ISSN: 2378-315X BBIJ

Biometrics & Biostatistics International Journal
Research Article
Volume 3 Issue 6 - 2016
On Poisson-Akash Distribution and its Applications
Rama Shanker1*, Hagos Fesshaye2 and Teklay Tesfazghi3
1Department of Statistics, Eritrea Institute of Technology, Eritrea
2Department of Economics, College of Business and Economics, Eritrea
3Department of Computer Engineering, Eritrea Institute of Technology, Eritrea
Received:April 25, 2016 | Published: May 11, 2016
*Corresponding author: Rama Shanker, Department of Statistics, Eritrea Institute of Technology, Asmara, Eritrea, Email:
Citation: Shanker R, Fesshaye H, Tesfazghi T (2016) On Poisson-Akash Distribution and its Applications. Biom Biostat Int J 3(6): 00075. DOI:10.15406/bbij.2016.03.00075

Abstract

A simple and interesting method for finding moments of ‘Poisson-Akash distribution (PAD)’ of Shanker [1], a Poisson mixture of Akash distribution introduced by Shanker [2] has been suggested. The first two moments about origin and the variance of PAD has been obtained and presented. The applications and the goodness of fit of PAD has been discussed using data-sets relating to ecology genetics, and thunderstorms and the fit has been compared with Poisson and Poisson-Lindley distribution, a Poisson mixture of Lindley [3] distribution, introduced by Sankaran [4] and the goodness of fit of PAD shows satisfactory fit in most of data-sets.

Keywords: Akash distribution; Poisson-Akash distribution; Lindley distribution; Poisson-Lindley distribution; Compounding; Moments; Estimation of parameter; Goodness of fit

Introduction

The probability mass function of Poisson-Akash distribution (PAD) having parameter θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde haaa@383A@  given by

P( X=x )= θ 3 θ 2 +2 x 2 +3x+( θ 2 +2θ+3 ) ( θ+1 ) x+3 ;x=0,1,2,...,θ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiuam aabmaabaGaamiwaiabg2da9iaadIhaaiaawIcacaGLPaaacqGH9aqp daWcaaqaaiabeI7aXnaaCaaabeqcfasaaiaaiodaaaaajuaGbaGaeq iUde3aaWbaaeqajuaibaGaaGOmaaaajuaGcqGHRaWkcaaIYaaaaiab gwSixpaalaaabaGaamiEamaaCaaabeqcfasaaiaaikdaaaqcfaOaey 4kaSIaaG4maiaadIhacqGHRaWkdaqadaqaaiabeI7aXnaaCaaabeqc fasaaiaaikdaaaqcfaOaey4kaSIaaGOmaiabeI7aXjabgUcaRiaaio daaiaawIcacaGLPaaaaeaadaqadaqaaiabeI7aXjabgUcaRiaaigda aiaawIcacaGLPaaadaahaaqabKqbGeaacaWG4bGaey4kaSIaaG4maa aaaaqcfaOaaGPaVlaaykW7caGG7aGaamiEaiabg2da9iaaicdacaGG SaGaaGymaiaacYcacaaIYaGaaiilaiaac6cacaGGUaGaaiOlaiaacY cacqaH4oqCcqGH+aGpcaaIWaaaaa@6FFF@ (1.1)

has been introduced by Shanker [1] for modeling various count data-sets. The PAD arises from Poisson distribution when its parameter λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4UdW gaaa@3838@ follows one parameter Akash distribution introduced by Shanker [2] having probability density function

f( λ,θ )= θ 3 θ 2 +2 ( 1+ λ 2 ) e θλ ;λ>0,θ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aabmaabaGaeq4UdWMaaiilaiaaykW7cqaH4oqCaiaawIcacaGLPaaa caaMe8UaaGPaVlabg2da9iaaysW7caaMc8+aaSaaaeaacqaH4oqCda ahaaqabKqbGeaacaaIZaaaaaqcfayaaiabeI7aXnaaCaaabeqcfasa aiaaikdaaaqcfaOaey4kaSIaaGPaVlaaikdaaaGaaGPaVpaabmaaba GaaGPaVlaaigdacqGHRaWkcaaMc8Uaeq4UdW2aaWbaaeqajuaibaGa aGOmaaaaaKqbakaawIcacaGLPaaacaaMc8UaaGPaVlaadwgadaahaa qabKqbGeaacqGHsislcqaH4oqCcaaMc8Uaeq4UdWgaaKqbakaaykW7 caaMc8Uaai4oaiabeU7aSjabg6da+iaaicdacaGGSaGaaGPaVlabeI 7aXjabg6da+iaaicdaaaa@7280@ (1.2)

We have

P( X=x )= 0 e λ λ x x! θ 3 θ 2 +2 ( 1+ λ 2 ) e θλ dλ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiuam aabmaabaGaamiwaiabg2da9iaadIhaaiaawIcacaGLPaaacqGH9aqp daWdXbqaamaalaaabaGaamyzamaaCaaabeqcfasaaiabgkHiTiabeU 7aSbaajuaGcqaH7oaBdaahaaqabKqbGeaacaWG4baaaaqcfayaaiaa dIhacaGGHaaaaaqcfasaaiaaicdaaeaacqGHEisPaKqbakabgUIiYd GaeyyXIC9aaSaaaeaacqaH4oqCdaahaaqabKqbGeaacaaIZaaaaaqc fayaaiabeI7aXnaaCaaabeqcfasaaiaaikdaaaqcfaOaey4kaSIaaG OmaaaadaqadaqaaiaaigdacqGHRaWkcqaH7oaBdaahaaqabKqbGeaa caaIYaaaaaqcfaOaayjkaiaawMcaaiaadwgadaahaaqabKqbGeaacq GHsislcqaH4oqCcaaMc8Uaeq4UdWgaaKqbakaadsgacqaH7oaBaaa@66FE@ (1.3)

