ISSN: 2378-315X BBIJ

Biometrics & Biostatistics International Journal
Research Article
Volume 5 Issue 2 - 2017
The Discrete Poisson-Garima Distribution
Rama Shanker*
Department of Statistics, Eritrea Institute of Technology, Eritrea
Received: December 13, 2017 | Published: February 13, 2017
*Corresponding author: Rama Shanker, Department of Statistics, Eritrea Institute of Technology, Eritrea, Email:
Citation: Shanker R (2017) The Discrete Poisson-Garima Distribution. Biom Biostat Int J 5(2): 00127. DOI: 10.15406/bbij.2017.05.00127

Abstract

In this paper, a discrete Poisson-Garima distribution has been obtained by compounding Poisson distribution with Garima distribution introduced by Shanker [1]. The general expression for the r th factorial moment has been derived and hence moments about origin and central moments have been obtained. The expression for coefficient of Variation, skewness, kurtosis and index of dispersion has been given. Maximum likelihood estimation and the method of moments have been discussed for estimating the parameter of the distribution. Two examples of real data set have been given to test the goodness of fit of the discrete Poisson-Garima distribution and the fit has been compared with Poisson and Poisson-Lindley distributions.

Keywords: Garima distribution; Poisson-Lindley distribution; Compounding; Moments; Skewness, Kurtosis; Index of dispersion; Estimation of parameter; Goodness of fit.

Introduction

Shanker [1] introduced a lifetime distribution named Garima distribution having probability density function

                                       f( x;θ )= θ θ+2 ( 1+θ+θx ) e θx ;x>0,θ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aabmaabaGaamiEaiaacUdacqaH4oqCaiaawIcacaGLPaaacqGH9aqp daWcaaqaaiabeI7aXbqaaiabeI7aXjabgUcaRiaaikdaaaWaaeWaae aacaaIXaGaey4kaSIaeqiUdeNaey4kaSIaeqiUdeNaaGPaVlaadIha aiaawIcacaGLPaaacaWGLbWaaWbaaeqajuaibaGaeyOeI0IaeqiUde NaamiEaaaajuaGcaaMc8UaaGPaVlaaykW7caaMc8Uaai4oaiaadIha cqGH+aGpcaaIWaGaaiilaiaaykW7caaMc8UaeqiUdeNaeyOpa4JaaG imaaaa@62D6@  (1.1)

to model behavioral science data. It has been shown by Shanker [1] that Garima distribution gives much better fit than exponential and Lindley [2] distributions and the new lifetime distributions introduced by Shanker [3-6]namely Shanker, Akash, Aradhana and Sujatha distributions. The first four moments about origin of Garima distribution obtained by Shanker [1] are given by

μ 1 = θ+3 θ( θ+2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaKqbGeaacaaIXaaajuaGbeaadaahaaqabeaacWaGGBOmGika aiabg2da9maalaaabaGaeqiUdeNaey4kaSIaaG4maaqaaiabeI7aXn aabmaabaGaeqiUdeNaey4kaSIaaGOmaaGaayjkaiaawMcaaaaaaaa@47DA@ , μ 2 = 2( θ+4 ) θ 2 ( θ+2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaKqbGeaacaaIYaaajuaGbeaadaahaaqabeaacWaGGBOmGika aiabg2da9maalaaabaGaaGOmamaabmaabaGaeqiUdeNaey4kaSIaaG inaaGaayjkaiaawMcaaaqaaiabeI7aXnaaCaaabeqcfasaaiaaikda aaqcfa4aaeWaaeaacqaH4oqCcqGHRaWkcaaIYaaacaGLOaGaayzkaa aaaaaa@4BBB@
μ 3 = 6( θ+5 ) θ 3 ( θ+2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaKqbGeaacaaIZaaajuaGbeaadaahaaqabeaacWaGGBOmGika aiabg2da9maalaaabaGaaGOnamaabmaabaGaeqiUdeNaey4kaSIaaG ynaaGaayjkaiaawMcaaaqaaiabeI7aXnaaCaaabeqcfasaaiaaioda aaqcfa4aaeWaaeaacqaH4oqCcqGHRaWkcaaIYaaacaGLOaGaayzkaa aaaaaa@4BC2@ , μ 4 = 24( θ+6 ) θ 4 ( θ+2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaKqbGeaacaaI0aaajuaGbeaadaahaaqabeaacWaGGBOmGika aiabg2da9maalaaabaGaaGOmaiaaisdadaqadaqaaiabeI7aXjabgU caRiaaiAdaaiaawIcacaGLPaaaaeaacqaH4oqCdaahaaqabKqbGeaa caaI0aaaaKqbaoaabmaabaGaeqiUdeNaey4kaSIaaGOmaaGaayjkai aawMcaaaaaaaa@4C7F@ .
The central moments of Garima distribution obtained by Shanker [1] are as follows

μ 2 = θ 2 +6θ+7 θ 2 ( θ+2 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaKqbGeaacaaIYaaajuaGbeaacqGH9aqpdaWcaaqaaiabeI7a XnaaCaaabeqcfasaaiaaikdaaaqcfaOaey4kaSIaaGOnaiabeI7aXj abgUcaRiaaiEdaaeaacqaH4oqCdaahaaqabKqbGeaacaaIYaaaaKqb aoaabmaabaGaeqiUdeNaey4kaSIaaGOmaaGaayjkaiaawMcaamaaCa aabeqcfasaaiaaikdaaaaaaaaa@4C6D@
μ 3 = 2( θ 3 +9 θ 2 +21θ+15 ) θ 3 ( θ+2 ) 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaKqbGeaacaaIZaaajuaGbeaacqGH9aqpdaWcaaqaaiaaikda daqadaqaaiabeI7aXnaaCaaabeqcfasaaiaaiodaaaqcfaOaey4kaS IaaGyoaiabeI7aXnaaCaaabeqcfasaaiaaikdaaaqcfaOaey4kaSIa aGOmaiaaigdacqaH4oqCcqGHRaWkcaaIXaGaaGynaaGaayjkaiaawM caaaqaaiabeI7aXnaaCaaabeqcfasaaiaaiodaaaqcfa4aaeWaaeaa cqaH4oqCcqGHRaWkcaaIYaaacaGLOaGaayzkaaWaaWbaaeqajuaiba GaaG4maaaaaaaaaa@551B@
μ 4 = 3( 3 θ 4 +36 θ 3 +134 θ 2 +204θ+111 ) θ 4 ( θ+2 ) 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaKqbGeaacaaI0aaajuaGbeaacqGH9aqpdaWcaaqaaiaaioda daqadaqaaiaaiodacqaH4oqCdaahaaqabKqbGeaacaaI0aaaaKqbak abgUcaRiaaiodacaaI2aGaeqiUde3aaWbaaeqajuaibaGaaG4maaaa juaGcqGHRaWkcaaIXaGaaG4maiaaisdacqaH4oqCdaahaaqabKqbGe aacaaIYaaaaKqbakabgUcaRiaaikdacaaIWaGaaGinaiabeI7aXjab gUcaRiaaigdacaaIXaGaaGymaaGaayjkaiaawMcaaaqaaiabeI7aXn aaCaaabeqcfasaaiaaisdaaaqcfa4aaeWaaeaacqaH4oqCcqGHRaWk caaIYaaacaGLOaGaayzkaaWaaWbaaeqajuaibaGaaGinaaaaaaaaaa@5E74@

Shanker [1] has studied various properties of Garima distribution including its shape, moments, skewness, kurtosis, hazard rate function, mean residual life function, stochastic ordering, mean deviations, order statistics, Bonferroni and Lorenz curves, entropy measure and stress-strength reliability, amongst others. The estimation of the parameter of Garima distribution has been discussed by Shanker [1] using both maximum likelihood estimation and the method of moments. In this paper, a Poisson mixture of Garima distribution named, Poisson-Garima distribution (PGD) has been proposed and its various statistical and mathematical properties have been investigated. The estimation of its parameter has been studied using maximum likelihood estimation and method of moments. Since Poisson-Lindley distribution (PLD), a Poisson mixture of Lindley [2] distribution and introduced by Sankaran [7], gives better fit than Poisson distribution, the Poisson-Garima distribution is expected to gives better fit than both Poisson and Poisson – Lindley distribution due to the fact that Garima distribution gives better fit than Lindley distribution. The goodness of fit of the Poisson-Garima distribution has been discussed and it has been compared with that of Poisson and Poisson-Lindley distributions.

