag qaamaaCaaabeqaaKqzadGaaG4maaaajuaGcqGHRaWkcaaI0aGaeqyS de2aaSbaaeaajugWaiaaikdaaKqbagqaaiabeI7aXnaaBaaabaqcLb macaaIXaaajuaGbeaadaahaaqabeaajugWaiaaikdaaaqcfaOaey4k aSIaaGynaiabeI7aXnaaBaaabaqcLbmacaaIXaaajuaGbeaadaahaa qabeaajugWaiaaikdaaaqcfaOaey4kaSIaaGymaiaaiAdacqaHXoqy daWgaaqaaKqzadGaaGymaaqcfayabaGaeqySde2aaSbaaeaajugWai aaikdaaKqbagqaaiabgUcaRiaaigdacaaIWaGaeqySde2aaSbaaeaa jugWaiaaigdaaKqbagqaaiabeI7aXnaaBaaabaqcLbmacaaIXaaaju aGbeaacqGHRaWkcaaIXaGaaGinaiabeg7aHnaaBaaabaqcLbmacaaI YaaajuaGbeaacqaH4oqCdaWgaaqaaKqzadGaaGOmaaqcfayabaGaey 4kaSIaaGOnaiabeg7aHnaaBaaabaqcLbmacaaIYaaajuaGbeaaaiaa wIcacaGLPaaacqaH4oqCdaWgaaqaaKqzadGaaGOmaaqcfayabaWaaW baaeqabaqcLbmacaaIYaaaaaqcfayaaiabgUcaRmaabmaabaGaeqiU de3aaSbaaeaajugWaiaaigdaaKqbagqaamaaCaaabeqaaKqzadGaaG ynaaaajuaGcqGHRaWkcaaIXaGaaGimaiabeg7aHnaaBaaabaqcLbma caaIYaaajuaGbeaacqaH4oqCdaWgaaqaaKqzadGaaGymaaqcfayaba WaaWbaaeqabaqcLbmacaaI0aaaaKqbakabgUcaRiaaikdacqaHXoqy 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LbmacaaIXaaajuaGbeaacqaHXoqydaWgaaqaaKqzadGaaGOmaaqcfa yabaGaeqiUde3aaSbaaeaajugWaiaaigdaaKqbagqaamaaCaaabeqa aKqzadGaaGOmaaaajuaGcqGHRaWkcaaIYaGaeqySde2aaSbaaeaaju gWaiaaikdaaKqbagqaaiabeI7aXnaaBaaabaqcLbmacaaIXaaajuaG beaadaahaaqabeaajugWaiaaiodaaaaajuaGcaGLOaGaayzkaaaaai aawUfacaGLDbaaaeaadaqadaqaaiabeI7aXnaaBaaabaqcLbmacaaI XaaajuaGbeaadaahaaqabeaajugWaiaaiodaaaqcfaOaey4kaSIaeq iUde3aaSbaaeaacaaIXaaabeaacqGHRaWkcaaIYaGaeqySde2aaSba aeaacaaIXaaabeaaaiaawIcacaGLPaaadaqadaqaaiabeI7aXnaaBa aabaqcLbmacaaIYaaajuaGbeaadaahaaqabeaajugWaiaaiodaaaqc faOaey4kaSIaeqiUde3aaSbaaeaajugWaiaaikdaaKqbagqaaiabgU caRiaaikdacqaHXoqydaWgaaqaaKqzadGaaGOmaaqcfayabaaacaGL OaGaayzkaaWaaeWaaeaacqaH4oqCdaWgaaqaaKqzadGaaGymaaqcfa yabaGaey4kaSIaeqiUde3aaSbaaeaajugWaiaaikdaaKqbagqaaaGa ayjkaiaawMcaamaaCaaabeqaaKqzadGaaGynaaaaaaaaaa@2702@

.