= θ 3 ( θ 2 +2 )x! 0 e ( θ+1 )λ [ λ x + λ x+2 ] dλ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyypa0 ZaaSaaaeaacqaH4oqCdaahaaqabKqbGeaacaaIZaaaaaqcfayaamaa bmaabaGaeqiUde3aaWbaaeqajuaibaGaaGOmaaaajuaGcqGHRaWkca aIYaaacaGLOaGaayzkaaGaaGPaVlaadIhacaGGHaaaamaapehabaGa amyzamaaCaaabeqcfasaaiabgkHiTKqbaoaabmaajuaibaGaeqiUde Naey4kaSIaaGymaaGaayjkaiaawMcaaiaaykW7cqaH7oaBaaqcfa4a amWaaeaacqaH7oaBdaahaaqabKqbGeaacaWG4baaaKqbakabgUcaRi abeU7aSnaaCaaabeqcfasaaiaadIhacqGHRaWkcaaIYaaaaaqcfaOa ay5waiaaw2faaaqcfasaaiaaicdaaeaacqGHEisPaKqbakabgUIiYd GaaGPaVlaadsgacqaH7oaBaaa@650B@

= θ 3 θ 2 +2 x 2 +3x+( θ 2 +2θ+3 ) ( θ+1 ) x+3 ;x=0,1,2,...,θ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyypa0 ZaaSaaaeaacqaH4oqCdaahaaqabKqbGeaacaaIZaaaaaqcfayaaiab eI7aXnaaCaaabeqcfasaaiaaikdaaaqcfaOaey4kaSIaaGOmaaaacq GHflY1daWcaaqaaiaadIhadaahaaqabKqbGeaacaaIYaaaaKqbakab gUcaRiaaiodacaWG4bGaey4kaSYaaeWaaeaacqaH4oqCdaahaaqabK qbGeaacaaIYaaaaKqbakabgUcaRiaaikdacqaH4oqCcqGHRaWkcaaI ZaaacaGLOaGaayzkaaaabaWaaeWaaeaacqaH4oqCcqGHRaWkcaaIXa aacaGLOaGaayzkaaWaaWbaaeqajuaibaGaamiEaiabgUcaRiaaioda aaaaaKqbakaaykW7caaMc8Uaai4oaiaadIhacqGH9aqpcaaIWaGaai ilaiaaigdacaGGSaGaaGOmaiaacYcacaGGUaGaaiOlaiaac6cacaGG SaGaeqiUdeNaeyOpa4JaaGimaaaa@6AC1@ (1.4)

This is the probability mass function of Poisson-Akash distribution (PAD)”.

It has been shown by Shanker [2] that the Akash 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, and a gamma (3, 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 +2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacqaH4oqCdaahaaqabKqbGeaacaaIYaaaaaqcfayaaiabeI7aXnaa CaaabeqcfasaaiaaikdaaaqcfaOaey4kaSIaaGOmaaaaaaa@3ED2@ and 2 θ 2 +2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacaaIYaaabaGaeqiUde3aaWbaaeqajuaibaGaaGOmaaaajuaGcqGH RaWkcaaIYaaaaaaa@3C3E@ respectively. Shanker [2] has discussed its mathematical and statistical properties including its shape, 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 the estimation of parameter and applications for modeling lifetime data from engineering and biomedical science.

Sankaran [3] 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@ (1.5)

by compounding Poisson distribution with Lindley distribution when the parameter λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4UdW gaaa@3838@ of Poisson distribution follows Lindley distribution, introduced by Lindley [5] having probability density function (p.d.f)

f( λ,θ )= θ 2 θ+1 ( 1+λ ) e θλ ;λ>0,θ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aabmaabaGaeq4UdWMaaiilaiabeI7aXbGaayjkaiaawMcaaiabg2da 9maalaaabaGaeqiUde3aaWbaaeqajuaibaGaaGOmaaaaaKqbagaacq aH4oqCcqGHRaWkcaaIXaaaamaabmaabaGaaGymaiabgUcaRiabeU7a SbGaayjkaiaawMcaaiaaykW7caWGLbWaaWbaaeqajuaibaGaeyOeI0 IaeqiUdeNaaGPaVlaaygW7cqaH7oaBaaqcfaOaaGPaVlaaykW7caGG 7aGaeq4UdWMaeyOpa4JaaGimaiaacYcacaaMc8UaeqiUdeNaeyOpa4 JaaGimaaaa@6162@ (1.6)

In this paper a simple and interesting method for finding moments of Poisson-Akash distribution (PAD) introduced by Shanker [5] has been suggested and hence the first two moments about origin and the variance has been presented. It seems that not much work has been done on the applications of PAD so far for count data arising in various fields of knowledge. The applications and goodness of fit of PAD have been discussed with various count data from ecology, genetics and thunderstorms and the goodness of fit of PAD has been compared with Poisson distribution and Poisson-Lindley distribution (PLD). The goodness of fit of PAD shows satisfactory fit in most of the data-sets.