Poisson-Garima Distribution

Assuming that the parameter λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4UdW gaaa@3838@  of Poisson distribution follows Garima distribution (1.1), the Poisson mixture of Garima distribution can be obtained as

P( X=x )= 0 e λ λ x x! θ θ+2 ( 1+θ+θλ ) e θλ dλ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiuam aabmaabaGaamiwaiabg2da9iaadIhaaiaawIcacaGLPaaacqGH9aqp daWdXbqaamaalaaabaGaamyzamaaCaaabeqcfasaaiabgkHiTiabeU 7aSbaajuaGcqaH7oaBdaahaaqabeaacaWG4baaaaqaaiaadIhacaGG HaaaaaqcfasaaiaaicdaaeaacqGHEisPaKqbakabgUIiYdGaeyyXIC 9aaSaaaeaacqaH4oqCaeaacqaH4oqCcqGHRaWkcaaIYaaaamaabmaa baGaaGymaiabgUcaRiabeI7aXjabgUcaRiabeI7aXjaaykW7cqaH7o aBaiaawIcacaGLPaaacaWGLbWaaWbaaeqajuaibaGaeyOeI0IaeqiU deNaaGPaVlabeU7aSbaajuaGcaWGKbGaeq4UdWgaaa@674C@ (2.1)
= θ ( θ+2 )x! 0 e ( θ+1 )λ [ ( 1+θ ) λ x +θ λ x+1 ]dλ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyypa0 ZaaSaaaeaacqaH4oqCaeaadaqadaqaaiabeI7aXjabgUcaRiaaikda aiaawIcacaGLPaaacaWG4bGaaiyiaaaadaWdXbqaaiaadwgadaahaa qabKqbGeaacqGHsisljuaGdaqadaqcfasaaiabeI7aXjabgUcaRiaa igdaaiaawIcacaGLPaaacqaH7oaBaaaabaGaaGimaaqaaiabg6HiLc qcfaOaey4kIipadaWadaqaamaabmaabaGaaGymaiabgUcaRiabeI7a XbGaayjkaiaawMcaaiaaykW7cqaH7oaBdaahaaqabKqbGeaacaWG4b aaaKqbakabgUcaRiabeI7aXjaaykW7cqaH7oaBdaahaaqabKqbGeaa caWG4bGaey4kaSIaaGymaaaaaKqbakaawUfacaGLDbaacaWGKbGaeq 4UdWgaaa@6620@
= θ θ+2 θx+( θ 2 +3θ+1 ) ( θ+1 ) x+2 ;x=0,1,2,3,...,θ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyypa0 ZaaSaaaeaacqaH4oqCaeaacqaH4oqCcqGHRaWkcaaIYaaaaiabgwSi xpaalaaabaGaeqiUdeNaamiEaiabgUcaRmaabmaabaGaeqiUde3aaW baaeqajuaibaGaaGOmaaaajuaGcqGHRaWkcaaIZaGaeqiUdeNaey4k aSIaaGymaaGaayjkaiaawMcaaaqaamaabmaabaGaeqiUdeNaey4kaS IaaGymaaGaayjkaiaawMcaamaaCaaabeqcfasaaiaadIhacqGHRaWk caaIYaaaaaaajuaGcaaMc8UaaGPaVlaaykW7caGG7aGaamiEaiabg2 da9iaaicdacaGGSaGaaGymaiaacYcacaaIYaGaaiilaiaaiodacaGG SaGaaiOlaiaac6cacaGGUaGaaiilaiabeI7aXjabg6da+iaaicdaaa a@6802@ (2.2)

This probability mass function (p.m.f.) has been named as “Poisson-Garima distribution (PGD)”.
It should be noted that Sankaran [7] obtained Poisson-Lindley distribution (PLD) having probability mass function (p.m.f)
P( X=x )= θ 2 ( x+θ+2 ) ( θ+1 ) x+3 ;x=0,1,2,...,θ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiuam aabmaabaGaamiwaiabg2da9iaadIhaaiaawIcacaGLPaaacqGH9aqp daWcaaqaaiabeI7aXnaaCaaabeqcfasaaiaaikdaaaqcfa4aaeWaae aacaWG4bGaey4kaSIaeqiUdeNaey4kaSIaaGOmaaGaayjkaiaawMca aaqaamaabmaabaGaeqiUdeNaey4kaSIaaGymaaGaayjkaiaawMcaam aaCaaabeqcfasaaiaadIhacqGHRaWkcaaIZaaaaaaajuaGcaaMc8Ua aGPaVlaacUdacaWG4bGaeyypa0JaaGimaiaacYcacaaIXaGaaiilai aaikdacaGGSaGaaiOlaiaac6cacaGGUaGaaiilaiabeI7aXjabg6da +iaaicdaaaa@5F91@   (2.3)

by compounding Poisson distribution with Lindley distribution, introduced by Lindley [2] having probability density function (p.d.f)

f( 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 aabmaabaGaamiEaiaacYcacqaH4oqCaiaawIcacaGLPaaacqGH9aqp daWcaaqaaiabeI7aXnaaCaaabeqcfasaaiaaikdaaaaajuaGbaGaeq iUdeNaey4kaSIaaGymaaaadaqadaqaaiaaigdacqGHRaWkcaWG4baa caGLOaGaayzkaaGaaGPaVlaadwgadaahaaqabKqbGeaacqaH4oqCca aMb8UaamiEaaaajuaGcaaMc8UaaGPaVlaacUdacaWG4bGaeyOpa4Ja aGimaiaacYcacaaMc8UaeqiUdeNaeyOpa4JaaGimaaaa@5C0E@  (2.4)

The graphs of the pmf of Poisson-Garima distribution (PGD) and Poisson-Lindley distribution (PLD) for varying values of the parameter are shown in the figure 1

Figure 1: Graphs of The Pmf Of PGD and PLD For Varying Values of the Parameter.

Moments and Related Measures of PGD

The r th factorial moment about origin of Poisson-Garima distribution (2.2) can be obtained as
μ ( r ) =E[ E( X ( r ) |λ ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaKqbGeaajuaGdaqadaqcfasaaiaadkhaaiaawIcacaGLPaaa aKqbagqaamaaCaaabeqaaiadacUHYaIOaaGaeyypa0Jaamyramaadm aabaGaamyramaabmaabaGaamiwamaaCaaabeqcfasaaKqbaoaabmaa juaibaGaamOCaaGaayjkaiaawMcaaaaajuaGcaGG8bGaeq4UdWgaca GLOaGaayzkaaaacaGLBbGaayzxaaaaaa@4D1D@  ,           where     X ( r ) =X( X1 )( X2 )...( Xr+1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiwam aaCaaabeqcfasaaKqbaoaabmaajuaibaGaamOCaaGaayjkaiaawMca aaaajuaGcqGH9aqpcaWGybWaaeWaaeaacaWGybGaeyOeI0IaaGymaa GaayjkaiaawMcaamaabmaabaGaamiwaiabgkHiTiaaikdaaiaawIca caGLPaaacaGGUaGaaiOlaiaac6cadaqadaqaaiaadIfacqGHsislca WGYbGaey4kaSIaaGymaaGaayjkaiaawMcaaaaa@4D78@  
= θ θ+2 0 [ x=0 x ( r ) e λ λ x x! ] ( 1+θ+θλ ) e θλ dλ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyypa0 ZaaSaaaeaacqaH4oqCaeaacqaH4oqCcqGHRaWkcaaIYaaaamaapeha baWaamWaaeaadaaeWbqaaiaadIhadaahaaqabKqbGeaajuaGdaqada qcfasaaiaadkhaaiaawIcacaGLPaaaaaqcfa4aaSaaaeaacaWGLbWa aWbaaeqajuaibaGaeyOeI0Iaeq4UdWgaaKqbakabeU7aSnaaCaaabe qcfasaaiaadIhaaaaajuaGbaGaamiEaiaacgcaaaaajuaibaGaamiE aiabg2da9iaaicdaaeaacqGHEisPaKqbakabggHiLdaacaGLBbGaay zxaaaajuaibaGaaGimaaqaaiabg6HiLcqcfaOaey4kIipadaqadaqa aiaaigdacqGHRaWkcqaH4oqCcqGHRaWkcqaH4oqCcaaMc8Uaeq4UdW gacaGLOaGaayzkaaGaamyzamaaCaaabeqcfasaaiabgkHiTiabeI7a XjaaykW7cqaH7oaBaaqcfaOaamizaiabeU7aSbaa@6E9D@
= θ θ+2 0 [ λ r x=r e λ λ xr ( xr )! ] ( 1+θ+θλ ) e θλ dλ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyypa0 ZaaSaaaeaacqaH4oqCaeaacqaH4oqCcqGHRaWkcaaIYaaaamaapeha baWaamWaaeaacqaH7oaBdaahaaqabKqbGeaacaWGYbaaaKqbaoaaqa habaWaaSaaaeaacaWGLbWaaWbaaeqajuaibaGaeyOeI0Iaeq4UdWga aKqbakabeU7aSnaaCaaabeqcfasaaiaadIhacqGHsislcaWGYbaaaa qcfayaamaabmaabaGaamiEaiabgkHiTiaadkhaaiaawIcacaGLPaaa caGGHaaaaaqcfasaaiaadIhacqGH9aqpcaaMc8UaamOCaaqaaiabg6 HiLcqcfaOaeyyeIuoaaiaawUfacaGLDbaaaKqbGeaacaaIWaaabaGa eyOhIukajuaGcqGHRiI8amaabmaabaGaaGymaiabgUcaRiabeI7aXj abgUcaRiabeI7aXjaaykW7cqaH7oaBaiaawIcacaGLPaaacaWGLbWa aWbaaeqajuaibaGaeyOeI0IaeqiUdeNaaGPaVlabeU7aSbaajuaGca WGKbGaeq4UdWgaaa@7428@
Taking x+r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEai abgUcaRiaadkhaaaa@395A@ in place of x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEaa aa@3781@ within bracket, we get
μ ( r ) = MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaKqbGeaajuaGdaqadaqcfasaaiaadkhaaiaawIcacaGLPaaa aKqbagqaamaaCaaabeqaaiadacUHYaIOaaGaeyypa0daaa@4063@ θ θ+2 0 λ r [ x=0 e λ λ x x! ] ( 1+θ+θλ ) e θλ dλ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacqaH4oqCaeaacqaH4oqCcqGHRaWkcaaIYaaaamaapehabaGaeq4U dW2aaWbaaeqajuaibaGaamOCaaaajuaGdaWadaqaamaaqahabaWaaS aaaeaacaWGLbWaaWbaaeqajuaibaGaeyOeI0Iaeq4UdWgaaKqbakab eU7aSnaaCaaabeqaaiaadIhaaaaabaGaamiEaiaacgcaaaaajuaiba GaamiEaiabg2da9iaaicdaaeaacqGHEisPaKqbakabggHiLdaacaGL BbGaayzxaaaajuaibaGaaGimaaqaaiabg6HiLcqcfaOaey4kIipada qadaqaaiaaigdacqGHRaWkcqaH4oqCcqGHRaWkcqaH4oqCcaaMc8Ua eq4UdWgacaGLOaGaayzkaaGaamyzamaaCaaabeqcfasaaiabgkHiTi abeI7aXjaaykW7cqaH7oaBaaqcfaOaamizaiabeU7aSbaa@6B4D@

