ISSN: 2378-315X BBIJ

Biometrics & Biostatistics International Journal
Research Article
Volume 5 Issue 1 - 2017
On Discrete Poisson-Shanker Distribution and Its Applications
Rama Shanker1*, Hagos Fesshaye2, Ravi Shanker3, Tekie Asehun Leonida4 and Simon Sium1
1Department of Statistics, Eritrea Institute of Technology, Eritrea
2Department of Economics, College of Business and Economics, Eritrea
3Department of Mathematics, GLA College NP University, India
4Department of Applied Mathematics, University of Twente, Netherlands
Received: December 13, 2016 | Published: January 18, 2017
*Corresponding author: Rama Shanker, Department of Statistics, Eritrea Institute of Technology, Asmara, Eritrea, Email:
Citation: Shanker R, Fesshaye H, Shanker R, Leonida TA, Sium S (2017) On Discrete Poisson-Shanker Distribution and Its Applications. Biom Biostat Int J 5(1): 00121. DOI: 10.15406/bbij.2017.05.00121

Abstract

A simple method for obtaining moments of Poisson-Shanker distribution (PSD) introduced by Shanker [1] has been proposed. The first four moments about origin and the variance have been obtained. The goodness of fit and the applications of the PSD have been discussed with count data from ecology, genetics and thunderstorms and the fit is compared with one parameter Poisson distribution (PD) and Poisson-Lindley distribution (PLD) introduced by Sankaran [2].

Keywords: Shanker distribution; Poisson-Shanker distribution; Poisson-Lindley distribution; Moments; Estimation of parameter; Applications

Introduction

The Poisson-Shanker distribution (PSD) defined by its probability mass function

P( X=x )= θ 2 θ 2 +1 x+( θ 2 +θ+1 ) ( θ+1 ) x+2 ;x=0,1,2,...,θ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiuam aabmaabaGaamiwaiabg2da9iaadIhaaiaawIcacaGLPaaacqGH9aqp daWcaaqaaiabeI7aXnaaCaaabeqcfasaaiaaikdaaaaajuaGbaGaeq iUde3aaWbaaeqajuaibaGaaGOmaaaajuaGcqGHRaWkcaaIXaaaamaa laaabaGaamiEaiabgUcaRmaabmaabaGaeqiUde3aaWbaaeqajuaiba GaaGOmaaaajuaGcqGHRaWkcqaH4oqCcqGHRaWkcaaIXaaacaGLOaGa ayzkaaaabaWaaeWaaeaacqaH4oqCcqGHRaWkcaaIXaaacaGLOaGaay zkaaWaaWbaaeqajuaibaGaamiEaiabgUcaRiaaikdaaaaaaKqbakaa ykW7caaMc8Uaai4oaiaadIhacqGH9aqpcaaIWaGaaiilaiaaigdaca GGSaGaaGOmaiaacYcacaGGUaGaaiOlaiaac6cacaGGSaGaeqiUdeNa eyOpa4JaaGimaaaa@68BE@                                  (1.1)

has been introduced by Shanker [1] for modeling count data-sets. Shanker [1] has shown that PSD is a Poisson mixture of Shanker distribution introduced by Shanker [3] when the parameter λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4UdW gaaa@3838@  of the Poisson distribution follows Shanker distribution of Shanker [3] having probability density function

f( λ;θ )= θ 2 θ 2 +1 ( θ+λ ) e θλ ;λ>0,θ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aabmaabaGaeq4UdWMaai4oaiabeI7aXbGaayjkaiaawMcaaiabg2da 9maalaaabaGaeqiUde3aaWbaaeqajuaibaGaaGOmaaaaaKqbagaacq aH4oqCdaahaaqabKqbGeaacaaIYaaaaKqbakabgUcaRiaaigdaaaWa aeWaaeaacqaH4oqCcqGHRaWkcqaH7oaBaiaawIcacaGLPaaacaWGLb WaaWbaaeqajuaibaGaeyOeI0IaeqiUdeNaaGPaVlabeU7aSbaajuaG caaMc8UaaGPaVlaaykW7caaMc8Uaai4oaiabeU7aSjabg6da+iaaic dacaGGSaGaaGPaVlaaykW7cqaH4oqCcqGH+aGpcaaIWaaaaa@6592@                                              (1.2)

We have        P( X=x )= 0 e λ λ x x! θ 2 θ 2 +1 ( θ+λ ) e θλ dλ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiuam aabmaabaGaamiwaiabg2da9iaadIhaaiaawIcacaGLPaaacqGH9aqp daWdXbqaamaalaaabaGaamyzamaaCaaabeqcfasaaiabgkHiTiabeU 7aSbaajuaGcqaH7oaBdaahaaqabKqbGeaacaWG4baaaaqcfayaaiaa dIhacaGGHaaaaaqcfasaaiaaicdaaeaacqGHEisPaKqbakabgUIiYd GaeyyXIC9aaSaaaeaacqaH4oqCdaahaaqabKqbGeaacaaIYaaaaaqc fayaaiabeI7aXnaaCaaabeqcfasaaiaaikdaaaqcfaOaey4kaSIaaG ymaaaadaqadaqaaiabeI7aXjabgUcaRiabeU7aSbGaayjkaiaawMca aiaadwgadaahaaqabKqbGeaacqGHsislcqaH4oqCcaaMc8Uaeq4UdW gaaKqbakaadsgacqaH7oaBaaa@665D@                                                         (1.3)

= θ 2 ( θ 2 +1 )x! 0 λ x ( θ+λ ) e ( θ+1 )λ dλ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyypa0 ZaaSaaaeaacqaH4oqCdaahaaqabKqbGeaacaaIYaaaaaqcfayaamaa bmaabaGaeqiUde3aaWbaaeqajuaibaGaaGOmaaaajuaGcqGHRaWkca aIXaaacaGLOaGaayzkaaGaaGPaVlaadIhacaGGHaaaamaapehabaGa eq4UdW2aaWbaaeqajuaibaGaamiEaaaaaeaacaaIWaaabaGaeyOhIu kajuaGcqGHRiI8amaabmaabaGaeqiUdeNaey4kaSIaeq4UdWgacaGL OaGaayzkaaGaamyzamaaCaaabeqcfasaaiabgkHiTKqbaoaabmaaju aibaGaeqiUdeNaey4kaSIaaGymaaGaayjkaiaawMcaaiaaykW7cqaH 7oaBaaqcfaOaamizaiabeU7aSbaa@6096@

= θ 2 θ 2 +1 x+( θ 2 +θ+1 ) ( θ+1 ) x+2 ;x=0,1,2,...,θ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyypa0 ZaaSaaaeaacqaH4oqCdaahaaqabKqbGeaacaaIYaaaaaqcfayaaiab eI7aXnaaCaaabeqcfasaaiaaikdaaaqcfaOaey4kaSIaaGymaaaada WcaaqaaiaadIhacqGHRaWkdaqadaqaaiabeI7aXnaaCaaabeqcfasa aiaaikdaaaqcfaOaey4kaSIaeqiUdeNaey4kaSIaaGymaaGaayjkai aawMcaaaqaamaabmaabaGaeqiUdeNaey4kaSIaaGymaaGaayjkaiaa wMcaamaaCaaabeqcfasaaiaadIhacqGHRaWkcaaIYaaaaaaajuaGca aMc8UaaGPaVlaacUdacaWG4bGaeyypa0JaaGimaiaacYcacaaIXaGa aiilaiaaikdacaGGSaGaaiOlaiaac6cacaGGUaGaaiilaiabeI7aXj abg6da+iaaicdaaaa@6380@ .                                  (1.4)

Which is the Poisson-Shanker  distribution (PSD), as given in (1.1).