It can be easily verified that at α 1 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqySde 2aaSbaaeaajugWaiaaigdaaKqbagqaaiabg2da9iaaicdaaaa@3C7B@  and α 2 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqySde 2aaSbaaeaajugWaiaaikdaaKqbagqaaiabg2da9iaaicdaaaa@3C7C@ , the above expression reduces to the corresponding expression for Shanker distribution introduced by Shanker [1].

Maximum Likelihood Estimation Of Parameters

Let ( x 1 , x 2 , x 3 ,..., x n ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacaWG4bWaaSbaaeaajugWaiaaigdaaKqbagqaaiaacYcacaaMc8Ua amiEamaaBaaabaqcLbmacaaIYaaajuaGbeaacaGGSaGaaGPaVlaadI hadaWgaaqaaKqzadGaaG4maaqcfayabaGaaiilaiaaykW7caaMc8Ua aiOlaiaac6cacaGGUaGaaGPaVlaaykW7caGGSaGaamiEamaaBaaaba qcLbmacaWGUbaajuaGbeaaaiaawIcacaGLPaaaaaa@54B4@  be a random sample of size n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaaaa@36E9@  from QSD (2.1)). The likelihood function, L MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamitaaaa@36C7@ of (2.1) is given by

L= ( θ 3 θ 3 +θ+2α ) n i=1 n ( θ+ x i +α x i 2 ) e nθ x ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamitai abg2da9maabmaabaWaaSaaaeaacqaH4oqCdaahaaqabeaajugWaiaa iodaaaaajuaGbaGaeqiUde3aaWbaaeqabaqcLbmacaaIZaaaaKqbak abgUcaRiabeI7aXjabgUcaRiaaikdacqaHXoqyaaaacaGLOaGaayzk aaWaaWbaaeqabaqcLbmacaWGUbaaaKqbaoaarahabaWaaeWaaeaacq aH4oqCcqGHRaWkcaWG4bWaaSbaaeaacaWGPbaabeaacqGHRaWkcqaH XoqycaaMc8UaamiEamaaBaaabaGaamyAaaqabaWaaWbaaeqabaqcLb macaaIYaaaaaqcfaOaayjkaiaawMcaaaqaaKqzadGaamyAaiabg2da 9iaaigdaaKqbagaajugWaiaad6gaaKqbakabg+GivdGaaGPaVlaadw gadaahaaqabeaajugWaiabgkHiTiaad6gacaaMc8UaeqiUdeNaaGPa VlqadIhagaqeaaaaaaa@6ED4@

The natural log likelihood function is thus obtained as

lnL=nln( θ 3 θ 3 +θ+2α )+ i=1 n ln( θ+ x i +α x i 2 ) nθ x ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaciiBai aac6gacaWGmbGaeyypa0JaamOBaiGacYgacaGGUbWaaeWaaeaadaWc aaqaaiabeI7aXnaaCaaabeqaaKqzadGaaG4maaaaaKqbagaacqaH4o qCdaahaaqabeaajugWaiaaiodaaaqcfaOaey4kaSIaeqiUdeNaey4k aSIaaGOmaiabeg7aHbaaaiaawIcacaGLPaaacqGHRaWkdaaeWbqaai GacYgacaGGUbWaaeWaaeaacqaH4oqCcqGHRaWkcaWG4bWaaSbaaeaa jugWaiaadMgaaKqbagqaaiabgUcaRiabeg7aHjaaykW7caWG4bWaaS baaeaajugWaiaadMgaaKqbagqaamaaCaaabeqaaKqzadGaaGOmaaaa aKqbakaawIcacaGLPaaaaeaajugWaiaadMgacqGH9aqpcaaIXaaaju aGbaqcLbmacaWGUbaajuaGcqGHris5aiabgkHiTiaad6gacaaMc8Ua eqiUdeNaaGPaVlqadIhagaqeaaaa@7348@