Moments of Pad

Using (1.3), the th moment about origin of PAD (1.1) can be obtained as

μ r =E[ E( X r |λ ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaKqbGeaacaWGYbaajuaGbeaadaahaaqabeaacWaGGBOmGika aiabg2da9iaadweadaWadaqaaiaadweadaqadaqaaiaadIfadaahaa qabKqbGeaacaWGYbaaaKqbakaacYhacqaH7oaBaiaawIcacaGLPaaa aiaawUfacaGLDbaaaaa@4893@ = θ 3 θ 2 +2 0 [ x=0 x r e λ λ x x! ] ( 1+ λ 2 ) e θλ dλ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyypa0 ZaaSaaaeaacqaH4oqCdaahaaqabKqbGeaacaaIZaaaaaqcfayaaiab eI7aXnaaCaaabeqcfasaaiaaikdaaaqcfaOaey4kaSIaaGOmaaaada WdXbqaamaadmaabaWaaabCaeaacaWG4bWaaWbaaeqabaGaamOCaaaa daWcaaqaaiaadwgadaahaaqabKqbGeaacqGHsislcqaH7oaBaaqcfa Oaeq4UdW2aaWbaaeqajuaibaGaamiEaaaaaKqbagaacaWG4bGaaiyi aaaaaKqbGeaacaWG4bGaeyypa0JaaGimaaqaaiabg6HiLcqcfaOaey yeIuoaaiaawUfacaGLDbaaaKqbGeaacaaIWaaabaGaeyOhIukajuaG cqGHRiI8amaabmaabaGaaGymaiabgUcaRiabeU7aSnaaCaaabeqcfa saaiaaikdaaaaajuaGcaGLOaGaayzkaaGaamyzamaaCaaabeqcfasa aiabgkHiTiabeI7aXjaaykW7cqaH7oaBaaqcfaOaamizaiabeU7aSb aa@6A92@

It is obvious 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 PAD (1.1) can be obtained as

μ 1 = θ 3 θ 2 +2 0 λ( 1+ λ 2 ) e θλ dλ= θ 2 +6 θ( θ 2 +2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaKqbGeaacaaIXaaajuaGbeaadaahaaqabeaacWaGGBOmGika aiabg2da9maalaaabaGaeqiUde3aaWbaaeqajuaibaGaaG4maaaaaK qbagaacqaH4oqCdaahaaqabKqbGeaacaaIYaaaaKqbakabgUcaRiaa ikdaaaWaa8qCaeaacqaH7oaBdaqadaqaaiaaigdacqGHRaWkcqaH7o aBdaahaaqabKqbGeaacaaIYaaaaaqcfaOaayjkaiaawMcaaaqcfasa aiaaicdaaeaacqGHEisPaKqbakabgUIiYdGaaGPaVlaadwgadaahaa qabKqbGeaacqGHsislcqaH4oqCcaaMc8Uaeq4UdWgaaKqbakaadsga cqaH7oaBcqGH9aqpdaWcaaqaaiabeI7aXnaaCaaabeqcfasaaiaaik daaaqcfaOaey4kaSIaaGOnaaqaaiabeI7aXnaabmaabaGaeqiUde3a aWbaaeqajuaibaGaaGOmaaaajuaGcqGHRaWkcaaIYaaacaGLOaGaay zkaaaaaaaa@6D98@

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 PAD (1.1) can be obtained as

μ 2 = θ 3 θ 2 +2 0 ( λ 2 +λ )( 1+ λ 2 ) e θλ dλ= θ 3 +2 θ 2 +6θ+24 θ 2 ( θ 2 +2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaKqbGeaacaaIYaaajuaGbeaadaahaaqabeaacWaGGBOmGika aiabg2da9maalaaabaGaeqiUde3aaWbaaeqajuaibaGaaG4maaaaaK qbagaacqaH4oqCdaahaaqabKqbGeaacaaIYaaaaKqbakabgUcaRiaa ikdaaaWaa8qCaeaadaqadaqaaiabeU7aSnaaCaaabeqcfasaaiaaik daaaqcfaOaey4kaSIaeq4UdWgacaGLOaGaayzkaaWaaeWaaeaacaaI XaGaey4kaSIaeq4UdW2aaWbaaeqajuaibaGaaGOmaaaaaKqbakaawI cacaGLPaaaaKqbGeaacaaIWaaabaGaeyOhIukajuaGcqGHRiI8aiaa ykW7caWGLbWaaWbaaeqajuaibaGaeyOeI0IaeqiUdeNaaGPaVlabeU 7aSbaajuaGcaWGKbGaeq4UdWMaeyypa0ZaaSaaaeaacqaH4oqCdaah aaqabKqbGeaacaaIZaaaaKqbakabgUcaRiaaikdacqaH4oqCdaahaa qabKqbGeaacaaIYaaaaKqbakabgUcaRiaaiAdacqaH4oqCcqGHRaWk caaIYaGaaGinaaqaaiabeI7aXnaaCaaabeqcfasaaiaaikdaaaqcfa 4aaeWaaeaacqaH4oqCdaahaaqabKqbGeaacaaIYaaaaKqbakabgUca RiaaikdaaiaawIcacaGLPaaaaaaaaa@7DED@