Shanker [3] has shown that the Shanker distribution (1.2) is a two component mixture of an exponential ( θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde haaa@383A@ ) distribution, a gamma (2, θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde haaa@383A@ ) distribution with their mixing proportions θ 2 θ 2 +1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacqaH4oqCdaahaaqabKqbGeaacaaIYaaaaaqcfayaaiabeI7aXnaa CaaabeqcfasaaiaaikdaaaqcfaOaey4kaSIaaGymaaaaaaa@3ED1@  and 1 θ 2 +1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacaaIXaaabaGaeqiUde3aaWbaaeqajuaibaGaaGOmaaaajuaGcqGH RaWkcaaIXaaaaaaa@3C3C@  respectively. Shanker [3] has discussed its various mathematical and statistical properties including its shape, moment generating function, moments, skewness, kurtosis, hazard rate function, mean residual life function, stochastic orderings, mean deviations, distribution of order statistics, Bonferroni and Lorenz curves, Renyi entropy measure, stress-strength reliability , amongst others along with estimation of parameter and applications. Shanker & Hagos [4] have detailed study on modeling lifetime data using one parameter Akash distribution introduced by Shanker [5], Shanker distribution of Shanker [3], Lindley [6] distribution and exponential distribution.

The probability mass function of Poisson-Lindley distribution (PLD) given by

P( X=x )= θ 2 ( x+θ+2 ) ( θ+1 ) x+3 ; MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiuam aabmaabaGaamiwaiabg2da9iaadIhaaiaawIcacaGLPaaacaaMe8Ua eyypa0JaaGjbVlaaykW7daWcaaqaaiabeI7aXnaaCaaabeqcfasaai aaikdaaaqcfaOaaGPaVpaabmaabaGaamiEaiabgUcaRiaaykW7cqaH 4oqCcqGHRaWkcaaMc8UaaGOmaaGaayjkaiaawMcaaaqaamaabmaaba GaeqiUdeNaey4kaSIaaGPaVlaaigdaaiaawIcacaGLPaaadaahaaqa bKqbGeaacaWG4bGaey4kaSIaaGPaVlaaiodaaaaaaKqbakaaysW7ca aMc8UaaGPaVlaaykW7caGG7aaaaa@6283@     x = 0, 1, 2,…,  θ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde NaeyOpa4JaaGimaaaa@39FC@ .                                        (1.5)

has been introduced by Sankaran [2] to model count data. The distribution arises from the Poisson distribution when its parameter λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4UdW gaaa@3838@ follows Lindley [6] distribution with its probability density function

f( λ,θ )= θ 2 θ+1 ( 1+λ ) e θλ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzai aaykW7daqadaqaaiabeU7aSjaacYcacaaMc8UaeqiUdehacaGLOaGa ayzkaaGaaGjbVlabg2da9iaaysW7caaMc8+aaSaaaeaacqaH4oqCda ahaaqabKqbGeaacaaIYaaaaaqcfayaaiabeI7aXjabgUcaRiaaykW7 caaIXaaaaiaaysW7daqadaqaaiaaigdacqGHRaWkcaaMc8Uaeq4UdW gacaGLOaGaayzkaaGaaGPaVlaaykW7caWGLbWaaWbaaeqajuaibaGa eyOeI0IaeqiUdeNaaGPaVlabeU7aSbaaaaa@6035@  ;    x>0,θ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEai abg6da+iaaicdacaaMc8UaaiilaiabeI7aXjabg6da+iaaicdaaaa@3EF6@                                           (1.6)

Shanker et al. [7] have critical study on modeling of lifetime data using exponential and Lindley [6] distributions and observed that in some data sets Lindley distribution gives better fit than exponential distribution while in some data sets exponential distribution gives better fit than Lindley distribution. Shanker & Hagos [8] have detailed study on Poisson-Lindley distribution and its applications to model count data from biological sciences.

In this paper a simple method of finding moments of Poisson-Shanker distribution (PSD) introduced by Shanker [1] has been suggested and hence the first four moments about origin and the variance have been presented. It seems that not much work has been done on the applications of PSD so far.  The PSD has been fitted to the some data sets relating to ecology, genetics and thunderstorms and the fit is compared with Poisson distribution (PD), and the Poisson-Lindley distribution (PLD). The goodness of fit of PSD shows satisfactory fit in majority of data sets.

Moments

Using (1.3) the r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCaaaa@36ED@ th moment about origin of PSD (1.1) can be obtained as

μ r =E[ E( X r |λ ) ]= θ 2 θ 2 +1 0 [ x=0 x r e λ λ x x! ] ( θ+λ ) e θλ dλ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaKqbGeaacaWGYbaajuaGbeaadaahaaqabeaacWaGGBOmGika aiabg2da9iaadweadaWadaqaaiaadweadaqadaqaaiaadIfadaahaa qabKqbGeaacaWGYbaaaKqbakaacYhacqaH7oaBaiaawIcacaGLPaaa aiaawUfacaGLDbaacqGH9aqpdaWcaaqaaiabeI7aXnaaCaaabeqcfa saaiaaikdaaaaajuaGbaGaeqiUde3aaWbaaeqajuaibaGaaGOmaaaa juaGcqGHRaWkcaaIXaaaamaapehabaWaamWaaeaadaaeWbqaaiaadI hadaahaaqabKqbGeaacaWGYbaaaKqbaoaalaaabaGaamyzamaaCaaa beqcfasaaiabgkHiTiabeU7aSbaajuaGcqaH7oaBdaahaaqabKqbGe aacaWG4baaaaqcfayaaiaadIhacaGGHaaaaaqcfasaaiaadIhacqGH 9aqpcaaIWaaabaGaeyOhIukajuaGcqGHris5aaGaay5waiaaw2faaa qaaiaaicdaaeaacqGHEisPaiabgUIiYdWaaeWaaeaacqaH4oqCcqGH RaWkcqaH7oaBaiaawIcacaGLPaaacaWGLbWaaWbaaeqajuaibaGaey OeI0IaeqiUdeNaaGPaVlabeU7aSbaajuaGcaWGKbGaeq4UdWgaaa@7C00@                               (2.1)