The maximum likelihood estimates (MLE) ( θ ^ , α ^ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacu aH4oqCgaqcaiaacYcacuaHXoqygaqcaaGaayjkaiaawMcaaaaa@3BA4@  of ( θ,α ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacq aH4oqCcaGGSaGaeqySdegacaGLOaGaayzkaaaaaa@3B84@  are then the solutions      of the following non-linear equations

lnL θ = 3n θ n( 3 θ 2 +1 ) θ 3 +θ+2α + i=1 n 1 θ+ x i +α x i 2 n x ¯ =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacqGHciITciGGSbGaaiOBaiaadYeaaeaacqGHciITcqaH4oqCaaGa eyypa0ZaaSaaaeaacaaIZaGaamOBaaqaaiabeI7aXbaacqGHsislda Wcaaqaaiaad6gacaaMc8+aaeWaaeaacaaIZaGaeqiUde3aaWbaaeqa baqcLbmacaaIYaaaaKqbakabgUcaRiaaigdaaiaawIcacaGLPaaaae aacqaH4oqCdaahaaqabeaajugWaiaaiodaaaqcfaOaey4kaSIaeqiU deNaey4kaSIaaGOmaiabeg7aHbaacqGHRaWkdaaeWbqaamaalaaaba GaaGymaaqaaiabeI7aXjabgUcaRiaadIhadaWgaaqaaKqzadGaamyA aaqcfayabaGaey4kaSIaeqySdeMaaGPaVlaadIhadaWgaaqaaKqzad GaamyAaaqcfayabaWaaWbaaeqabaqcLbmacaaIYaaaaaaaaKqbagaa jugWaiaadMgacqGH9aqpcaaIXaaajuaGbaqcLbmacaWGUbaajuaGcq GHris5aiabgkHiTiaad6gacaaMc8UabmiEayaaraGaeyypa0JaaGim aaaa@7A1A@
lnL α = 2n θ 3 +θ+2α + i=1 n x i 2 θ+ x i +α x i 2 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacqGHciITciGGSbGaaiOBaiaadYeaaeaacqGHciITcqaHXoqyaaGa eyypa0ZaaSaaaeaacqGHsislcaaIYaGaamOBaaqaaiabeI7aXnaaCa aabeqaaKqzadGaaG4maaaajuaGcqGHRaWkcqaH4oqCcqGHRaWkcaaI YaGaeqySdegaaiabgUcaRmaaqahabaWaaSaaaeaacaWG4bWaaSbaae aacaWGPbaabeaadaahaaqabeaajugWaiaaikdaaaaajuaGbaGaeqiU deNaey4kaSIaamiEamaaBaaabaqcLbmacaWGPbaajuaGbeaacqGHRa WkcqaHXoqycaaMc8UaamiEamaaBaaabaqcLbmacaWGPbaajuaGbeaa daahaaqabeaajugWaiaaikdaaaaaaaqcfayaaKqzadGaamyAaiabg2 da9iaaigdaaKqbagaajugWaiaad6gaaKqbakabggHiLdGaeyypa0Ja aGimaaaa@6CF6@

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

These two natural log likelihood equations do not seem to be solved directly because they are not in closed forms. However, the Fisher’s scoring method can be applied to solve these equations. For, we have