The expression within the bracket is clearly unity and hence we have
μ ( r ) = MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaKqbGeaajuaGdaqadaqcfasaaiaadkhaaiaawIcacaGLPaaa aKqbagqaamaaCaaabeqaaiadacUHYaIOaaGaeyypa0daaa@4063@   θ θ+2 0 λ r ( 1+θ+θλ ) e θλ dλ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacqaH4oqCaeaacqaH4oqCcqGHRaWkcaaIYaaaamaapehabaGaeq4U dW2aaWbaaeqajuaibaGaamOCaaaaaeaacaaIWaaabaGaeyOhIukaju aGcqGHRiI8amaabmaabaGaaGymaiabgUcaRiabeI7aXjabgUcaRiab eI7aXjaaykW7cqaH7oaBaiaawIcacaGLPaaacaWGLbWaaWbaaeqaju aibaGaeyOeI0IaeqiUdeNaaGPaVlabeU7aSbaajuaGcaWGKbGaeq4U dWgaaa@589D@
Using gamma integral and some algebraic simplification, we get finally a general expression for the r th factorial moment of PGD (2.2) as
μ ( r ) = r!( θ+r+2 ) θ r ( θ+2 ) ;r=1,2,3,.... MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeY7aTn aaBaaajuaibaqcfa4aaeWaaKqbGeaacaWGYbaacaGLOaGaayzkaaaa juaGbeaadaahaaqabeaacWaGGBOmGikaaiabg2da9maalaaabaGaam OCaiaacgcadaqadaqaaiabeI7aXjabgUcaRiaadkhacqGHRaWkcaaI YaaacaGLOaGaayzkaaaabaGaeqiUde3aaWbaaeqajuaibaGaamOCaa aajuaGdaqadaqaaiabeI7aXjabgUcaRiaaikdaaiaawIcacaGLPaaa aaGaaGPaVlaaykW7caGG7aGaamOCaiabg2da9iaaigdacaGGSaGaaG OmaiaacYcacaaIZaGaaiilaiaac6cacaGGUaGaaiOlaiaac6caaaa@5E00@   (3.1)