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 the fourth moments about origin of the Poisson distribution, the third and the fourth moments about origin of the PAD (1.1) can thus be obtained as

μ 3 = θ 4 +6 θ 3 +12 θ 2 +72θ+120 θ 3 ( θ 2 +2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaKqbGeaacaaIZaaajuaGbeaadaahaaqabeaacWaGGBOmGika aiabg2da9maalaaabaGaeqiUde3aaWbaaeqajuaibaGaaGinaaaaju aGcqGHRaWkcaaI2aGaeqiUde3aaWbaaeqajuaibaGaaG4maaaajuaG cqGHRaWkcaaIXaGaaGOmaiabeI7aXnaaCaaabeqcfasaaiaaikdaaa qcfaOaey4kaSIaaG4naiaaikdacqaH4oqCcqGHRaWkcaaIXaGaaGOm aiaaicdaaeaacqaH4oqCdaahaaqabKqbGeaacaaIZaaaaKqbaoaabm aabaGaeqiUde3aaWbaaeqajuaibaGaaGOmaaaajuaGcqGHRaWkcaaI YaaacaGLOaGaayzkaaaaaaaa@5CD2@ (2.4)

μ 4 = θ 5 +14 θ 4 +42 θ 3 +192 θ 2 +720θ+720 θ 4 ( θ 2 +2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaKqbGeaacaaI0aaajuaGbeaadaahaaqabeaacWaGGBOmGika aiabg2da9maalaaabaGaeqiUde3aaWbaaeqajuaibaGaaGynaaaaju aGcqGHRaWkcaaIXaGaaGinaiabeI7aXnaaCaaabeqcfasaaiaaisda aaqcfaOaey4kaSIaaGinaiaaikdacqaH4oqCdaahaaqabKqbGeaaca aIZaaaaKqbakabgUcaRiaaigdacaaI5aGaaGOmaiabeI7aXnaaCaaa beqcfasaaiaaikdaaaqcfaOaey4kaSIaaG4naiaaikdacaaIWaGaeq iUdeNaey4kaSIaaG4naiaaikdacaaIWaaabaGaeqiUde3aaWbaaeqa juaibaGaaGinaaaajuaGdaqadaqaaiabeI7aXnaaCaaabeqcfasaai aaikdaaaqcfaOaey4kaSIaaGOmaaGaayjkaiaawMcaaaaaaaa@64BF@ (2.5)

The variance of the PAD (1.1) can thus be obtained as

μ 2 = θ 5 + θ 4 +8 θ 3 +16 θ 2 +12θ+12 θ 2 ( θ 2 +2 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaKqbGeaacaaIYaaajuaGbeaacqGH9aqpdaWcaaqaaiabeI7a XnaaCaaabeqcfasaaiaaiwdaaaqcfaOaey4kaSIaeqiUde3aaWbaae qajuaibaGaaGinaaaajuaGcqGHRaWkcaaI4aGaeqiUde3aaWbaaeqa juaibaGaaG4maaaajuaGcqGHRaWkcaaIXaGaaGOnaiabeI7aXnaaCa aabeqcfasaaiaaikdaaaqcfaOaey4kaSIaaGymaiaaikdacqaH4oqC cqGHRaWkcaaIXaGaaGOmaaqaaiabeI7aXnaaCaaabeqcfasaaiaaik daaaqcfa4aaeWaaeaacqaH4oqCdaahaaqabKqbGeaacaaIYaaaaKqb akabgUcaRiaaikdaaiaawIcacaGLPaaadaahaaqabKqbGeaacaaIYa aaaaaaaaa@5E4D@ (2.6)

It has been shown by Shanker [5] that PAD (1.1) has increasing hazard rate, unimodal and always over-dispersed, and thus is a suitable model for count data which are over-dispersed

Parameter Estimation of Pad

Maximum Likelihood Estimate (MLE) of the Parameter: Let ( x 1 , x 2 ,..., x n ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacaWG4bWaaSbaaKqbGeaacaaIXaaajuaGbeaacaGGSaGaamiEamaa BaaajuaibaGaaGOmaaqcfayabaGaaiilaiaac6cacaGGUaGaaiOlai aacYcacaWG4bWaaSbaaKqbGeaacaWGUbaajuaGbeaaaiaawIcacaGL Paaaaaa@442B@ be a random sample of size n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaaaa@36E9@ from the PAD (1.1) 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 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Aaaaa@36E6@ 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=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamitaaaa@36C7@ of the PAD (1.1) can be given by