2 lnL θ 2 = 3n θ 2 + n( 3 θ 4 6 θ 3 +5 θ 2 12θα+1 ) α 2 ( θ 3 +θ+2α ) 2 i=1 n 1 ( θ+ x i +α x i 2 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaaba GaeyOaIy7aaWbaaeqajyaGbaGaaGOmaaaajuaGciGGSbGaaiOBaiaa dYeaaeaacqGHciITcqaH4oqCdaahaaqabKGbagaacaaIYaaaaaaaju aGcqGH9aqpcqGHsisldaWcaaqaaiaaiodacaWGUbaabaGaeqiUdexc ga4aaWbaaeqabaGaaGOmaaaaaaqcfaOaey4kaSYaaSaaaeaacaWGUb WaaeWaaeaacaaIZaGaeqiUdexcga4aaWbaaeqabaGaaGinaaaajuaG cqGHsislcaaI2aGaeqiUde3aaWbaaeqajyaGbaGaaG4maaaajuaGcq GHRaWkcaaI1aGaeqiUde3aaWbaaeqajyaGbaGaaGOmaaaajuaGcqGH sislcaaIXaGaaGOmaiabeI7aXjaaykW7cqaHXoqycqGHRaWkcaaIXa aacaGLOaGaayzkaaGaeqySde2aaWbaaeqajyaGbaGaaGOmaaaaaKqb agaadaqadaqaaiabeI7aXnaaCaaabeqcgayaaiaaiodaaaqcfaOaey 4kaSIaeqiUdeNaey4kaSIaaGOmaiabeg7aHbGaayjkaiaawMcaaKGb aoaaCaaabeqaaiaaikdaaaaaaKqbakabgkHiTmaaqahabaWaaSaaae aacaaIXaaabaWaaeWaaeaacqaH4oqCcqGHRaWkcaWG4bqcga4aaSba aeaacaWGPbaabeaajuaGcqGHRaWkcqaHXoqycaaMc8UaamiEaKGbao aaBaaabaGaamyAaaqabaqcfa4aaWbaaeqajyaGbaGaaGOmaaaaaKqb akaawIcacaGLPaaajyaGdaahaaqabeaacaaIYaaaaaaaaeaacaWGPb Gaeyypa0JaaGymaaqaaiaad6gaaKqbakabggHiLdaaaa@8F86@
2 lnL θα = 2n( 3 θ 2 +1 ) ( θ 3 +θ+2α ) 2 i=1 n x i 2 ( θ+ x i +α x i 2 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xMi=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaaba GaeyOaIy7aaWbaaeqajyaGbaGaaGOmaaaajuaGciGGSbGaaiOBaiaa dYeaaeaacqGHciITcqaH4oqCcaaMc8UaeyOaIyRaeqySdegaaiabg2 da9maalaaabaGaaGOmaiaad6gacaaMc8+aaeWaaeaacaaIZaGaeqiU de3aaWbaaeqajyaGbaGaaGOmaaaajuaGcqGHRaWkcaaIXaaacaGLOa GaayzkaaaabaWaaeWaaeaacqaH4oqCdaahaaqabKGbagaacaaIZaaa aKqbakabgUcaRiabeI7aXjabgUcaRiaaikdacqaHXoqyaiaawIcaca GLPaaadaahaaqabKGbagaacaaIYaaaaaaajuaGcqGHsisldaaeWbqa amaalaaabaGaamiEaKGbaoaaBaaabaGaamyAaaqabaWaaWbaaeqaba GaaGOmaaaaaKqbagaadaqadaqaaiabeI7aXjabgUcaRiaadIhajyaG daWgaaqaaiaadMgaaeqaaKqbakabgUcaRiabeg7aHjaaykW7caWG4b qcga4aaSbaaeaacaWGPbaabeaadaahaaqabeaacaaIYaaaaaqcfaOa ayjkaiaawMcaaKGbaoaaCaaabeqaaiaaikdaaaaaaaqaaiaadMgacq GH9aqpcaaIXaaabaGaamOBaaqcfaOaeyyeIuoaaaa@7902@
2 lnL α 2 = 4n ( θ 3 +θ+2α ) 2 i=1 n x i 4 ( θ+ x i +α x i 2 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xMi=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaaba GaeyOaIy7aaWbaaeqajyaGbaGaaGOmaaaajuaGciGGSbGaaiOBaiaa dYeaaeaacqGHciITcqaHXoqyjyaGdaahaaqabeaacaaIYaaaaaaaju aGcqGH9aqpdaWcaaqaaiaaisdacaWGUbaabaWaaeWaaeaacqaH4oqC jyaGdaahaaqabeaacaaIZaaaaKqbakabgUcaRiabeI7aXjabgUcaRi aaikdacqaHXoqyaiaawIcacaGLPaaajyaGdaahaaqabeaacaaIYaaa aaaajuaGcqGHsisldaaeWbqaamaalaaabaGaamiEamaaBaaabaGaam yAaaqabaqcga4aaWbaaeqabaGaaGinaaaaaKqbagaadaqadaqaaiab eI7aXjabgUcaRiaadIhadaWgaaqcgayaaiaadMgaaKqbagqaaiabgU caRiabeg7aHjaaykW7caWG4bqcga4aaSbaaeaacaWGPbaabeaadaah aaqabeaacaaIYaaaaaqcfaOaayjkaiaawMcaaKGbaoaaCaaabeqaai aaikdaaaaaaaqaaiaadMgacqGH9aqpcaaIXaaabaGaamOBaaqcfaOa eyyeIuoaaaa@6D3B@