It is clear that the expression under the bracket in (2.1) is the r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCaaaa@36ED@ th moment about origin of the Poisson distribution. Taking r=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOCai abg2da9iaaigdaaaa@393C@  in (2.1) and using the first moment about origin of the Poisson distribution, the first moment about origin of the PSD (1.1) can be obtained as

μ 1 = θ 2 θ 2 +1 0 λ( θ+λ ) e θλ dλ= θ 2 +2 θ( θ 2 +1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaKqbGeaacaaIXaaajuaGbeaadaahaaqabeaacWaGGBOmGika aiabg2da9maalaaabaGaeqiUde3aaWbaaeqajuaibaGaaGOmaaaaaK qbagaacqaH4oqCdaahaaqabKqbGeaacaaIYaaaaKqbakabgUcaRiaa igdaaaWaa8qCaeaacqaH7oaBdaqadaqaaiabeI7aXjabgUcaRiabeU 7aSbGaayjkaiaawMcaaiaaykW7aeaacaaIWaaabaGaeyOhIukacqGH RiI8aiaadwgadaahaaqabKqbGeaacqGHsislcqaH4oqCcaaMc8Uaeq 4UdWgaaKqbakaadsgacqaH7oaBcqGH9aqpdaWcaaqaaiabeI7aXnaa CaaabeqcfasaaiaaikdaaaqcfaOaey4kaSIaaGOmaaqaaiabeI7aXn aabmaabaGaeqiUde3aaWbaaeqajuaibaGaaGOmaaaajuaGcqGHRaWk caaIXaaacaGLOaGaayzkaaaaaaaa@6C36@                                                

Again taking   r=2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOCai abg2da9iaaikdaaaa@393D@   in (2.1) and using the second moment about origin of the Poisson distribution, the second moment about origin of the PSD (1.1) can be obtained as

μ 2 = θ 2 θ 2 +1 0 ( λ 2 +λ )( θ+λ ) e θλ dλ= θ 3 +2 θ 2 +2θ+6 θ 2 ( θ 2 +1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaKqbGeaacaaIYaaajuaGbeaadaahaaqabeaacWaGGBOmGika aiabg2da9maalaaabaGaeqiUde3aaWbaaeqajuaibaGaaGOmaaaaaK qbagaacqaH4oqCdaahaaqabKqbGeaacaaIYaaaaKqbakabgUcaRiaa igdaaaWaa8qCaeaadaqadaqaaiabeU7aSnaaCaaabeqcfasaaiaaik daaaqcfaOaey4kaSIaeq4UdWgacaGLOaGaayzkaaWaaeWaaeaacqaH 4oqCcqGHRaWkcqaH7oaBaiaawIcacaGLPaaacaaMc8oabaGaaGimaa qaaiabg6HiLcGaey4kIipacaWGLbWaaWbaaeqajuaibaGaeyOeI0Ia eqiUdeNaaGPaVlabeU7aSbaajuaGcaWGKbGaeq4UdWMaeyypa0ZaaS aaaeaacqaH4oqCdaahaaqabKqbGeaacaaIZaaaaKqbakabgUcaRiaa ikdacqaH4oqCdaahaaqabKqbGeaacaaIYaaaaKqbakabgUcaRiaaik dacqaH4oqCcqGHRaWkcaaI2aaabaGaeqiUde3aaWbaaeqajuaibaGa aGOmaaaajuaGdaqadaqaaiabeI7aXnaaCaaabeqcfasaaiaaikdaaa qcfaOaey4kaSIaaGymaaGaayjkaiaawMcaaaaaaaa@7BD1@                                                                   

Similarly, taking r=3and4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOCai abg2da9iaaiodacaaMc8UaaGPaVlaabggacaqGUbGaaeizaiaaykW7 caaMc8UaaGinaaaa@42E4@ in (2.1) and using the third and fourth moments about origin of the Poisson distribution, the third and the fourth moments about origin of the PSD (1.1) are obtained as

μ 3 = θ 4 +6 θ 3 +8 θ 2 +18θ+24 θ 3 ( θ 2 +1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaKqbGeaacaaIZaaajuaGbeaadaahaaqabeaacWaGGBOmGika aiabg2da9maalaaabaGaeqiUde3aaWbaaeqajuaibaGaaGinaaaaju aGcqGHRaWkcaaI2aGaeqiUde3aaWbaaeqajuaibaGaaG4maaaajuaG cqGHRaWkcaaI4aGaeqiUde3aaWbaaeqajuaibaGaaGOmaaaajuaGcq GHRaWkcaaIXaGaaGioaiabeI7aXjabgUcaRiaaikdacaaI0aaabaGa eqiUde3aaWbaaeqajuaibaGaaG4maaaajuaGdaqadaqaaiabeI7aXn aaCaaabeqcfasaaiaaikdaaaqcfaOaey4kaSIaaGymaaGaayjkaiaa wMcaaaaaaaa@5B65@                                                                     

  μ 4 = θ 5 +14 θ 4 +38 θ 3 +66 θ 2 +144θ+120 θ 4 ( θ 2 +1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaKqbGeaacaaI0aaajuaGbeaadaahaaqabeaacWaGGBOmGika aiabg2da9maalaaabaGaeqiUde3aaWbaaeqajuaibaGaaGynaaaaju aGcqGHRaWkcaaIXaGaaGinaiabeI7aXnaaCaaabeqcfasaaiaaisda aaqcfaOaey4kaSIaaG4maiaaiIdacqaH4oqCdaahaaqabKqbGeaaca aIZaaaaKqbakabgUcaRiaaiAdacaaI2aGaeqiUde3aaWbaaeqajuai baGaaGOmaaaajuaGcqGHRaWkcaaIXaGaaGinaiaaisdacqaH4oqCcq GHRaWkcaaIXaGaaGOmaiaaicdaaeaacqaH4oqCdaahaaqabKqbGeaa caaI0aaaaKqbaoaabmaabaGaeqiUde3aaWbaaeqajuaibaGaaGOmaa aajuaGcqGHRaWkcaaIXaaacaGLOaGaayzkaaaaaaaa@6403@                                                  