Substituting r=1,2,3,and4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOCai abg2da9iaaigdacaGGSaGaaGOmaiaacYcacaaIZaGaaiilaiaaykW7 caqGHbGaaeOBaiaabsgacaaMc8UaaGPaVlaaisdaaaa@44E0@  in (3.1), the first four factorial moments can be obtained and using the relationship between factorial moments and moments about origin, the first four moments about origin of the PGD (2.2) are obtained as
μ 1 = θ+3 θ( θ+2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiVd0 MbauaadaWgaaqcfasaaiaaigdaaKqbagqaaiabg2da9maalaaabaGa eqiUdeNaey4kaSIaaG4maaqaaiabeI7aXnaabmaabaGaeqiUdeNaey 4kaSIaaGOmaaGaayjkaiaawMcaaaaaaaa@44DC@  
μ 2 = θ 2 +5θ+8 θ 2 ( θ+2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiVd0 MbauaadaWgaaqcfasaaiaaikdaaKqbagqaaiabg2da9maalaaabaGa eqiUde3aaWbaaeqajuaibaGaaGOmaaaajuaGcqGHRaWkcaaI1aGaeq iUdeNaey4kaSIaaGioaaqaaiabeI7aXnaaCaaabeqcfasaaiaaikda aaqcfa4aaeWaaeaacqaH4oqCcqGHRaWkcaaIYaaacaGLOaGaayzkaa aaaaaa@4B6D@  
μ 3 = θ 3 +9 θ 2 +30θ+30 θ 3 ( θ+2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiVd0 MbauaadaWgaaqcfasaaiaaiodaaKqbagqaaiabg2da9maalaaabaGa eqiUde3aaWbaaeqajuaibaGaaG4maaaajuaGcqGHRaWkcaaI5aGaeq iUde3aaWbaaeqajuaibaGaaGOmaaaajuaGcqGHRaWkcaaIZaGaaGim aiabeI7aXjabgUcaRiaaiodacaaIWaaabaGaeqiUde3aaWbaaeqaju aibaGaaG4maaaajuaGdaqadaqaaiabeI7aXjabgUcaRiaaikdaaiaa wIcacaGLPaaaaaaaaa@51D2@  
μ 4 = θ 4 +17 θ 3 +92 θ 2 +204θ+144 θ 4 ( θ+2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiVd0 MbauaadaWgaaqcfasaaiaaisdaaKqbagqaaiabg2da9maalaaabaGa eqiUde3aaWbaaeqajuaibaGaaGinaaaajuaGcqGHRaWkcaaIXaGaaG 4naiabeI7aXnaaCaaabeqcfasaaiaaiodaaaqcfaOaey4kaSIaaGyo aiaaikdacqaH4oqCdaahaaqabKqbGeaacaaIYaaaaKqbakabgUcaRi aaikdacaaIWaGaaGinaiabeI7aXjabgUcaRiaaigdacaaI0aGaaGin aaqaaiabeI7aXnaaCaaabeqcfasaaiaaisdaaaqcfa4aaeWaaeaacq aH4oqCcqGHRaWkcaaIYaaacaGLOaGaayzkaaaaaaaa@59BD@
Using the relationship between moments about mean and the moments about origin, the moments about mean of the PGD (2.2) are obtained as
μ 2 = σ 2 = θ 3 +6 θ 2 +12θ+7 θ 2 ( θ+2 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaKqbGeaacaaIYaaajuaGbeaacqGH9aqpcqaHdpWCdaahaaqa bKqbGeaacaaIYaaaaKqbakabg2da9maalaaabaGaeqiUde3aaWbaae qajuaibaGaaG4maaaajuaGcqGHRaWkcaaI2aGaeqiUde3aaWbaaeqa juaibaGaaGOmaaaajuaGcqGHRaWkcaaIXaGaaGOmaiabeI7aXjabgU caRiaaiEdaaeaacqaH4oqCdaahaaqabKqbGeaacaaIYaaaaKqbaoaa bmaabaGaeqiUdeNaey4kaSIaaGOmaaGaayjkaiaawMcaamaaCaaabe qcfasaaiaaikdaaaaaaaaa@567A@  
μ 3 = θ 5 +10 θ 4 +42 θ 3 +87 θ 2 +84θ+30 θ 3 ( θ+2 ) 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaKqbGeaacaaIZaaajuaGbeaacqGH9aqpdaWcaaqaaiabeI7a XnaaCaaabeqcfasaaiaaiwdaaaqcfaOaey4kaSIaaGymaiaaicdacq aH4oqCdaahaaqabKqbGeaacaaI0aaaaKqbakabgUcaRiaaisdacaaI YaGaeqiUde3aaWbaaeqajuaibaGaaG4maaaajuaGcqGHRaWkcaaI4a GaaG4naiabeI7aXnaaCaaabeqcfasaaiaaikdaaaqcfaOaey4kaSIa aGioaiaaisdacqaH4oqCcqGHRaWkcaaIZaGaaGimaaqaaiabeI7aXn aaCaaabeqcfasaaiaaiodaaaqcfa4aaeWaaeaacqaH4oqCcqGHRaWk caaIYaaacaGLOaGaayzkaaWaaWbaaeqajuaibaGaaG4maaaaaaaaaa@5EF4@  
μ 4 = ( θ 7 +19 θ 6 +148 θ 5 +607 θ 4 +1402 θ 3 +1816 θ 2 +1224θ+333 ) θ 4 ( θ+2 ) 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaKqbGeaacaaI0aaajuaGbeaacqGH9aqpdaWcaaqaamaabmaa baGaeqiUde3aaWbaaeqajuaibaGaaG4naaaajuaGcqGHRaWkcaaIXa GaaGyoaiabeI7aXnaaCaaabeqcfasaaiaaiAdaaaqcfaOaey4kaSIa aGymaiaaisdacaaI4aGaeqiUde3aaWbaaeqajuaibaGaaGynaaaaju aGcqGHRaWkcaaI2aGaaGimaiaaiEdacqaH4oqCdaahaaqabKqbGeaa caaI0aaaaKqbakabgUcaRiaaigdacaaI0aGaaGimaiaaikdacqaH4o qCdaahaaqabKqbGeaacaaIZaaaaKqbakabgUcaRiaaigdacaaI4aGa aGymaiaaiAdacqaH4oqCdaahaaqabKqbGeaacaaIYaaaaKqbakabgU caRiaaigdacaaIYaGaaGOmaiaaisdacqaH4oqCcqGHRaWkcaaIZaGa aG4maiaaiodaaiaawIcacaGLPaaaaeaacqaH4oqCdaahaaqabKqbGe aacaaI0aaaaKqbaoaabmaabaGaeqiUdeNaey4kaSIaaGOmaaGaayjk aiaawMcaamaaCaaabeqcfasaaiaaisdaaaaaaaaa@7287@
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 aadaGcaaqaaiabek7aInaaBaaajuaibaGaaGymaaqcfayabaaabeaa aiaawIcacaGLPaaaaaa@3B56@ , coefficient of Kurtosis ( β 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacqaHYoGydaWgaaqcfasaaiaaikdaaKqbagqaaaGaayjkaiaawMca aaaa@3B47@  and index of dispersion ( γ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacqaHZoWzaiaawIcacaGLPaaaaaa@39B4@ of the PGD (2.2) are thus obtained as
C.V= σ μ 1 = θ 3 +6 θ 2 +12θ+7 θ+3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4qai aac6cacaWGwbGaeyypa0ZaaSaaaeaacqaHdpWCaeaacuaH8oqBgaqb amaaBaaajuaibaGaaGymaaqcfayabaaaaiabg2da9maalaaabaWaaO aaaeaacqaH4oqCdaahaaqabKqbGeaacaaIZaaaaKqbakabgUcaRiaa iAdacqaH4oqCdaahaaqabKqbGeaacaaIYaaaaKqbakabgUcaRiaaig dacaaIYaGaeqiUdeNaey4kaSIaaG4naaqabaaabaGaeqiUdeNaey4k aSIaaG4maaaaaaa@517C@  
β 1 = μ 3 μ 2 3/2 = θ 5 +10 θ 4 +42 θ 3 +87 θ 2 +84θ+30 ( θ 3 +6 θ 2 +12θ+7 ) 3/2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaOaaae aacqaHYoGydaWgaaqcfasaaiaaigdaaKqbagqaaaqabaGaeyypa0Za aSaaaeaacqaH8oqBdaWgaaqcfasaaiaaiodaaKqbagqaaaqaaiabeY 7aTnaaBaaajuaibaGaaGOmaaqcfayabaWaaWbaaeqajuaibaqcfa4a aSGbaKqbGeaacaaIZaaabaGaaGOmaaaaaaaaaKqbakabg2da9maala aabaGaeqiUde3aaWbaaeqajuaibaGaaGynaaaajuaGcqGHRaWkcaaI XaGaaGimaiabeI7aXnaaCaaabeqcfasaaiaaisdaaaqcfaOaey4kaS IaaGinaiaaikdacqaH4oqCdaahaaqabKqbGeaacaaIZaaaaKqbakab gUcaRiaaiIdacaaI3aGaeqiUde3aaWbaaeqajuaibaGaaGOmaaaaju aGcqGHRaWkcaaI4aGaaGinaiabeI7aXjabgUcaRiaaiodacaaIWaaa baWaaeWaaeaacqaH4oqCdaahaaqabKqbGeaacaaIZaaaaKqbakabgU caRiaaiAdacqaH4oqCdaahaaqabKqbGeaacaaIYaaaaKqbakabgUca RiaaigdacaaIYaGaeqiUdeNaey4kaSIaaG4naaGaayjkaiaawMcaam aaCaaabeqcfasaaKqbaoaalyaajuaibaGaaG4maaqaaiaaikdaaaaa aaaaaaa@72A9@  
β 2 = μ 4 μ 2 2 = ( θ 7 +19 θ 6 +148 θ 5 +607 θ 4 +1402 θ 3 +1816 θ 2 +1224θ+333 ) ( θ 3 +6 θ 2 +12θ+7 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOSdi 2aaSbaaKqbGeaacaaIYaaajuaGbeaacqGH9aqpdaWcaaqaaiabeY7a TnaaBaaajuaibaGaaGinaaqcfayabaaabaGaeqiVd02aaSbaaKqbGe aacaaIYaaajuaGbeaadaahaaqabKqbGeaacaaIYaaaaaaajuaGcqGH 9aqpdaWcaaqaamaabmaabaGaeqiUde3aaWbaaeqajuaibaGaaG4naa aajuaGcqGHRaWkcaaIXaGaaGyoaiabeI7aXnaaCaaabeqcfasaaiaa iAdaaaqcfaOaey4kaSIaaGymaiaaisdacaaI4aGaeqiUde3aaWbaae qajuaibaGaaGynaaaajuaGcqGHRaWkcaaI2aGaaGimaiaaiEdacqaH 4oqCdaahaaqabKqbGeaacaaI0aaaaKqbakabgUcaRiaaigdacaaI0a GaaGimaiaaikdacqaH4oqCdaahaaqabKqbGeaacaaIZaaaaKqbakab gUcaRiaaigdacaaI4aGaaGymaiaaiAdacqaH4oqCdaahaaqabKqbGe aacaaIYaaaaKqbakabgUcaRiaaigdacaaIYaGaaGOmaiaaisdacqaH 4oqCcqGHRaWkcaaIZaGaaG4maiaaiodaaiaawIcacaGLPaaaaeaada qadaqaaiabeI7aXnaaCaaabeqcfasaaiaaiodaaaqcfaOaey4kaSIa aGOnaiabeI7aXnaaCaaabeqcfasaaiaaikdaaaqcfaOaey4kaSIaaG ymaiaaikdacqaH4oqCcqGHRaWkcaaI3aaacaGLOaGaayzkaaWaaWba aeqajuaibaGaaGOmaaaaaaaaaa@830D@  
γ= σ 2 μ 1 = θ 3 +6 θ 2 +12θ+7 θ( θ+2 )( θ+3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4SdC Maeyypa0ZaaSaaaeaacqaHdpWCdaahaaqabKqbGeaacaaIYaaaaaqc fayaaiabeY7aTnaaBaaajuaibaGaaGymaaqcfayabaWaaWbaaeqaba Gamai4gkdiIcaaaaGaeyypa0ZaaSaaaeaacqaH4oqCdaahaaqabKqb GeaacaaIZaaaaKqbakabgUcaRiaaiAdacqaH4oqCdaahaaqabKqbGe aacaaIYaaaaKqbakabgUcaRiaaigdacaaIYaGaeqiUdeNaey4kaSIa aG4naaqaaiabeI7aXnaabmaabaGaeqiUdeNaey4kaSIaaGOmaaGaay jkaiaawMcaamaabmaabaGaeqiUdeNaey4kaSIaaG4maaGaayjkaiaa wMcaaaaaaaa@5D72@
To study the nature and behavior of μ 1 , μ 2 ,C.V, β 1 , β 2 andγ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaKqbGeaacaaIXaaajuaGbeaadaahaaqabeaacWaGGBOmGika aiaacYcacaaMc8UaaGPaVlabeY7aTnaaBaaajuaibaGaaGOmaaqcfa yabaGaaiilaiaaykW7caaMc8Uaae4qaiaab6cacaqGwbGaaiilaiaa ykW7caaMc8+aaOaaaeaacqaHYoGydaWgaaqcfasaaiaaigdaaKqbag qaaaqabaGaaiilaiaaykW7caaMc8UaeqOSdi2aaSbaaKqbGeaacaaI YaaajuaGbeaacaaMc8UaaGPaVlaabggacaqGUbGaaeizaiaaykW7ca aMc8Uaeq4SdCgaaa@62A5@  of PGD and PLD, values of these characteristics for varying values of parameter θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde haaa@383A@  have been computed and presented in table 1