L= ( θ 3 θ 2 +2 ) n 1 ( θ+1 ) x=1 k f x ( x+3 ) x=1 k [ x 3 +3x+( θ 2 +2θ+3 ) ] f x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamitai abg2da9maabmaabaWaaSaaaeaacqaH4oqCdaahaaqabKqbGeaacaaI ZaaaaaqcfayaaiabeI7aXnaaCaaabeqcfasaaiaaikdaaaqcfaOaey 4kaSIaaGOmaaaaaiaawIcacaGLPaaadaahaaqabKqbGeaacaWGUbaa aKqbaoaalaaabaGaaGymaaqaamaabmaabaGaeqiUdeNaey4kaSIaaG ymaaGaayjkaiaawMcaamaaCaaabeqaamaaqahabaGaamOzamaaBaaa juaibaGaamiEaaqcfayabaWaaeWaaeaacaWG4bGaey4kaSIaaG4maa GaayjkaiaawMcaaaqcfasaaiaadIhacqGH9aqpcaaIXaaabaGaam4A aaqcfaOaeyyeIuoaaaaaamaarahabaWaamWaaeaacaWG4bWaaWbaae qajuaibaGaaG4maaaajuaGcqGHRaWkcaaIZaGaamiEaiabgUcaRmaa bmaabaGaeqiUde3aaWbaaeqajuaibaGaaGOmaaaajuaGcqGHRaWkca aIYaGaeqiUdeNaey4kaSIaaG4maaGaayjkaiaawMcaaaGaay5waiaa w2faamaaCaaabeqcfasaceaaDlGaamOzaKqbaoaaBaaajuaibaGaam iEaaqabaaaaaqaaiaadIhacqGH9aqpcaaIXaaabaGaam4AaaqcfaOa ey4dIunaaaa@736B@

The log likelihood function is thus obtained as

logL=nlog( θ 3 θ 2 +2 ) x=1 k f x ( x+3 ) log( θ+1 )+ x=1 k f x log[ x 2 +x+( θ 2 +2θ+3 ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaciiBai aac+gacaGGNbGaamitaiabg2da9iaad6gaciGGSbGaai4BaiaacEga daqadaqaamaalaaabaGaeqiUde3aaWbaaeqajuaibaGaaG4maaaaaK qbagaacqaH4oqCdaahaaqabKqbGeaacaaIYaaaaKqbakabgUcaRiaa ikdaaaaacaGLOaGaayzkaaGaeyOeI0YaaabCaeaacaWGMbWaaSbaaK qbGeaacaWG4baajuaGbeaadaqadaqaaiaadIhacqGHRaWkcaaIZaaa caGLOaGaayzkaaaajuaibaGaamiEaiabg2da9iaaigdaaeaacaWGRb aajuaGcqGHris5aiGacYgacaGGVbGaai4zamaabmaabaGaeqiUdeNa ey4kaSIaaGymaaGaayjkaiaawMcaaiabgUcaRmaaqahabaGaamOzam aaBaaajuaibaGaamiEaaqcfayabaGaciiBaiaac+gacaGGNbWaamWa aeaacaWG4bWaaWbaaeqajuaibaGaaGOmaaaajuaGcqGHRaWkcaWG4b Gaey4kaSYaaeWaaeaacqaH4oqCdaahaaqabKqbGeaacaaIYaaaaKqb akabgUcaRiaaikdacqaH4oqCcqGHRaWkcaaIZaaacaGLOaGaayzkaa aacaGLBbGaayzxaaaajuaibaGaamiEaiabg2da9iaaigdaaeaacaWG RbaajuaGcqGHris5aaaa@7CFE@

The first derivative of the log likelihood function is given by

dlogL dθ = 3n θ 2nθ θ 2 +2 n( x ¯ +3 ) θ+1 + x=1 k 2( θ+1 ) f x [ x 2 +x+( θ 2 +2θ+3 ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacaWGKbGaciiBaiaac+gacaGGNbGaamitaaqaaiaadsgacqaH4oqC aaGaeyypa0ZaaSaaaeaacaaIZaGaamOBaaqaaiabeI7aXbaacqGHsi sldaWcaaqaaiaaikdacaWGUbGaeqiUdehabaGaeqiUde3aaWbaaeqa juaibaGaaGOmaaaajuaGcqGHRaWkcaaIYaaaaiabgkHiTmaalaaaba GaamOBamaabmaabaGabmiEayaaraGaey4kaSIaaG4maaGaayjkaiaa wMcaaaqaaiabeI7aXjabgUcaRiaaigdaaaGaey4kaSYaaabCaeaada WcaaqaaiaaikdadaqadaqaaiabeI7aXjabgUcaRiaaigdaaiaawIca caGLPaaacaWGMbWaaSbaaKqbGeaacaWG4baajuaGbeaaaeaadaWada qaaiaadIhadaahaaqabKqbGeaacaaIYaaaaKqbakabgUcaRiaadIha cqGHRaWkdaqadaqaaiabeI7aXnaaCaaabeqcfasaaiaaikdaaaqcfa Oaey4kaSIaaGOmaiabeI7aXjabgUcaRiaaiodaaiaawIcacaGLPaaa aiaawUfacaGLDbaaaaaajuaibaGaamiEaiabg2da9iaaigdaaeaaca WGRbaajuaGcqGHris5aaaa@7602@

where x ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiEayaara aaaa@370B@ 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 PAD (1.1) 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 thus given by the solution of the non-linear equation

3n θ 2nθ θ 2 +2 n( x ¯ +3 ) θ+1 + x=1 k 2( θ+1 ) f x [ x 2 +x+( θ 2 +2θ+3 ) ] =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacaaIZaGaamOBaaqaaiabeI7aXbaacqGHsisldaWcaaqaaiaaikda caWGUbGaeqiUdehabaGaeqiUde3aaWbaaeqajuaibaGaaGOmaaaaju aGcqGHRaWkcaaIYaaaaiabgkHiTmaalaaabaGaamOBamaabmaabaGa bmiEayaaraGaey4kaSIaaG4maaGaayjkaiaawMcaaaqaaiabeI7aXj abgUcaRiaaigdaaaGaey4kaSYaaabCaeaadaWcaaqaaiaaikdadaqa daqaaiabeI7aXjabgUcaRiaaigdaaiaawIcacaGLPaaacaWGMbWaaS baaKqbGeaacaWG4baajuaGbeaaaeaadaWadaqaaiaadIhadaahaaqa bKqbGeaacaaIYaaaaKqbakabgUcaRiaadIhacqGHRaWkdaqadaqaai abeI7aXnaaCaaabeqcfasaaiaaikdaaaqcfaOaey4kaSIaaGOmaiab eI7aXjabgUcaRiaaiodaaiaawIcacaGLPaaaaiaawUfacaGLDbaaaa aajuaibaGaamiEaiabg2da9iaaigdaaeaacaWGRbaajuaGcqGHris5 aiabg2da9iaaicdaaaa@6F83@