The solution of following equations gives MLE’s ( θ ^ , α ^ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacuaH4oqCgaqcaiaacYcacuaHXoqygaqcaaGaayjkaiaawMcaaaaa @3C32@  of ( θ,α ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacqaH4oqCcaGGSaGaeqySdegacaGLOaGaayzkaaaaaa@3C12@  of QSD

[ 2 lnL θ 2 2 lnL θα 2 lnL θα 2 lnL α 2 ] θ ^ = θ 0 α ^ = α 0 [ θ ^ θ 0 α ^ α 0 ]= [ lnL θ lnL α ] θ ^ = θ 0 α ^ = α 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xMi=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaadmaaba qbaeqabiGaaaqaamaalaaabaGaeyOaIy7aaWbaaeqajyaGbaGaaGOm aaaajuaGciGGSbGaaiOBaiaadYeaaeaacqGHciITcqaH4oqCdaahaa qabKGbagaacaaIYaaaaaaaaKqbagaadaWcaaqaaiabgkGi2oaaCaaa beqcgayaaiaaikdaaaqcfaOaciiBaiaac6gacaWGmbaabaGaeyOaIy RaeqiUdeNaaGPaVlabgkGi2kabeg7aHbaaaeaadaWcaaqaaiabgkGi 2oaaCaaabeqcgayaaiaaikdaaaqcfaOaciiBaiaac6gacaWGmbaaba GaeyOaIyRaeqiUdeNaaGPaVlabgkGi2kabeg7aHbaaaeaadaWcaaqa aiabgkGi2oaaCaaabeqcgayaaiaaikdaaaqcfaOaciiBaiaac6gaca WGmbaabaGaeyOaIyRaeqySde2aaWbaaeqajyaGbaGaaGOmaaaaaaaa aaqcfaOaay5waiaaw2faaKGbaoaaBaaaeaqabeaacuaH4oqCgaqcai abg2da9iabeI7aXnaaBaaabaGaaGimaaqabaaabaGafqySdeMbaKaa cqGH9aqpcqaHXoqydaWgaaqaaiaaicdaaeqaaaaabeaajuaGdaWada qaauaabeqaceaaaeaacuaH4oqCgaqcaiabgkHiTiabeI7aXnaaBaaa jyaGbaGaaGimaaqcfayabaaabaGafqySdeMbaKaacqGHsislcqaHXo qydaWgaaqcgayaaiaaicdaaKqbagqaaaaaaiaawUfacaGLDbaacqGH 9aqpdaWadaqaauaabeqaceaaaeaadaWcaaqaaiabgkGi2kGacYgaca GGUbGaamitaaqaaiabgkGi2kabeI7aXbaaaeaadaWcaaqaaiabgkGi 2kGacYgacaGGUbGaamitaaqaaiabgkGi2kabeg7aHbaaaaaacaGLBb Gaayzxaaqcga4aaSbaaeaafaqabeGabaaabaGafqiUdeNbaKaacqGH 9aqpcqaH4oqCdaWgaaqaaiaaicdaaeqaaaqaaiqbeg7aHzaajaGaey ypa0JaeqySde2aaSbaaeaacaaIWaaabeaaaaaabeaaaaa@A14B@

where θ 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde 3aaSbaaKqbGeaacaaIWaaabeaaaaa@3943@ and α 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqySde 2cdaWgaaqcfayaaKqzadGaaGimaaqcfayabaaaaa@3B53@ are the initial values of θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde haaa@383A@  and α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqySde gaaa@3823@ , respectively. These equations are solved iteratively till sufficiently close values of θ ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaaaaa@384A@  and α ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqySde MbaKaaaaa@3833@  are obtained.