 

Values of θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde haaa@383A@  for Poisson-Garima Distribution

1

2

3

4

5

6

μ 1 ' MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaKqbGeaacaaIXaaajuaGbeaacaGGNaaaaa@3A7D@

1.333333

0.625

0.4

0.291667

0.228571

0.1875

μ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaKqbGeaacaaIYaaajuaGbeaaaaa@39D3@

2.888889

0.984375

0.551111

0.373264

0.279184

0.221788

CV

1.274755

1.587451

1.855921

2.094697

2.311655

2.511701

β 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaOaaae aacqaHYoGydaWgaaqcfasaaiaaigdaaKqbagqaaaqabaaaaa@39CD@

1.915904

2.147798

2.355147

2.54717

2.727407

2.897852

β 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOSdi 2aaSbaaKqbGeaacaaIYaaajuaGbeaaaaa@39BE@

8.210059

9.335601

10.36498

11.36106

12.34549

13.32641

γ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4SdC gaaa@382B@

2.166667

1.575

1.377778

1.279762

1.221429

1.18287

 

Values of θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde haaa@383A@ for Poisson-Lindley Distribution

 

1

2

3

4

5

6

μ 1 ' MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaKqbGeaacaaIXaaajuaGbeaacaGGNaaaaa@3A7D@

1.5

0.666667

0.416667

0.3

0.233333

0.190476

μ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaKqbGeaacaaIYaaajuaGbeaaaaa@39D3@

3.25

1.055556

0.576389

0.385

0.285556

0.225624

CV

1.20185

1.541104

1.822087

2.068279

2.290174

2.493742

β 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaOaaae aacqaHYoGydaWgaaqcfasaaiaaigdaaKqbagqaaaqabaaaaa@39CD@

1.792108

2.083265

2.314307

2.517935

2.704839

2.87957

β 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOSdi 2aaSbaaKqbGeaacaaIYaaajuaGbeaaaaa@39BE@

7.532544

8.941828

10.10611

11.17187

12.19654

13.203

γ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4SdC gaaa@382B@

2.166667

1.583333

1.383333

1.283333

1.22381

1.184524

Table 1: Values of μ 1 , μ 2 ,C.V, β 1 , β 2 andγ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaKqbGeaacaaIXaaajuaGbeaadaahaaqabeaacWaGGBOmGika aiaacYcacaaMc8UaaGPaVlabeY7aTnaaBaaajuaibaGaaGOmaaqcfa yabaGaaiilaiaaykW7caaMc8Uaae4qaiaab6cacaqGwbGaaiilaiaa ykW7caaMc8+aaOaaaeaacqaHYoGydaWgaaqcfasaaiaaigdaaKqbag qaaaqabaGaaiilaiaaykW7caaMc8UaeqOSdi2aaSbaaKqbGeaacaaI YaaajuaGbeaacaaMc8UaaGPaVlaabggacaqGUbGaaeizaiaaykW7ca aMc8Uaeq4SdCgaaa@62A5@ of PGD and PLD for varying values of the parameter θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde haaa@383A@

No. of errors per group

Observed Frequency

Expected Frequency

PD

PLD

PGD

0
1
2
3
4

35
11
8
4
2

27.4
21.5
8.4 2.2 0.5 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiGaaqaabe qaaiaaiIdacaGGUaGaaGinaaqaaiaaikdacaGGUaGaaGOmaaqaaiaa icdacaGGUaGaaGynaaaacaGL9baaaaa@3DA2@

33.0
15.3
6.8 2.9 2.0 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiGaaqaabe qaaiaaiAdacaGGUaGaaGioaaqaaiaaikdacaGGUaGaaGyoaaqaaiaa ikdacaGGUaGaaGimaaaacaGL9baaaaa@3DA8@

33.3
15.1
6.6
2.9 2.1 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiGaaqaabe qaaiaaikdacaGGUaGaaGyoaaqaaiaaikdacaGGUaGaaGymaaaacaGL 9baaaaa@3B74@

Total

60

60.0

60.0

60.0

ML estimate

 

θ ^ =1.628413 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpcaaIXaGaaiOlaiaaiAdacaaIYaGaaGioaiaaisda caaIXaGaaG4maaaa@3F31@

θ ^ =1.7434 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpcaaIXaGaaiOlaiaaiEdacaaI0aGaaG4maiaaisda aaa@3DB7@

θ ^ =1.628413 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpcaaIXaGaaiOlaiaaiAdacaaIYaGaaGioaiaaisda caaIXaGaaG4maaaa@3F31@

χ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4Xdm 2aaWbaaeqajuaibaGaaGOmaaaaaaa@3947@

 

7.98

2.20

1.71

d.f.

 

1

1

2

p-value

 

0.0047

0.1380

0.4253

Table 2: Distribution of mistakes in copying groups of random digits.

No. of errors per Group

Observed Frequency

Expected Frequency

PD

PLD

PGD

0
1
2
3
4
5

33
12
6
3
1
1

26.4
19.8
7.4 1.8 0.3 0.3 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaK qzGeabaeqakeaajugibiaaiEdacaGGUaGaaGinaaGcbaqcLbsacaaI XaGaaiOlaiaaiIdaaOqaaKqzGeGaaGimaiaac6cacaaIZaaakeaaju gibiaaicdacaGGUaGaaG4maaaakiaaw2haaaaa@4359@

31.5
14.2
6.1 2.5 1.0 0.7 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaiAdacaGGUaGaaGymaaqaaiaaikdacaGGUaGaaGynaaqa aiaaigdacaGGUaGaaGimaaqaaiaaicdacaGGUaGaaG4naaaacaGL9b aaaaa@4058@

31.7
14.0
6.0 2.5 1.0 0.8 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaiAdacaGGUaGaaGimaaqaaiaaikdacaGGUaGaaGynaaqa aiaaigdacaGGUaGaaGimaaqaaiaaicdacaGGUaGaaGioaaaacaGL9b aaaaa@4058@

Total

56

56.0

56.0

56.0

ML estimate

 

θ ^ =0.7500 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpcaaIWaGaaiOlaiaaiEdacaaI1aGaaGimaiaaicda aaa@3DB0@

θ ^ =1.8081 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpcaaIXaGaaiOlaiaaiIdacaaIWaGaaGioaiaaigda aaa@3DB6@

θ ^ =1.695033 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpcaaIXaGaaiOlaiaaiAdacaaI5aGaaGynaiaaicda caaIZaGaaG4maaaa@3F33@

χ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4Xdm 2aaWbaaeqajuaibaGaaGOmaaaaaaa@3947@

 

4.87

0.53

0.38

d.f.

 

1

1

1

p-value

 

0.0273

0.4660

0.5376

Table 3: Distribution of Pyrausta nublilalis.

The graph of the coefficient of variation (C.V), coefficient of skewness ( β 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aadaGcaaqaaiabek7aInaaBaaajuaibaGaaGymaaqcfayabaaabeaa aiaawIcacaGLPaaaaaa@3B56@ , coefficient of kurtosis ( β 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacqaHYoGydaWgaaqcfasaaiaaikdaaKqbagqaaaGaayjkaiaawMca aaaa@3B47@ , and index of dispersion ( γ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacqaHZoWzaiaawIcacaGLPaaaaaa@39B4@  of PGD and PLD are presented in figure 2.