The variance of Poisson-Shanker distribution can thus be obtained as

μ 2 = σ 2 = θ 5 + θ 4 +3 θ 3 +4 θ 2 +2θ+2 θ 2 ( θ 2 +1 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaKqbGeaacaaIYaaajuaGbeaacqGH9aqpcqaHdpWCdaahaaqa bKqbGeaacaaIYaaaaKqbakabg2da9maalaaabaGaeqiUde3aaWbaae qajuaibaGaaGynaaaajuaGcqGHRaWkcqaH4oqCdaahaaqabKqbGeaa caaI0aaaaKqbakabgUcaRiaaiodacqaH4oqCdaahaaqabKqbGeaaca aIZaaaaKqbakabgUcaRiaaisdacqaH4oqCdaahaaqabKqbGeaacaaI YaaaaKqbakabgUcaRiaaikdacqaH4oqCcqGHRaWkcaaIYaaabaGaeq iUde3aaWbaaeqajuaibaGaaGOmaaaajuaGdaqadaqaaiabeI7aXnaa CaaabeqcfasaaiaaikdaaaqcfaOaey4kaSIaaGymaaGaayjkaiaawM caamaaCaaabeqcfasaaiaaikdaaaaaaaaa@6077@                      

Estimation of Parameter

Maximum likelihood estimate (MLE) of the parameter: Suppose ( x 1 , x 2 ,..., x n ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacaWG4bWaaSbaaKqbGeaacaaIXaaajuaGbeaacaGGSaGaamiEamaa BaaajuaibaGaaGOmaaqcfayabaGaaiilaiaac6cacaGGUaGaaiOlai aacYcacaWG4bWaaSbaaKqbGeaacaWGUbaajuaGbeaaaiaawIcacaGL Paaaaaa@442B@ is a random sample of size n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOBaa aa@3777@ from the PSD (1.1) and suppose 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 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4Aaa aa@3774@ is the largest observed value having non-zero frequency. The likelihood function L MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamitaa aa@3755@ of the PSD (1.1) is given by

L= ( θ 2 θ 2 +1 ) n 1 ( θ+1 ) x=1 k f x ( x+2 ) x=1 k [ x+( θ 2 +θ+1 ) ] f x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamitai abg2da9maabmaabaWaaSaaaeaacqaH4oqCdaahaaqabKqbGeaacaaI YaaaaaqcfayaaiabeI7aXnaaCaaabeqcfasaaiaaikdaaaqcfaOaey 4kaSIaaGymaaaaaiaawIcacaGLPaaadaahaaqabKqbGeaacaWGUbaa aKqbaoaalaaabaGaaGymaaqaamaabmaabaGaeqiUdeNaey4kaSIaaG ymaaGaayjkaiaawMcaamaaCaaabeqaamaaqahabaGaamOzamaaBaaa juaibaGaamiEaaqcfayabaWaaeWaaeaacaWG4bGaey4kaSIaaGOmaa GaayjkaiaawMcaaaqcfasaaiaadIhacqGH9aqpcaaIXaaabaGaam4A aaqcfaOaeyyeIuoaaaaaamaarahabaWaamWaaeaacaWG4bGaey4kaS YaaeWaaeaacqaH4oqCdaahaaqabKqbGeaacaaIYaaaaKqbakabgUca RiabeI7aXjabgUcaRiaaigdaaiaawIcacaGLPaaaaiaawUfacaGLDb aadaahaaqabeaacaWGMbWaaSbaaKqbGeaacaWG4baajuaGbeaaaaaa juaibaGaamiEaiabg2da9iaaigdaaeaacaWGRbaajuaGcqGHpis1aa aa@6D91@

The log likelihood function is thus obtained as

logL=nlog( θ 2 θ 2 +1 ) x=1 k f x ( x+2 ) log( θ+1 )+ x=1 k f x log[ x+( θ 2 +θ+1 ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaciiBai aac+gacaGGNbGaamitaiabg2da9iaad6gaciGGSbGaai4BaiaacEga daqadaqaamaalaaabaGaeqiUde3aaWbaaeqajuaibaGaaGOmaaaaaK qbagaacqaH4oqCdaahaaqabKqbGeaacaaIYaaaaKqbakabgUcaRiaa igdaaaaacaGLOaGaayzkaaGaeyOeI0YaaabCaeaacaWGMbWaaSbaaK qbGeaacaWG4baajuaGbeaadaqadaqaaiaadIhacqGHRaWkcaaIYaaa caGLOaGaayzkaaaajuaibaGaamiEaiabg2da9iaaigdaaeaacaWGRb aajuaGcqGHris5aiGacYgacaGGVbGaai4zamaabmaabaGaeqiUdeNa ey4kaSIaaGymaaGaayjkaiaawMcaaiabgUcaRmaaqahabaGaamOzam aaBaaajuaibaGaamiEaaqcfayabaGaciiBaiaac+gacaGGNbWaamWa aeaacaWG4bGaey4kaSYaaeWaaeaacqaH4oqCdaahaaqabKqbGeaaca aIYaaaaKqbakabgUcaRiabeI7aXjabgUcaRiaaigdaaiaawIcacaGL PaaaaiaawUfacaGLDbaaaKqbGeaacaWG4bGaeyypa0JaaGymaaqaai aadUgaaKqbakabggHiLdaaaa@78C4@

The first derivative of the log likelihood function is given by

dlogL dθ = 2n θ( θ 2 +1 ) n( x ¯ +2 ) θ+1 + x=1 k ( 2θ+1 ) f x x+( θ 2 +θ+1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacaWGKbGaciiBaiaac+gacaGGNbGaamitaaqaaiaadsgacqaH4oqC aaGaeyypa0ZaaSaaaeaacaaIYaGaamOBaaqaaiabeI7aXnaabmaaba GaeqiUde3aaWbaaeqajuaibaGaaGOmaaaajuaGcqGHRaWkcaaIXaaa caGLOaGaayzkaaaaaiabgkHiTmaalaaabaGaamOBamaabmaabaGabm iEayaaraGaey4kaSIaaGOmaaGaayjkaiaawMcaaaqaaiabeI7aXjab gUcaRiaaigdaaaGaey4kaSYaaabCaeaadaWcaaqaamaabmaabaGaaG OmaiabeI7aXjabgUcaRiaaigdaaiaawIcacaGLPaaacaWGMbWaaSba aKqbGeaacaWG4baajuaGbeaaaeaacaWG4bGaey4kaSYaaeWaaeaacq aH4oqCdaahaaqabKqbGeaacaaIYaaaaKqbakabgUcaRiabeI7aXjab gUcaRiaaigdaaiaawIcacaGLPaaaaaaajuaibaGaamiEaiabg2da9i aaigdaaeaacaWGRbaajuaGcqGHris5aaaa@6CFD@

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@  of PSD (1.1) is the solution of the following non-linear equation