Data Analysis

In this section, the goodness of fit of QSD has been discussed with two real lifetime data sets from engineering and the fit has been compared with one parameter exponential, Lindley and Shanker distributions. The following two data sets have been considered.

Data set 1

This data set is the strength data of glass of the aircraft window reported by Fuller et al [6].

18.83 20.8 21.657 23.03 23.23 24.05 24.321 25.5 25.52 25.8 26.69 26.77 26.78
27.05 27.67 29.9 31.11 33.2 33.73 33.76 33.89 34.76 35.75 35.91 36.98 37.08
37.09 39.58 44.045 45.29 45.381

Data set 2

The following data represent the tensile strength, measured in GPa, of 69 carbon fibers tested under tension at gauge lengths of 20mm, Bader and Priest [7].

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

In order to compare the considered distributions, values of 2lnL MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyOeI0 IaaGOmaiGacYgacaGGUbGaamitaaaa@3AE2@ , AIC(Akaike Information Criterion) and K-S Statistic ( Kolmogorov-Smirnov Statistic) for the data sets have been computed and presented in Table 1. The formula for AIC and K-S Statistic is defined as follow:

AIC=2lnL+2k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyqai aadMeacaWGdbGaeyypa0JaeyOeI0IaaGOmaiGacYgacaGGUbGaamit aiabgUcaRiaaikdacaWGRbaaaa@40D2@ and K-S= Sup x | F n ( x ) F 0 ( x ) | MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4sai aab2cacaWGtbGaeyypa0ZaaCbeaeaacaqGtbGaaeyDaiaabchaaeaa jugWaiaadIhaaKqbagqaamaaemaabaGaamOraSWaaSbaaKqbagaaju gWaiaad6gaaKqbagqaamaabmaabaGaamiEaaGaayjkaiaawMcaaiab gkHiTiaadAeadaWgaaqaaKqzadGaaGimaaqcfayabaWaaeWaaeaaca WG4baacaGLOaGaayzkaaaacaGLhWUaayjcSdaaaa@503B@ , where k= MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4Aai abg2da9aaa@387A@ number of parameters, n= MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOBai abg2da9aaa@387D@  sample size, F n ( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOram aaBaaabaqcLbmacaWGUbaajuaGbeaadaqadaqaaiaadIhaaiaawIca caGLPaaaaaa@3CA5@ is the empirical distribution function and F 0 ( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOram aaBaaabaqcLbmacaaIWaaajuaGbeaadaqadaqaaiaadIhaaiaawIca caGLPaaaaaa@3C6C@  is the theoretical cumulative distribution function.. The best distribution corresponds to lower values of 2lnL MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyOeI0 IaaGOmaiGacYgacaGGUbGaamitaaaa@3AE2@ , AIC and K-S statistic. It can be easily seen from table 1 that the QSD gives better fit than one parameter exponential, Lindley and Shanker distributions and hence it can be considered as an important distribution for modeling lifetime data from engineering.