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 to solve above non-linear equation to get maximum likelihood estimate of the parameter.

Method of moment estimate (MOME) of the Parameter:

Let ( x 1 , x 2 ,..., x n ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacaWG4bWaaSbaaKqbGeaacaaIXaaajuaGbeaacaGGSaGaamiEamaa BaaajuaibaGaaGOmaaqcfayabaGaaiilaiaac6cacaGGUaGaaiOlai aacYcacaWG4bWaaSbaaKqbGeaacaWGUbaajuaGbeaaaiaawIcacaGL Paaaaaa@442B@ be a random sample of size n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaaaa@36E9@ from the PAD (1.1). 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 PAD (1.1) is the solution of the following cubic equation

x ¯ θ 3 θ 2 +2 x ¯ θ6=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOabmiEay aaraGaeqiUde3aaWbaaeqajuaibaGaaG4maaaajuaGcqGHsislcqaH 4oqCdaahaaqabKqbGeaacaaIYaaaaKqbakabgUcaRiaaikdaceWG4b GbaebacqaH4oqCcqGHsislcaaI2aGaeyypa0JaaGimaaaa@46FD@

where x ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiEayaara aaaa@370B@ is the sample mean.

Applications and Goodness of Fit of Pad

When events seem to occur at random, Poisson distribution is a suitable statistical model. Examples of events where Poisson distribution is a suitable model includes the number of customers arriving at a service point, the number of telephone calls arriving at an exchange , the number of fatal traffic accidents per week in a given state, the number of radioactive particle emissions per unit of time, the number of meteorites that collide with a test satellite during a single orbit, the number of organisms per unit volume of some fluid, the number of defects per unit of some materials, the number of flaws per unit length of some wire, are some amongst others. Further, the conditions for using Poisson distribution are the independence of events and equality of mean and variance, which are rarely satisfied completely in biomedical science and thunderstorms due to the fact that the occurrences of successive events in biomedical science and thunderstorms are dependent. Negative binomial distribution is the appropriate choice for the situation where successive events are dependent but negative binomial distribution requires higher degree of over-dispersion Johnson et al [6].  In biomedical science and thunderstorms, these conditions are not fully satisfied. Generally, the count data in biomedical science and thunderstorms are either over-dispersed or under-dispersed. The main reason for selecting PLD and PAD to fit count data from biomedical science and thunderstorms are that these two distributions are always over-dispersed and PAD has some flexibility over PLD.

Applications in ecology

Ecology is the branch of biology which deals with the relations and interactions between organisms and their environment, including their organisms. Since the organisms and their environment in the nature are complex, dynamic, interdependent, mutually reactive and interrelated, ecology deals with the various principles which govern such relationship between organisms and their environment. Firstly Fisher et al. [7] discussed the applications of Logarithmic series distribution (LSD) to model count data in the science of ecology. Later, Kempton [8] who fitted the generalized form of Fisher’s Logarithmic series distribution (LSD) to model insect data and concluded that it gives a superior fit as compared to ordinary Logarithmic series distribution (LSD). He also concluded that it gives better explanation for the data having exceptionally long tail. Tripathi & Gupta [9] proposed another generalization of the Logarithmic series distribution (LSD) which is flexible to describe short-tailed as well as long-tailed data and fitted it to insect data and found that it gives better fit as compared to ordinary Logarithmic series distribution. Shanker [10], Mishra & Shanker [11] have discussed applications of generalized logarithmic series distributions (GLSD) to models data in ecology. Shanker & Hagos [12] have tried to fit PLD for data relating to ecology and observed that PLD gives satisfactory fit.

In this section we have tried to fit Poisson distribution (PD), Poisson -Lindley distribution (PLD) and Poisson-Akash distribution (PAD) to many count data from biological sciences using maximum likelihood estimates. The data were on haemocytometer yeast cell counts per square, on European red mites on apple leaves and European corn borers per plant.

It is obvious from above tables that in Table 4.1.1, PD gives better fit than PLD and PSD; in Table 4.1.2 PAD gives better fit than PD and PLD while in Table 4.1.3, PLD gives better fit than PD and PAD.