2n θ( θ 2 +1 ) n( x ¯ +2 ) θ+1 + x=1 k ( 2θ+1 ) f x x+( θ 2 +θ+1 ) =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacaaIYaGaamOBaaqaaiabeI7aXnaabmaabaGaeqiUde3aaWbaaeqa juaibaGaaGOmaaaajuaGcqGHRaWkcaaIXaaacaGLOaGaayzkaaaaai abgkHiTmaalaaabaGaamOBamaabmaabaGabmiEayaaraGaey4kaSIa aGOmaaGaayjkaiaawMcaaaqaaiabeI7aXjabgUcaRiaaigdaaaGaey 4kaSYaaabCaeaadaWcaaqaamaabmaabaGaaGOmaiabeI7aXjabgUca RiaaigdaaiaawIcacaGLPaaacaWGMbWaaSbaaKqbGeaacaWG4baaju aGbeaaaeaacaWG4bGaey4kaSYaaeWaaeaacqaH4oqCdaahaaqabKqb GeaacaaIYaaaaKqbakabgUcaRiabeI7aXjabgUcaRiaaigdaaiaawI cacaGLPaaaaaaajuaibaGaamiEaiabg2da9iaaigdaaeaacaWGRbaa juaGcqGHris5aiabg2da9iaaicdaaaa@667E@          

This non-linear equation can be solved by any numerical iteration methods such as Newton-Raphson method, Bisection method, Regula-Falsi method etc. In this paper, Newton-Raphson method has been used for estimating the parameter.

Shanker [1] has showed that the MLE of θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde haaa@383A@  of PSD (1.1) is consistent and asymptotically normal.

Method of moment estimate (MOME) of the parameter: Equating the population mean to the corresponding sample mean, 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@  of PSD (1.1) is the solution of the following cubic equation

x ¯ θ 3 θ 2 + x ¯ θ2=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOabmiEay aaraGaeqiUde3aaWbaaeqajuaibaGaaG4maaaajuaGcqGHsislcqaH 4oqCdaahaaqabKqbGeaacaaIYaaaaKqbakabgUcaRiqadIhagaqeai abeI7aXjabgkHiTiaaikdacqGH9aqpcaaIWaaaaa@463D@                                       

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

Goodness of Fit and Applications

Since the condition for the applications for Poisson distribution is the independence of events and equality of mean and variance, this condition is rarely satisfied completely in biological and medical science due to the fact that the occurrences of successive events are dependent. Further, the negative binomial distribution is a possible alternative to the Poisson distribution when successive events are possibly dependent, (see Johnson et al. [9]), but for fitting negative binomial distribution (NBD) to the count data, mean should be less than the variance (over-dispersion). In biological and medical sciences, these conditions are not fully satisfied. Generally, the count data in biological science and medical science are either over-dispersed or under-dispersed. The main reason for selecting PLD and PSD to fit data from biological science and thunderstorms are that these two distributions are always over-dispersed and PSD has some flexibility over PLD.

Count data from ecology and biological sciences

In this section we fit Poisson distribution (PD), Poisson -Lindley distribution (PLD) and Poisson-Shanker distribution (PSD) to many count data from ecology and biological sciences using maximum likelihood estimate. The data were on haemocytometer yeast cell counts per square, on European red mites on apple leaves and European corn borers per plant. Recall that Shanker & Hagos [7] have fitted Poisson-Lindley distribution(PLD) to the same data sets.

It is obvious from above tables that in Table 1, PD gives better fit than PLD and PSD; in Table 2, PSD gives better fit than PD and PLD while in Table 3, PLD gives better fit than PD and PSD.

Number of Yeast Cells per Square

Observed Frequency

Expected Frequency

PD

PLD

PSD

0

213

202.1

234.0

233.2

1

128

138.0

99.4

99.6

2

37

47.1

40.5

41.0

3

18

10.7 1.8 0.2 0.1 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaigdacaaIWaGaaiOlaiaaiEdaaeaacaaIXaGaaiOlaiaa iIdaaeaacaaIWaGaaiOlaiaaikdaaeaacaaIWaGaaiOlaiaaigdaaa GaayzFaaaaaa@4110@

16.0 6.2 2.4 1.5 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaigdacaaI2aGaaiOlaiaaicdaaeaacaaI2aGaaiOlaiaa ikdaaeaacaaIYaGaaiOlaiaaisdaaeaacaaIXaGaaiOlaiaaiwdaaa GaayzFaaaaaa@4117@

16.3 6.7 2.3 0.9 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaigdacaaI2aGaaiOlaiaaiodaaeaacaaI2aGaaiOlaiaa iEdaaeaacaaIYaGaaiOlaiaaiodaaeaacaaIWaGaaiOlaiaaiMdaaa GaayzFaaaaaa@4121@

4

3

5

1

6

0

Total

400

400.0

400.0

400.0

ML Estimate

θ ^ =0.6825 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpcaaIWaGaaiOlaiaaiAdacaaI4aGaaGOmaiaaiwda aaa@3DB9@

θ ^ =1.950236 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpcaaIXaGaaiOlaiaaiMdacaaI1aGaaGimaiaaikda caaIZaGaaGOnaaaa@3F32@

θ ^ =1.795126 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpcaaIXaGaaiOlaiaaiEdacaaI5aGaaGynaiaaigda caaIYaGaaGOnaaaa@3F37@

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

10.08

11.04

12.25

d.f.

2

2

2

p-value

0.0065

0.0040

0.0023

Table 1: Observed and expected number of Haemocytometer yeast cell counts per square observed by Gosset [10].

Number mites per Leaf

Observed Frequency

Expected Frequency

PD

PLD

PSD

0

38

25.3

35.8

36.0

1

17

29.1

20.7

20.6

2

10

16.7

11.4

11.2

3

9

6.4 1.8 0.4 0.2 0.1 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaiAdacaGGUaGaaGinaaqaaiaaigdacaGGUaGaaGioaaqa aiaaicdacaGGUaGaaGinaaqaaiaaicdacaGGUaGaaGOmaaqaaiaaic dacaGGUaGaaGymaaaacaGL9baaaaa@4283@

6.0

6.0

4

3

3.1 1.6 0.8 0.6 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaiodacaGGUaGaaGymaaqaaiaaigdacaGGUaGaaGOnaaqa aiaaicdacaGGUaGaaGioaaqaaiaaicdacaGGUaGaaGOnaaaacaGL9b aaaaa@405B@

3.1 1.6 0.8 0.7 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaiodacaGGUaGaaGymaaqaaiaaigdacaGGUaGaaGOnaaqa aiaaicdacaGGUaGaaGioaaqaaiaaicdacaGGUaGaaG4naaaacaGL9b aaaaa@405C@

5

2

6

1

7+

0

Total

80

80.0

80.0

80.0

ML Estimate

θ ^ =1.15 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpcaaIXaGaaiOlaiaaigdacaaI1aaaaa@3C37@

θ ^ =1.255891 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpcaaIXaGaaiOlaiaaikdacaaI1aGaaGynaiaaiIda caaI5aGaaGymaaaa@3F37@

θ ^ =1.219731 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpcaaIXaGaaiOlaiaaikdacaaIXaGaaGyoaiaaiEda caaIZaGaaGymaaaa@3F30@

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

18.27

2.47

2.37

d.f.