Figure 2: Graphs of (C.V), ( β 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqbaoaakaaakeaajugibiabek7aITWaaSbaaeaajugWaiaaigda aSqabaaabeaaaOGaayjkaiaawMcaaaaa@3D19@ , ( β 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeGaeqOSdiwcfa4aaSbaaSqaaKqzadGaaGOmaaWcbeaaaOGa ayjkaiaawMcaaaaa@3D00@ , and ( γ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeGaeq4SdCgakiaawIcacaGLPaaaaaa@3A57@ PGD and PLD for Varying Values of the Parameter θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH4o qCaaa@383B@ .

Statistical Properties of PGD

The PGD (1.3) is always over dispersed ( σ 2 >μ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacqaHdpWCdaahaaqabKqbGeaacaaIYaaaaKqbakabg6da+iabeY7a TbGaayjkaiaawMcaaaaa@3E28@ .
We have
σ 2 = θ 3 +6 θ 2 +12θ+7 θ 2 ( θ+2 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4Wdm 3aaWbaaeqajuaibaGaaGOmaaaajuaGcqGH9aqpdaWcaaqaaiabeI7a XnaaCaaabeqcfasaaiaaiodaaaqcfaOaey4kaSIaaGOnaiabeI7aXn aaCaaabeqcfasaaiaaikdaaaqcfaOaey4kaSIaaGymaiaaikdacqaH 4oqCcqGHRaWkcaaI3aaabaGaeqiUde3aaWbaaeqajuaibaGaaGOmaa aajuaGdaqadaqaaiabeI7aXjabgUcaRiaaikdaaiaawIcacaGLPaaa daahaaqabKqbGeaacaaIYaaaaaaaaaa@5225@
= θ+3 θ( θ+2 ) [ θ 3 +6 θ 2 +12θ+7 θ( θ+2 )( θ+3 ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyypa0 ZaaSaaaeaacqaH4oqCcqGHRaWkcaaIZaaabaGaeqiUde3aaeWaaeaa cqaH4oqCcqGHRaWkcaaIYaaacaGLOaGaayzkaaaaamaadmaabaWaaS aaaeaacqaH4oqCdaahaaqabKqbGeaacaaIZaaaaKqbakabgUcaRiaa iAdacqaH4oqCdaahaaqabKqbGeaacaaIYaaaaKqbakabgUcaRiaaig dacaaIYaGaeqiUdeNaey4kaSIaaG4naaqaaiabeI7aXnaabmaabaGa eqiUdeNaey4kaSIaaGOmaaGaayjkaiaawMcaamaabmaabaGaeqiUde Naey4kaSIaaG4maaGaayjkaiaawMcaaaaaaiaawUfacaGLDbaaaaa@5CEA@
= θ+3 θ( θ+2 ) [ 1+ θ 2 +6θ+7 θ( θ+2 )( θ+3 ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyypa0 ZaaSaaaeaacqaH4oqCcqGHRaWkcaaIZaaabaGaeqiUde3aaeWaaeaa cqaH4oqCcqGHRaWkcaaIYaaacaGLOaGaayzkaaaaamaadmaabaGaaG ymaiabgUcaRmaalaaabaGaeqiUde3aaWbaaeqajuaibaGaaGOmaaaa juaGcqGHRaWkcaaI2aGaeqiUdeNaey4kaSIaaG4naaqaaiabeI7aXn aabmaabaGaeqiUdeNaey4kaSIaaGOmaaGaayjkaiaawMcaamaabmaa baGaeqiUdeNaey4kaSIaaG4maaGaayjkaiaawMcaaaaaaiaawUfaca GLDbaaaaa@58DD@
=μ[ 1+ θ 2 +6θ+7 θ( θ+2 )( θ+3 ) ]>μ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyypa0 JaeqiVd02aamWaaeaacaaIXaGaey4kaSYaaSaaaeaacqaH4oqCdaah aaqabKqbGeaacaaIYaaaaKqbakabgUcaRiaaiAdacqaH4oqCcqGHRa WkcaaI3aaabaGaeqiUde3aaeWaaeaacqaH4oqCcqGHRaWkcaaIYaaa caGLOaGaayzkaaWaaeWaaeaacqaH4oqCcqGHRaWkcaaIZaaacaGLOa GaayzkaaaaaaGaay5waiaaw2faaiabg6da+iabeY7aTbaa@5359@
This shows that PGD (2.2) is always over dispersed.

Unimodality and increasing hazard rate

Since
P( x+1;θ ) P( x;θ ) = θ( x+1 )+( θ 2 +3θ+1 ) ( θ+1 )[ θx+( θ 2 +3θ+1 ) ] = 1 θ+1 [ 1+ θ θx+( θ 2 +3θ+1 ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacaWGqbWaaeWaaeaacaWG4bGaey4kaSIaaGymaiaacUdacqaH4oqC aiaawIcacaGLPaaaaeaacaWGqbWaaeWaaeaacaWG4bGaai4oaiabeI 7aXbGaayjkaiaawMcaaaaacqGH9aqpdaWcaaqaaiabeI7aXnaabmaa baGaamiEaiabgUcaRiaaigdaaiaawIcacaGLPaaacqGHRaWkdaqada qaaiabeI7aXnaaCaaabeqcfasaaiaaikdaaaqcfaOaey4kaSIaaG4m aiabeI7aXjabgUcaRiaaigdaaiaawIcacaGLPaaaaeaadaqadaqaai abeI7aXjabgUcaRiaaigdaaiaawIcacaGLPaaadaWadaqaaiabeI7a XjaadIhacqGHRaWkdaqadaqaaiabeI7aXnaaCaaabeqcfasaaiaaik daaaqcfaOaey4kaSIaaG4maiabeI7aXjabgUcaRiaaigdaaiaawIca caGLPaaaaiaawUfacaGLDbaaaaGaeyypa0ZaaSaaaeaacaaIXaaaba GaeqiUdeNaey4kaSIaaGymaaaadaWadaqaaiaaigdacqGHRaWkdaWc aaqaaiabeI7aXbqaaiabeI7aXjaadIhacqGHRaWkdaqadaqaaiabeI 7aXnaaCaaabeqcfasaaiaaikdaaaqcfaOaey4kaSIaaG4maiabeI7a XjabgUcaRiaaigdaaiaawIcacaGLPaaaaaaacaGLBbGaayzxaaaaaa@8174@  is decreasing function in x, P( x;θ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiuam aabmaabaGaamiEaiaacUdacqaH4oqCaiaawIcacaGLPaaaaaa@3C54@ is log-concave. Therefore, the PGD has an increasing hazard rate and thus unimodal. Detailed discussion relationship between log-concavity, unimodality and increasing hazard rate of discrete distribution can be seen in Grandell [8].