Applications in genetics

Genetics is the branch of biological science which deals with heredity and variation. Heredity includes those traits or characteristics which are transmitted from generation to generation, and is therefore fixed for a particular individual. Variation, on the other hand, is mainly of two types, namely hereditary and environmental. Hereditary variation refers to differences in inherited traits whereas environmental variations are those which are mainly due to environment. Much quantitative works seem to be done in genetics but so far no works has been done on fitting of PAD for count data in genetics. The segregation of chromosomes has been studied using statistical tool, mainly chi-square ( χ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4Xdm 2aaWbaaeqajuaibaGaaGOmaaaaaaa@3947@ ).  In the analysis of data observed on chemically induced chromosome aberrations in cultures of human leukocytes, Loeschke & Kohler [13] suggested the negative binomial distribution while Janardan & Schaeffer [14] suggested modified Poisson distribution. Shanker [10], Mishra & Shanker [11] have discussed applications of generalized Logarithmic series distributions (GLSD) to model data in mortality, ecology and genetics. Shanker & Hagos [12] have detailed study on the applications of PLD to model data from genetics.  In this section an attempt has been made to fit to data relating to genetics using PAD, PLD and PD using maximum likelihood estimate. Also an attempt has been made to fit PAD, PLD, and PD to the data of Catcheside et al. [15,16] in Tables 4.2.2, 4.2.3, and 4.2.4.

Number of Yeast Cells Per Square

Observed Frequency

Expected Frequency

PD

PLD

PAD

0

213

202.1

234

236.8

1

128

138.0

99.4

95.6

2

37

47.1

40.5

39.9

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@

16.6 6.7 2.7 1.7 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaigdacaaI2aGaaiOlaiaaiAdaaeaacaaI2aGaaiOlaiaa iEdaaeaacaaIYaGaaiOlaiaaiEdaaeaacaaIXaGaaiOlaiaaiEdaaa GaayzFaaaaaa@4127@

4

3

5

1

6

0

Total

400.0

400.0

400.0

Estimate of Parameter

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 ^ =2.260342 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpcaaIYaGaaiOlaiaaikdacaaI2aGaaGimaiaaioda caaI0aGaaGOmaaaa@3F2B@

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

10.08

11.04

14.68

d.f.

2

2

2

p-value

0.0065

0.004

0.0006

Table 4.1.1: Observed and expected number of Haemocytometer yeast cell counts per square observed by Gosset [18].

Number of Red Mites Per Leaf

Observed Frequency

Expected Frequency

PD

PLD

PAD

0

38

25.3

35.8

36.3

1

17

29.1

20.7

20.1

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
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@

6.1
3.2 1.6 0.8 0.7 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaiodacaGGUaGaaGOmaaqaaiaaigdacaGGUaGaaGOnaaqa aiaaicdacaGGUaGaaGioaaqaaiaaicdacaGGUaGaaG4naaaacaGL9b aaaaa@405D@

4

3

5

2

6

1

7+

0

Total

80

80.0

80.0

80.0

Estimate of Parameter

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 NbaKaacqGH9aqpqaaaaaaaaaWdbiaaigdacaGGUaGaaGOmaiaaiwda caaI1aGaaGioaiaaiMdacaaIXaaaaa@3F57@

θ ^ =1.620588 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpcaaIXaGaaiOlaiaaiAdacaaIYaGaaGimaiaaiwda caaI4aGaaGioaaaa@3F36@

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

18.27

2.47

2.07

d.f.

2

3

3

p-value

0.0001

0.4807

0.558

Table 4.1.2: Observed and expected number of red mites on Apple leaves.

Number of Corn- Borer Per Plant

Observed Frequency

Expected Frequency

PD

PLD

PAD

0

188

169.4

194.0

196.3

1

83

109.8

79.5

76.5

2

36

35.6

31.3

30.8

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@

12.4 4.9 3.1 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaigdacaaIYaGaaiOlaiaaisdaaeaacaaI0aGaaiOlaiaa iMdaaeaacaaIZaGaaiOlaiaaigdaaaGaayzFaaaaaa@3EED@

4

2

5

1

Total

324

324.0

324.0

324.0

Estimate of Parameter

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 ^ =2.345109 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpcaaIYaGaaiOlaiaaiodacaaI0aGaaGynaiaaigda caaIWaGaaGyoaaaa@3F30@

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

15.19

1.29

2.33

d.f.

2

2

2

p-value

0.0005

0.5247

0.3119

Table 4.1.3: Observed and expected number of European corn-borer of Mc Guire et al. [19].

It is obvious from the fitting of PAD, PLD, and PD that PAD gives much closer fit in Tables 4.2.1, 4.2.2 and 4.2.3 but in Table 4.2.4, PLD better fit than PD and PAD.

Number of Aberrations

Observed Frequency

Expected Frequency

PD

PLD

PAD

0

268

231.3

257.0

260.4

1

87

126.7

93.4

89.7

2

26

34.7

32.8

32.1

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
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@

11.5
4.1 1.4 0.5 0.3 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaisdacaGGUaGaaGymaaqaaiaaigdacaGGUaGaaGinaaqa aiaaicdacaGGUaGaaGynaaqaaiaaicdacaGGUaGaaG4maaaacaGL9b aaaaa@4054@

4

4

5

2

6

1

7+

3

Total

400

400.0

400.0

400.0

Estimate of Parameter

θ ^ =0.5475 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpqaaaaaaaaaWdbiaaicdacaGGUaGaaGynaiaaisda caaI3aGaaGynaaaa@3DD9@

θ ^ =2.380442 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpqaaaaaaaaaWdbiaaikdacaGGUaGaaG4maiaaiIda caaIWaGaaGinaiaaisdacaaIYaaaaa@3F4F@

θ ^ =2.659408 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpqaaaaaaaaaWdbiaaikdacaGGUaGaaGOnaiaaiwda caaI5aGaaGinaiaaicdacaaI4aaaaa@3F5A@

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

38.21

6.21

4.17

d.f.