2

3

3

p-value

0.0001

0.4807

0.4992

Table 2: Observed and expected number of red mites on Apple leaves, available in Fisher et al. [11].

Number of bores per Plant

Observed Frequency

Expected Frequency

PD

PLD

PSD

0

188

169.4

194.0

195.0

1

83

109.8

79.5

78.4

2

36

35.6

31.3

31.0

3

14

7.8 1.2 0.2 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaiEdacaGGUaGaaGioaaqaaiaaigdacaGGUaGaaGOmaaqa aiaaicdacaGGUaGaaGOmaaaacaGL9baaaaa@3E2F@

12.0 4.5 2.7 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaigdacaaIYaGaaiOlaiaaicdaaeaacaaI0aGaaiOlaiaa iwdaaeaacaaIYaGaaiOlaiaaiEdaaaGaayzFaaaaaa@3EEA@

12.1 4.6 2.9 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaigdacaaIYaGaaiOlaiaaigdaaeaacaaI0aGaaiOlaiaa iAdaaeaacaaIYaGaaiOlaiaaiMdaaaGaayzFaaaaaa@3EEE@

4

2

5

1

Total

324

324.0

324.0

324.0

ML Estimate

θ ^ =0.648148 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpcaaIWaGaaiOlaiaaiAdacaaI0aGaaGioaiaaigda caaI0aGaaGioaaaa@3F37@

θ ^ =2.043252 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpcaaIYaGaaiOlaiaaicdacaaI0aGaaG4maiaaikda caaI1aGaaGOmaaaa@3F2A@

θ ^ =1.879553 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpcaaIXaGaaiOlaiaaiIdacaaI3aGaaGyoaiaaiwda caaI1aGaaG4maaaa@3F3E@

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

15.19

1.29

1.67

d.f.

2

2

2

p-value

0.0005

0.5247

0.4338

Table 3: Observed and expected number of European corn- borer of Mc Guire et al. [12].

Number of Aberrations

Observed Frequency

Expected Frequency

PD

PLD

PSD

0

268

231.3

257.0

258.3

1

87

126.7

93.4

92.1

2

26

34.7

32.8

32.4

3

9

6.3 0.8 0.1 0.1 0.1 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaiAdacaGGUaGaaG4maaqaaiaaicdacaGGUaGaaGioaaqa aiaaicdacaGGUaGaaGymaaqaaiaaicdacaGGUaGaaGymaaqaaiaaic dacaGGUaGaaGymaaaacaGL9baaaaa@427D@

11.2

11.3

4

4

3.8 1.2 0.4 0.2 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaiodacaGGUaGaaGioaaqaaiaaigdacaGGUaGaaGOmaaqa aiaaicdacaGGUaGaaGinaaqaaiaaicdacaGGUaGaaGOmaaaacaGL9b aaaaa@4056@

3.9 1.3 0.5 1.5 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaiodacaGGUaGaaGyoaaqaaiaaigdacaGGUaGaaG4maaqa aiaaicdacaGGUaGaaGynaaqaaiaaigdacaGGUaGaaGynaaaacaGL9b aaaaa@405D@

5

2

6

1

7+

3

Total

400

400.0

400.0

400.0

ML Estimate

θ ^ =0.5475 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpcaaIWaGaaiOlaiaaiwdacaaI0aGaaG4naiaaiwda aaa@3DB9@

θ ^ =2.380442 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpcaaIYaGaaiOlaiaaiodacaaI4aGaaGimaiaaisda caaI0aGaaGOmaaaa@3F2F@

θ ^ =2.162674 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpcaaIYaGaaiOlaiaaigdacaaI2aGaaGOmaiaaiAda caaI3aGaaGinaaaa@3F34@

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

38.21

6.21

3.45

d.f.

2

3

3

p-value

0.0000

0.1018

0.3273

Table 4: Distribution of number of Chromatid aberrations (0.2 g chinon 1, 24 hours).

Count data from genetics

In this section we fit PSD, PLD and PD using maximum likelihood estimate to count data relating to genetics. Recall that Shanker & Hagos [8] have fitted Poisson-Lindley distribution to the same data sets. The data set in Table 4 is available in Loeschke & Kohler [13], and Janardan & Schaeffer [14]. The data sets in Tables 5-7 are available in Catcheside et al. [15,16].

Class/Exposure ( μg|kg MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 Maam4zaiaacYhacaWGRbGaam4zaaaa@3C02@ )

Observed Frequency

Expected Frequency

PD

PLD

PSD

0

413

374.0

405.7

407.1

1

124

177.4

133.6

131.9

2

42

42.1

42.6

42.3

3

15

6.6 0.8 0.1 0.0 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaiAdacaGGUaGaaGOnaaqaaiaaicdacaGGUaGaaGioaaqa aiaaicdacaGGUaGaaGymaaqaaiaaicdacaGGUaGaaGimaaaacaGL9b aaaaa@4057@

13.3

13.5

4

5

4.1 1.2 0.5 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaisdacaGGUaGaaGymaaqaaiaaigdacaGGUaGaaGOmaaqa aiaaicdacaGGUaGaaGynaaaacaGL9baaaaa@3E28@

4.3 1.3 0.6 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaisdacaGGUaGaaG4maaqaaiaaigdacaGGUaGaaG4maaqa aiaaicdacaGGUaGaaGOnaaaacaGL9baaaaa@3E2C@

5

0

6

2

Total

601

601.0

601.0

601.0

ML Estimate

θ ^ =0.47421 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpcaaIWaGaaiOlaiaaisdacaaI3aGaaGinaiaaikda caaIXaaaaa@3E70@

θ ^ =2.685373 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpcaaIYaGaaiOlaiaaiAdacaaI4aGaaGynaiaaioda caaI3aGaaG4maaaa@3F3A@

θ ^ =2.419447 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpcaaIYaGaaiOlaiaaisdacaaIXaGaaGyoaiaaisda caaI0aGaaG4naaaa@3F37@

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

48.17

1.34

0.82

d.f.

2

3

3

p-value

0.0000

0.7196

0.8446

Table 5: Mammalian cytogenetic dosimetry lesions in rabbit lymphoblast induced by streptonigrin (NSC-45383), Exposure-60 μg|kg MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 Maam4zaiaacYhacaWGRbGaam4zaaaa@3C02@ .