Data Sets

Distributions

ML Estimates

Standard Errors

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

AIC

K-S Statistic

1

QSD

θ ^ =0.097330 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpcaaIWaGaaiOlaiaaicdacaaI5aGaaG4naiaaioda caaIZaGaaGimaaaa@3F2E@

0.0101017

240.53

244.53

0.298

α ^ =13.623065 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqySde MbaKaacqGH9aqpcaaIXaGaaG4maiaac6cacaaI2aGaaGOmaiaaioda caaIWaGaaGOnaiaaiwdaaaa@3FD5@

52.81378

Shanker

θ ^ =0.6471636 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpcaaIWaGaaiOlaiaaiAdacaaI0aGaaG4naiaaigda caaI2aGaaG4maiaaiAdaaaa@3FF3@

0.0082

252.35

254.35

0.358

Lindley

θ ^ =0.062990 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpcaaIWaGaaiOlaiaaicdacaaI2aGaaGOmaiaaiMda caaI5aGaaGimaaaa@3F32@

0.008

253.98

255.98

0.365

Exponential

θ ^ =0.032449 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpcaaIWaGaaiOlaiaaicdacaaIZaGaaGOmaiaaisda caaI0aGaaGyoaaaa@3F2E@

0.005822

274.53

276.53

0.458

2

QSD

θ ^ =1.20552 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpcaaIXaGaaiOlaiaaikdacaaIWaGaaGynaiaaiwda caaIYaaaaa@3E6D@

0.083861

186.78

190.78

0.314

α ^ =49.73844 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqySde MbaKaacqGH9aqpcaaI0aGaaGyoaiaac6cacaaI3aGaaG4maiaaiIda caaI0aGaaGinaaaa@3F28@

34.58363

Shanker

θ ^ =0.658030 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpcaaIWaGaaiOlaiaaiAdacaaI1aGaaGioaiaaicda caaIZaGaaGimaaaa@3F2E@

0.052373

233

235

0.369

Lindley

θ ^ =0.65450 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpcaaIWaGaaiOlaiaaiAdacaaI1aGaaGinaiaaiwda caaIWaaaaa@3E72@

0.058031

238.38

240.38

0.401

Exponential

θ ^ =0.407942 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpcaaIWaGaaiOlaiaaisdacaaIWaGaaG4naiaaiMda caaI0aGaaGOmaaaa@3F32@

0.04911

261.73

263.73

0.448

Table 1: MLE’s, 2ln L MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyOeI0 ceaaaaaaaaa8qacaaIYaGaaGPaVlaabYgacaqGUbGaaeiiaiaadYea aaa@3D2C@ , Standard Error, AIC, and K-S Statistic of the fitted distributions of data sets 1 and 2.

Concluding Remarks

 A two-parameter quasi Shanker distribution (QSD), of which one parameter Shanker distribution introduced by Shanker [1] is a particular case, has been suggested and investigated. Its mathematical properties including moments, 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. For estimating its parameters method of maximum likelihood estimation has been discussed. Finally, two numerical examples of real lifetime data sets has been presented to test the goodness of fit of QSD over exponential, Lindley and Shanker distributions and the fit by QSD has been quite satisfactory. Therefore, QSD can be recommended as an important two-parameter lifetime distribution.

References

  1. Shanker R (2015) Shanker distribution and Its Applications. International Journal of Statistics and Applications 5(6): 338-348.
  2. Lindley DV (1958) Fiducial distributions and Bayes’ theorem, Journal of the Royal Statistical Society. Series B 20(1): 102-107.
  3. Shanker R (2016) The discrete Poisson-Shanker distribution. Jacobs Journal of Biostatistics 2(2): 41-21.
  4. Shaked M, Shanthikumar JG (1994) Stochastic Orders and Their Applications. Academic Press. New York.
  5. Bonferroni CE (1930) Elementi di Statistca generale, Seeber, Firenze.
  6. Fuller EJ, Frieman S, Quinn J, Quinn G, Carter W (1994) Fracture mechanics approach to the design of glass aircraft windows: A case study, SPIE Proc 2286: 419-430.
  7. Bader MG, Priest AM (1982) Statistical aspects of fiber and bundle strength in hybrid composites. In; hayashi T, Kawata K Umekawa S (Eds.), Progressin Science in Engineering Composites, ICCM-IV, Tokyo, 1129-1136.
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