Generating Functions

Probability generating function: The probability generating function of the PGD (2.2) can be obtained as
P X ( t )=E( t X )= θ ( θ+2 ) ( θ+1 ) 2 [ θ x=0 x ( t θ+1 ) x +( θ 2 +3θ+1 ) x=0 ( t θ+1 ) x ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiuam aaBaaajuaibaGaamiwaaqcfayabaWaaeWaaeaacaWG0baacaGLOaGa ayzkaaGaeyypa0JaamyramaabmaabaGaamiDamaaCaaabeqcfasaai aadIfaaaaajuaGcaGLOaGaayzkaaGaeyypa0ZaaSaaaeaacqaH4oqC aeaadaqadaqaaiabeI7aXjabgUcaRiaaikdaaiaawIcacaGLPaaada qadaqaaiabeI7aXjabgUcaRiaaigdaaiaawIcacaGLPaaadaahaaqa bKqbGeaacaaIYaaaaaaajuaGdaWadaqaaiabeI7aXnaaqahabaGaam iEamaabmaabaWaaSaaaeaacaWG0baabaGaeqiUdeNaey4kaSIaaGym aaaaaiaawIcacaGLPaaadaahaaqabKqbGeaacaWG4baaaKqbakabgU caRmaabmaabaGaeqiUde3aaWbaaeqajuaibaGaaGOmaaaajuaGcqGH RaWkcaaIZaGaeqiUdeNaey4kaSIaaGymaaGaayjkaiaawMcaamaaqa habaWaaeWaaeaadaWcaaqaaiaadshaaeaacqaH4oqCcqGHRaWkcaaI XaaaaaGaayjkaiaawMcaamaaCaaabeqcfasaaiaadIhaaaaabaGaam iEaiabg2da9iaaicdaaeaacqGHEisPaKqbakabggHiLdaajuaibaGa amiEaiabg2da9iaaicdaaeaacqGHEisPaKqbakabggHiLdaacaGLBb Gaayzxaaaaaa@7BFF@
= θ ( θ+2 ) ( θ+1 ) 2 [ θ( θ+1 )t ( θ+1t ) 2 + ( θ 2 +3θ+1 )( θ+1 ) θ+1t ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyypa0 ZaaSaaaeaacqaH4oqCaeaadaqadaqaaiabeI7aXjabgUcaRiaaikda aiaawIcacaGLPaaadaqadaqaaiabeI7aXjabgUcaRiaaigdaaiaawI cacaGLPaaadaahaaqabKqbGeaacaaIYaaaaaaajuaGdaWadaqaamaa laaabaGaeqiUde3aaeWaaeaacqaH4oqCcqGHRaWkcaaIXaaacaGLOa GaayzkaaGaamiDaaqaamaabmaabaGaeqiUdeNaey4kaSIaaGymaiab gkHiTiaadshaaiaawIcacaGLPaaadaahaaqabKqbGeaacaaIYaaaaa aajuaGcqGHRaWkdaWcaaqaamaabmaabaGaeqiUde3aaWbaaeqajuai baGaaGOmaaaajuaGcqGHRaWkcaaIZaGaeqiUdeNaey4kaSIaaGymaa GaayjkaiaawMcaamaabmaabaGaeqiUdeNaey4kaSIaaGymaaGaayjk aiaawMcaaaqaaiabeI7aXjabgUcaRiaaigdacqGHsislcaWG0baaaa Gaay5waiaaw2faaaaa@6B5E@
= θ ( θ+2 )( θ+1 ) [ θt ( θ+1t ) 2 + θ 2 +3θ+1 θ+1t ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyypa0 ZaaSaaaeaacqaH4oqCaeaadaqadaqaaiabeI7aXjabgUcaRiaaikda aiaawIcacaGLPaaadaqadaqaaiabeI7aXjabgUcaRiaaigdaaiaawI cacaGLPaaaaaWaamWaaeaadaWcaaqaaiabeI7aXjaadshaaeaadaqa daqaaiabeI7aXjabgUcaRiaaigdacqGHsislcaWG0baacaGLOaGaay zkaaWaaWbaaeqajuaibaGaaGOmaaaaaaqcfaOaey4kaSYaaSaaaeaa cqaH4oqCdaahaaqabKqbGeaacaaIYaaaaKqbakabgUcaRiaaiodacq aH4oqCcqGHRaWkcaaIXaaabaGaeqiUdeNaey4kaSIaaGymaiabgkHi TiaadshaaaaacaGLBbGaayzxaaaaaa@5E83@
= θ 3 +( 4t ) θ 2 +2( 2t )θ+( 1t ) ( θ+1 )( θ+2 ) ( θ+1t ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyypa0 ZaaSaaaeaacqaH4oqCdaahaaqabKqbGeaacaaIZaaaaKqbakabgUca RmaabmaabaGaaGinaiabgkHiTiaadshaaiaawIcacaGLPaaacqaH4o qCdaahaaqabKqbGeaacaaIYaaaaKqbakabgUcaRiaaikdadaqadaqa aiaaikdacqGHsislcaWG0baacaGLOaGaayzkaaGaeqiUdeNaey4kaS YaaeWaaeaacaaIXaGaeyOeI0IaamiDaaGaayjkaiaawMcaaaqaamaa bmaabaGaeqiUdeNaey4kaSIaaGymaaGaayjkaiaawMcaamaabmaaba GaeqiUdeNaey4kaSIaaGOmaaGaayjkaiaawMcaamaabmaabaGaeqiU deNaey4kaSIaaGymaiabgkHiTiaadshaaiaawIcacaGLPaaadaahaa qabKqbGeaacaaIYaaaaaaaaaa@615C@

Moment Generating Function: The moment generating function of the PGD (2.2) is thus given by
M X ( t )= θ 3 +( 4 e t ) θ 2 +2( 2 e t )θ+( 1 e t ) ( θ+1 )( θ+2 ) ( θ+1 e t ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamytam aaBaaajuaibaGaamiwaaqcfayabaWaaeWaaeaacaWG0baacaGLOaGa ayzkaaGaeyypa0ZaaSaaaeaacqaH4oqCdaahaaqabKqbGeaacaaIZa aaaKqbakabgUcaRmaabmaabaGaaGinaiabgkHiTiaadwgadaahaaqa bKqbGeaacaWG0baaaaqcfaOaayjkaiaawMcaaiabeI7aXnaaCaaabe qcfasaaiaaikdaaaqcfaOaey4kaSIaaGOmamaabmaabaGaaGOmaiab gkHiTiaadwgadaahaaqabKqbGeaacaWG0baaaaqcfaOaayjkaiaawM caaiabeI7aXjabgUcaRmaabmaabaGaaGymaiabgkHiTiaadwgadaah aaqabKqbGeaacaWG0baaaaqcfaOaayjkaiaawMcaaaqaamaabmaaba GaeqiUdeNaey4kaSIaaGymaaGaayjkaiaawMcaamaabmaabaGaeqiU deNaey4kaSIaaGOmaaGaayjkaiaawMcaamaabmaabaGaeqiUdeNaey 4kaSIaaGymaiabgkHiTiaadwgadaahaaqabeaacaWG0baaaaGaayjk aiaawMcaamaaCaaabeqcfasaaiaaikdaaaaaaaaa@6CCE@ .