Class/Exposure ( μg|kg MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 Maam4zaiaacYhacaWGRbGaam4zaaaa@3C02@ )

Observed Frequency

Expected Frequency

PD

PLD

PSD

0

200

172.5

191.8

192.7

1

57

95.4

70.3

69.4

2

30

26.4

24.9

24.6

3

7

4.9 0.7 0.1 0.0 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaisdacaGGUaGaaGyoaaqaaiaaicdacaGGUaGaaG4naaqa aiaaicdacaGGUaGaaGymaaqaaiaaicdacaGGUaGaaGimaaaacaGL9b aaaaa@4057@

8.6 2.9 1.0 0.5 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaiIdacaGGUaGaaGOnaaqaaiaaikdacaGGUaGaaGyoaaqa aiaaigdacaGGUaGaaGimaaqaaiaaicdacaGGUaGaaGynaaaacaGL9b aaaaa@4061@

8.7 3.0 1.0 0.6 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaiIdacaGGUaGaaG4naaqaaiaaiodacaGGUaGaaGimaaqa aiaaigdacaGGUaGaaGimaaqaaiaaicdacaGGUaGaaGOnaaaacaGL9b aaaaa@405B@

4

4

5

0

6

2

Total

300

300.0

300.0

300.0

ML Estimate

θ ^ =0.55333 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpcaaIWaGaaiOlaiaaiwdacaaI1aGaaG4maiaaioda caaIZaaaaa@3E71@

θ ^ =2.353339 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpcaaIYaGaaiOlaiaaiodacaaI1aGaaG4maiaaioda caaIZaGaaGyoaaaa@3F34@

θ ^ =2.138048 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpcaaIYaGaaiOlaiaaigdacaaIZaGaaGioaiaaicda caaI0aGaaGioaaaa@3F32@

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

29.68

3.91

3.66

d.f.

2

2

2

p-value

0.0000

0.1415

0.1604

Table 6: Mammalian cytogenetic dosimetry lesions in rabbit lymphoblast induced by streptonigrin (NSC-45383), Exposure-70 μg|kg MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 Maam4zaiaacYhacaWGRbGaam4zaaaa@3C02@ .

Class/Exposure ( μg|kg MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 Maam4zaiaacYhacaWGRbGaam4zaaaa@3C02@ )

Observed Frequency

Expected Frequency

PD

PLD

PSD

0

155

127.8

158.3

159.3

1

83

109.0

77.2

76.3

2

33

46.5

35.9

35.4

3

14

13.2 2.8 0.5 0.2 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaigdacaaIZaGaaiOlaiaaikdaaeaacaaIYaGaaiOlaiaa iIdaaeaacaaIWaGaaiOlaiaaiwdaaeaacaaIWaGaaiOlaiaaikdaaa GaayzFaaaaaa@4113@

16.1

16.1

4

11

7.1 3.1 2.3 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaiEdacaGGUaGaaGymaaqaaiaaiodacaGGUaGaaGymaaqa aiaaikdacaGGUaGaaG4maaaacaGL9baaaaa@3E2C@

7.2 3.2 2.5 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaiEdacaGGUaGaaGOmaaqaaiaaiodacaGGUaGaaGOmaaqa aiaaikdacaGGUaGaaGynaaaacaGL9baaaaa@3E30@

5

3

6

1

Total

300

300.0

300.0

300.0

ML Estimate

θ ^ =0.853333 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpcaaIWaGaaiOlaiaaiIdacaaI1aGaaG4maiaaioda caaIZaGaaG4maaaa@3F31@

θ ^ =1.617611 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpcaaIXaGaaiOlaiaaiAdacaaIXaGaaG4naiaaiAda caaIXaGaaGymaaaa@3F2F@

θ ^ =1.520805 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpcaaIXaGaaiOlaiaaiwdacaaIYaGaaGimaiaaiIda caaIWaGaaGynaaaa@3F2D@

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

24.97

1.51

1.48

d.f.

2

3

3

p-value

0.0000

0.6799

0.6868

Table 7: Mammalian cytogenetic dosimetry lesions in rabbit lymphoblast induced by streptonigrin (NSC-45383), Exposure -90 μg|kg MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 Maam4zaiaacYhacaWGRbGaam4zaaaa@3C02@ .

It is obvious from the fitting of PSD, PLD, and PD that both PSD and PLD gives much satisfactory fit than PD. Further, PSD gives much closer fit than both PLD and PD in almost all data sets.

Count data from thunderstorms

In this section, we fit PSD, PLD and PD to count data from thunderstorms available in Falls et al. [17].

It is obvious from the fitting of PSD, PLD and PD to thunderstorms data that PLD gives better fit than both PSD and PD in Table 8, 9 and 11 while PSD gives better fit than both PLD and PD in Table 10.

No. of Thunderstorms

Observed Frequency

Expected Frequency

PD

PLD

PSD

0

187

155.6

185.3

186.4

1

77

117.0

83.5

82.3

2

40

43.9

35.9

35.5

3

17

11.0 2.1 0.3 0.1 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaigdacaaIXaGaaiOlaiaaicdaaeaacaaIYaGaaiOlaiaa igdaaeaacaaIWaGaaiOlaiaaiodaaeaacaaIWaGaaiOlaiaaigdaaa GaayzFaaaaaa@4105@

15.0

15.0

4

6

6.1 2.5 1.7 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaiAdacaGGUaGaaGymaaqaaiaaikdacaGGUaGaaGynaaqa aiaaigdacaGGUaGaaG4naaaacaGL9baaaaa@3E31@

6.3 2.6 1.9 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaiAdacaGGUaGaaG4maaqaaiaaikdacaGGUaGaaGOnaaqa aiaaigdacaGGUaGaaGyoaaaacaGL9baaaaa@3E36@

5

2

6

1

Total

330

330.0

330.0

330.0

ML Estimate

θ ^ =0.751515 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpcaaIWaGaaiOlaiaaiEdacaaI1aGaaGymaiaaiwda caaIXaGaaGynaaaa@3F30@

θ ^ =1.804268 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpcaaIXaGaaiOlaiaaiIdacaaIWaGaaGinaiaaikda caaI2aGaaGioaaaa@3F35@

θ ^ =1.679053 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpcaaIXaGaaiOlaiaaiAdacaaI3aGaaGyoaiaaicda caaI1aGaaG4maaaa@3F37@

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

31.93

1.43

1.48

d.f.

2

3

3

p-value

0.0000

0.6985

0.6869

Table 8: Observed and expected number of days that experienced X thunderstorms events at Cape Kennedy, Florida for the 11-year period of record for the month of June, January 1957 to December 1967, Falls et al. [17].

No. of Thunderstorms

Observed Frequency

Expected Frequency

PD

PLD

PSD

0

177

142.3

177.7

178.7

1

80

124.4

88.0

86.9

2

47

54.3

41.5

41.0

3

26

15.8 3.5 0.7 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaigdacaaI1aGaaiOlaiaaiIdaaeaacaaIZaGaaiOlaiaa iwdaaeaacaaIWaGaaiOlaiaaiEdaaaGaayzFaaaaaa@3EF2@

18.9

18.9

4

9

8.4 6.5 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaiIdacaGGUaGaaGinaaqaaiaaiAdacaGGUaGaaGynaaaa caGL9baaaaa@3C0B@

8.6 6.9 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaiIdacaGGUaGaaGOnaaqaaiaaiAdacaGGUaGaaGyoaaaa caGL9baaaaa@3C11@

5

2

Total

341

341.0

341.0

341.0

ML Estimate

θ ^ =0.873900 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpcaaIWaGaaiOlaiaaiIdacaaI3aGaaG4maiaaiMda caaIWaGaaGimaaaa@3F33@

θ ^ =1.583536 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpcaaIXaGaaiOlaiaaiwdacaaI4aGaaG4maiaaiwda caaIZaGaaGOnaaaa@3F37@

θ ^ =1.497274 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpcaaIXaGaaiOlaiaaisdacaaI5aGaaG4naiaaikda caaI3aGaaGinaaaa@3F3A@

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

39.74

5.15

5.41

d.f.

2

3

3

p-value

0.0000

0.1611

0.1441

Table 9: Observed and expected number of days that experienced X thunderstorms events at Cape Kennedy, Florida for the 11-year period of record for the month of July, January 1957 to December 1967, Falls et al. [17].

No. of Thunderstorms

Observed Frequency

Expected Frequency

PD

PLD

PSD

0

185

151.8

184.8

186.0

1

89

122.9

87.2

86.1

2

30

49.7

39.3

38.8

3

24

13.4 2.7 0.5 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaigdacaaIZaGaaiOlaiaaisdaaeaacaaIYaGaaiOlaiaa iEdaaeaacaaIWaGaaiOlaiaaiwdaaaGaayzFaaaaaa@3EEB@

17.1

17.1

4

10

7.3 5.3 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaiEdacaGGUaGaaG4maaqaaiaaiwdacaGGUaGaaG4maaaa caGL9baaaaa@3C06@

7.4 5.6 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaiEdacaGGUaGaaGinaaqaaiaaiwdacaGGUaGaaGOnaaaa caGL9baaaaa@3C0A@

5

3

Total

341

341.0

341.0

341.0

ML estimate

θ ^ =0.809384 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpcaaIWaGaaiOlaiaaiIdacaaIWaGaaGyoaiaaioda caaI4aGaaGinaaaa@3F38@

θ ^ =1.693425 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpcaaIXaGaaiOlaiaaiAdacaaI5aGaaG4maiaaisda caaIYaGaaGynaaaa@3F36@

θ ^ =1.586731 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpcaaIXaGaaiOlaiaaiwdacaaI4aGaaGOnaiaaiEda caaIZaGaaGymaaaa@3F37@

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

49.49

5.03

4.87

d.f.

2

3

3

p-value

0.0000

0.1696

0.1816

Table 10: Observed and expected number of days that experienced X thunderstorms events at Cape Kennedy, Florida for the 11-year period of record for the month of August, January 1957 to December 1967, Falls et al. [17].

No. of Thunderstorms

Observed Frequency

Expected Frequency

PD

PLD

PSD

0

549

547.5

547.5

550.8

1

246

364.8

259.0

255.7

2

117

148.2

116.9

115.5

3

67

40.1

51.2

51.1

4

25

8.1 1.3 0.3 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaiIdacaGGUaGaaGymaaqaaiaaigdacaGGUaGaaG4maaqa aiaaicdacaGGUaGaaG4maaaacaGL9baaaaa@3E2B@

21.9

22.3

5

7

9.2 6.3 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaiMdacaGGUaGaaGOmaaqaaiaaiAdacaGGUaGaaG4maaaa caGL9baaaaa@3C08@

9.6 7.0 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaiMdacaGGUaGaaGOnaaqaaiaaiEdacaGGUaGaaGimaaaa caGL9baaaaa@3C0A@

6

1

Total

1012

1012.0

1012.0

1012.0

ML Estimate

θ ^ =0.812253 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpcaaIWaGaaiOlaiaaiIdacaaIXaGaaGOmaiaaikda caaI1aGaaG4maaaa@3F2D@

θ ^ =1.688990 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpcaaIXaGaaiOlaiaaiAdacaaI4aGaaGioaiaaiMda caaI5aGaaGimaaaa@3F41@

θ ^ =1.582475 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpcaaIXaGaaiOlaiaaiwdacaaI4aGaaGOmaiaaisda caaI3aGaaGynaaaa@3F38@

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

141.42

9.60

10.09

d.f.

3

4

4

p-value

0.0000

0.0477

0.0389

Table 11: Observed and expected number of days that experienced X thunderstorms events at Cape Kennedy, Florida for the 11-year period of record for the summer, January 1957 to December 1967, Falls et al. [17].

Concluding Remarks

In the present paper, a simple and interesting method for finding moments of Poisson-Shanker distribution (PSD) has been suggested and thus the first four moments about origin and the variance have been obtained. The goodness of fit of PSD has been discussed with several data from ecology, genetics and thunderstorms and the fit has been compared with Poisson distribution (PD) and Poisson-Lindley distribution (PLD).

References

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  3. Shanker R (2015) Shanker distribution and its applications. International Journal of Statistics and Applications5(6): 338-348.
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  12. Mc Guire JU, Brindley TA, Bancroft TA (1957) The distribution of European corn-borer larvae pyrausta in field corn. Biometrics 13(1): 65-78.
  13. Loeschke V, Kohler W (1976) Deterministic and Stochastic models of the negative binomial distribution and the analysis of chromosomal aberrations in human leukocytes. Biometrische Zeitschrift 18(6): 427-451.
  14. Janardan KG, Schaeffer DJ (1977) Models for the analysis of chromosomal aberrations in human leukocytes. Biometrical Journal 19(8): 599-612.
  15. Catcheside DG, Lea DE, Thoday JM (1946) Types of chromosome structural change induced by the irradiation on Tradescantia microspores. J Genet 47: 113-136.
  16. Catcheside DG, Lea DE, Thoday JM (1946) The production of chromosome structural changes in Tradescantia microspores in relation to dosage, intensity and temperature.  J Genet 47: 137-149.
  17. Falls LW, Williford WO, Carter MC (1971) Probability distributions for thunderstorm activity at Cape Kennedy, Florida. Journal of Applied Meteorology 10(1): 97-104.
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