Estimation of Parameter

Maximum likelihood estimate (MLE): Let x 1 , x 2 ,..., x n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEam aaBaaajuaibaGaaGymaaqcfayabaGaaiilaiaadIhadaWgaaqcfasa aiaaikdaaKqbagqaaiaacYcacaGGUaGaaiOlaiaac6cacaGGSaGaam iEamaaBaaajuaibaGaamOBaaqcfayabaaaaa@42A2@ be a random sample of size n from the PGD (2.2) and let f x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aaBaaajuaibaGaamiEaaqcfayabaaaaa@3949@ be the observed frequency in the sample corresponding to X=x(x=1,2,3,...,k) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiwai abg2da9iaadIhacaaMc8UaaGPaVlaacIcacaWG4bGaeyypa0JaaGym aiaacYcacaaIYaGaaiilaiaaiodacaGGSaGaaiOlaiaac6cacaGGUa GaaiilaiaadUgacaGGPaaaaa@47D0@ such that x=1 k f x =n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaabCae aacaWGMbWaaSbaaKqbGeaacaWG4baajuaGbeaaaKqbGeaacaWG4bGa eyypa0JaaGymaaqaaiaadUgaaKqbakabggHiLdGaeyypa0JaamOBaa aa@41D6@ , where k is the largest observed value having non-zero frequency. The likelihood function L of the PGD (2.2) is given by
L= ( θ θ+2 ) n 1 ( θ+1 ) x=1 k ( x+2 ) f x x=1 k [ θx+( θ 2 +3θ+1 ) ] f x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamitai abg2da9maabmaabaWaaSaaaeaacqaH4oqCaeaacqaH4oqCcqGHRaWk caaIYaaaaaGaayjkaiaawMcaamaaCaaabeqcfasaaiaad6gaaaqcfa 4aaSaaaeaacaaIXaaabaWaaeWaaeaacqaH4oqCcqGHRaWkcaaIXaaa caGLOaGaayzkaaWaaWbaaeqabaWaaabCaeaadaqadaqaaiaadIhacq GHRaWkcaaIYaaacaGLOaGaayzkaaGaamOzamaaBaaajuaibaGaamiE aaqcfayabaaajuaibaGaamiEaiabg2da9iaaigdaaeaacaWGRbaaju aGcqGHris5aaaaaaWaaebCaeaadaWadaqaaiabeI7aXjaadIhacqGH RaWkdaqadaqaaiabeI7aXnaaCaaabeqcfasaaiaaikdaaaqcfaOaey 4kaSIaaG4maiabeI7aXjabgUcaRiaaigdaaiaawIcacaGLPaaaaiaa wUfacaGLDbaadaahaaqabKqbGeaacaWGMbqcfa4aaSbaaKqbGeaaca WG4baabeaaaaaabaGaamiEaiabg2da9iaaigdaaeaacaWGRbaajuaG cqGHpis1aaaa@6CD1@
The log likelihood function is obtained as
logL=nlog( θ θ+2 ) x=1 k f x ( x+2 ) log( θ+1 )+ x=1 k f x log[ θx+( θ 2 +3θ+1 ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaciiBai aac+gacaGGNbGaamitaiabg2da9iaad6gaciGGSbGaai4BaiaacEga daqadaqaamaalaaabaGaeqiUdehabaGaeqiUdeNaey4kaSIaaGOmaa aaaiaawIcacaGLPaaacqGHsisldaaeWbqaaiaadAgadaWgaaqcfasa aiaadIhaaKqbagqaamaabmaabaGaamiEaiabgUcaRiaaikdaaiaawI cacaGLPaaaaKqbGeaacaWG4bGaeyypa0JaaGymaaqaaiaadUgaaKqb akabggHiLdGaciiBaiaac+gacaGGNbWaaeWaaeaacqaH4oqCcqGHRa WkcaaIXaaacaGLOaGaayzkaaGaey4kaSYaaabCaeaacaWGMbWaaSba aKqbGeaacaWG4baajuaGbeaaciGGSbGaai4BaiaacEgadaWadaqaai abeI7aXjaadIhacqGHRaWkdaqadaqaaiabeI7aXnaaCaaabeqcfasa aiaaikdaaaqcfaOaey4kaSIaaG4maiabeI7aXjabgUcaRiaaigdaai aawIcacaGLPaaaaiaawUfacaGLDbaaaKqbGeaacaWG4bGaeyypa0Ja aGymaaqaaiaadUgaaKqbakabggHiLdaaaa@7804@
The first derivative of the log likelihood function is given by
dlogL dθ = 2n θ( θ+2 ) n( x ¯ +2 ) θ+1 + x=1 k ( x+2θ+3 ) f x θx+( θ 2 +3θ+1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacaWGKbGaciiBaiaac+gacaGGNbGaamitaaqaaiaadsgacqaH4oqC aaGaeyypa0ZaaSaaaeaacaaIYaGaamOBaaqaaiabeI7aXnaabmaaba GaeqiUdeNaey4kaSIaaGOmaaGaayjkaiaawMcaaaaacqGHsisldaWc aaqaaiaad6gadaqadaqaaiqadIhagaqeaiabgUcaRiaaikdaaiaawI cacaGLPaaaaeaacqaH4oqCcqGHRaWkcaaIXaaaaiabgUcaRmaaqaha baWaaSaaaeaadaqadaqaaiaadIhacqGHRaWkcaaIYaGaeqiUdeNaey 4kaSIaaG4maaGaayjkaiaawMcaaiaadAgadaWgaaqcfasaaiaadIha aKqbagqaaaqaaiabeI7aXjaadIhacqGHRaWkdaqadaqaaiabeI7aXn aaCaaabeqcfasaaiaaikdaaaqcfaOaey4kaSIaaG4maiabeI7aXjab gUcaRiaaigdaaiaawIcacaGLPaaaaaaajuaibaGaamiEaiabg2da9i aaigdaaeaacaWGRbaajuaGcqGHris5aaaa@6FB8@
where x ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOabmiEay aaraaaaa@3799@  is the sample mean.
The maximum likelihood estimate (MLE), θ ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaaaaa@384A@  of θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde haaa@383A@  is the solution of the equation dlogL dθ =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacaWGKbGaciiBaiaac+gacaGGNbGaamitaaqaaiaadsgacqaH4oqC aaGaeyypa0JaaGimaaaa@3F7D@ and is given by the solution of the non-linear equation
2n θ( θ+2 ) n( x ¯ +2 ) θ+1 + x=1 k ( x+2θ+3 ) f x θx+( θ 2 +3θ+1 ) =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacaaIYaGaamOBaaqaaiabeI7aXnaabmaabaGaeqiUdeNaey4kaSIa aGOmaaGaayjkaiaawMcaaaaacqGHsisldaWcaaqaaiaad6gadaqada qaaiqadIhagaqeaiabgUcaRiaaikdaaiaawIcacaGLPaaaaeaacqaH 4oqCcqGHRaWkcaaIXaaaaiabgUcaRmaaqahabaWaaSaaaeaadaqada qaaiaadIhacqGHRaWkcaaIYaGaeqiUdeNaey4kaSIaaG4maaGaayjk aiaawMcaaiaadAgadaWgaaqcfasaaiaadIhaaKqbagqaaaqaaiabeI 7aXjaadIhacqGHRaWkdaqadaqaaiabeI7aXnaaCaaabeqcfasaaiaa ikdaaaqcfaOaey4kaSIaaG4maiabeI7aXjabgUcaRiaaigdaaiaawI cacaGLPaaaaaaajuaibaGaamiEaiabg2da9iaaigdaaeaacaWGRbaa juaGcqGHris5aiabg2da9iaaicdaaaa@6939@  
This non-linear equation can be solved by any numerical iteration methods such as Newton- Raphson, Bisection method, Regula –Falsi method etc

Method of Moment Estimate (Mome): Let x 1 , x 2 ,..., x n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEam aaBaaajuaibaGaaGymaaqcfayabaGaaiilaiaadIhadaWgaaqcfasa aiaaikdaaKqbagqaaiaacYcacaGGUaGaaiOlaiaac6cacaGGSaGaam iEamaaBaaajuaibaGaamOBaaqcfayabaaaaa@42A2@ be a random sample of size n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOBaa aa@3777@ from the PGD (2.2). Equating the first population moment about origin to the corresponding sample moment, the MOME θ ˜ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaGaaaaa@3849@ of θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde haaa@383A@ is given by
θ ˜ = ( 12 x ¯ )+ 4 x ¯ 2 +8 x ¯ +1 2 x ¯ ; x ¯ >0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaGaacqGH9aqpdaWcaaqaamaabmaabaGaaGymaiabgkHiTiaaikda ceWG4bGbaebaaiaawIcacaGLPaaacqGHRaWkdaGcaaqaaiaaisdace WG4bGbaebadaahaaqabKqbGeaacaaIYaaaaKqbakabgUcaRiaaiIda ceWG4bGbaebacqGHRaWkcaaIXaaabeaaaeaacaaIYaGabmiEayaara aaaiaaykW7caaMc8UaaGPaVlaacUdaceWG4bGbaebacqGH+aGpcaaI Waaaaa@511E@  
where x ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOabmiEay aaraaaaa@3799@ is the sample mean.

Applications

The PGD has been fitted to a number of data - sets to test its goodness of fit over Poisson distribution (PD) and Poisson-Lindley distribution (PLD. The parameter has been estimated using maximum likelihood estimation. Two examples of observed data-sets, for which the PD, PLD and PGD has been fitted, are presented. The first data-set is due to Kemp and kemp [9] regarding the distribution of mistakes in copying groups of random digits and the second data-set is due to Beall [10] regarding the distribution of Pyrausta nublilalis.

Conclusions

A discrete Poisson-Garima distribution has been proposed by compounding Poisson distribution with Garima distribution introduced by Shanker [1]. Expression for r th factorial moment about origin has been derived and hence moments about origin and central moments have been given. The nature and behavior of coefficient of Variation, skewness, kurtosis and index of dispersion of the proposed distribution have been studied for varying values of the parameter. The estimation of parameter has been discussed using both maximum likelihood estimation and method of moments. The goodness of fit of the proposed distribution has been discussed with two examples of real data set and fit has been compared with Poisson and Poisson-Lindley distributions. The goodness of fit of the Poisson – Garima distribution shows that it gives better fit than both Poisson and Poisson-Lindley distribution and hence it can be considered as an important distribution to model discrete data over these two discrete distributions.

References

  1. Shanker R (2016) Garima distribution and its application to model behavioral science data, Biometrics & Biostatistics International Journal 4(7): 1-9
  2. Lindley DV (1958) Fiducial distributions and Bayes’ theorem, Journal of the Royal Statistical Society, Series B 20(1): 102- 107.
  3. Shanker R (2015) a Shanker Distribution and Its Applications, International Journal of Statistics and Applications 5(6): 338 - 348
  4. Shanker R (2015) b Akash Distribution and Its Applications, International Journal of Probability and Statistics4(3): 65-75.
  5. Shanker R (2016) a Aradhana Distribution and Its Applications, International Journal of Statistics and Applications 6(1): 23 - 34.
  6. Shanker R (2016) b Sujatha Distribution and Its Applications, Statistics in Transition-New series 17(3): 1-20.
  7. Sankaran M (1970) The discrete Poisson-Lindley distribution, Biometrics 26(1): 145-149.
  8. Grandell J (1997) Mixed Poisson Processes, Chapman& Hall, London.
  9. Kemp CD Kemp AW (1965) Some properties of the Hermite distribution, Biometrika, 52(3): 381-394.
  10. Beall G (1940) The fit and significance of contagious distributions when applied to observations on larval insects, Ecology 21: 460-474.
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