2

3

3

p-value

0.0000

0.1018

0.2437

Table 4.2.1: Distribution of number of Chromatid aberrations (0.2 g chinon 1, 24 hours).

Class/Exposure
(μg|kg) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaaiikai abeY7aTjaadEgacaGG8bGaam4AaiaadEgacaGGPaaaaa@3D5B@

Observed Frequency

Expected Frequency

PD

PLD

PAD

0

413

374.0

405.7

409.5

1

124

177.4

133.6

128.7

2

42

42.1

42.6

42.1

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

13.9
4.6 1.5 0.7 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaisdacaGGUaGaaGOnaaqaaiaaigdacaGGUaGaaGynaaqa aiaaicdacaGGUaGaaG4naaaacaGL9baaaaa@3E32@

4

5

5

0

6

2

Total

601

601.0

601.0

601.0

Estimate of Parameter

θ ^ =0.47421 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpqaaaaaaaaaWdbiaaicdacaGGUaGaaGinaiaaiEda caaI0aGaaGOmaiaaigdaaaa@3E90@

θ ^ =2.685373 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpqaaaaaaaaaWdbiaaikdacaGGUaGaaGOnaiaaiIda caaI1aGaaG4maiaaiEdacaaIZaaaaa@3F5A@

θ ^ =2.915059 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpqaaaaaaaaaWdbiaaikdacaGGUaGaaGyoaiaaigda caaI1aGaaGimaiaaiwdacaaI5aaaaa@3F57@

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

48.17

1.34

0.29

d.f.

2

3

3

p-value

0.0000

0.7196

0.9619

Table 4.2.2: 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=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaaiikai abeY7aTjaadEgacaGG8bGaam4AaiaadEgacaGGPaaaaa@3D5B@

Observed Frequency

Expected Frequency

PD

PLD

PAD

0

200

172.5

191.8

194.1

1

57

95.4

70.3

67.6

2

30

26.4

24.9

24.5

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@

8.9 3.2 1.1 0.6 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaiIdacaGGUaGaaGyoaaqaaiaaiodacaGGUaGaaGOmaaqa aiaaigdacaGGUaGaaGymaaqaaiaaicdacaGGUaGaaGOnaaaacaGL9b aaaaa@4060@

4

4

5

0

6

2

Total

300

300.0

300.0

300.0

Estimate of Parameter

θ ^ =0.55333 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpqaaaaaaaaaWdbiaaicdacaGGUaGaaGynaiaaiwda caaIZaGaaG4maiaaiodaaaa@3E91@

θ ^ =2.353339 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpqaaaaaaaaaWdbiaaikdacaGGUaGaaG4maiaaiwda caaIZaGaaG4maiaaiodacaaI5aaaaa@3F54@

θ ^ =2.626739 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpqaaaaaaaaaWdbiaaikdacaGGUaGaaGOnaiaaikda caaI2aGaaG4naiaaiodacaaI5aaaaa@3F5B@

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

29.68

3.91

3.12

d.f.

2

2

2

p-value

0.0000

0.1415

0.2101

Table 4.2.3: 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=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaaiikai abeY7aTjaadEgacaGG8bGaam4AaiaadEgacaGGPaaaaa@3D5B@

Observed Frequency

Expected Frequency

PD

PLD

PAD

0

155

127.8

158.3

160.7

1

83

109.0

77.2

74.3

2

33

46.5

35.9

35.3

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

16.5
7.5 3.3 2.4 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaiEdacaGGUaGaaGynaaqaaiaaiodacaGGUaGaaG4maaqa aiaaikdacaGGUaGaaGinaaaacaGL9baaaaa@3E33@

4

11

5

3

6

1

Total

300

300.0

300.0

300.0

Estimate of Parameter

θ ^ =0.853333 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpqaaaaaaaaaWdbiaaicdacaGGUaGaaGioaiaaiwda caaIZaGaaG4maiaaiodacaaIZaaaaa@3F51@

θ ^ =1.617611 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpqaaaaaaaaaWdbiaaigdacaGGUaGaaGOnaiaaigda caaI3aGaaGOnaiaaigdacaaIXaaaaa@3F4F@

θ ^ =1.963313 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpqaaaaaaaaaWdbiaaigdacaGGUaGaaGyoaiaaiAda caaIZaGaaG4maiaaigdacaaIZaaaaa@3F52@

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

24.97

1.51

1.98

d.f.

2

3

3

p-value

0.0000

0.6799

0.5766

Table 4.2.4: 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@ .

Applications in thunderstorms

In thunderstorm activity, the occurrence of successive thunderstorm events (THE’s) is often dependent process which means that the occurrence of a THE indicates that the atmosphere is unstable and the conditions are favorable for the formation of further thunderstorm activity. The negative binomial distribution (NBD) is a possible alternative to the Poisson distribution when successive events are possibly dependent Johnson et al. [6]. The theoretical and empirical justification for using the NBD to describe THE activity has been fully explained and discussed by Falls et al. [17]. Further, for fitting Poisson distribution to the count data equality of mean and variance should be satisfied. Similarly, for fitting NBD to the count data, mean should be less than the variance. In THE, these conditions are not fully satisfied. As a model to describe the frequencies of thunderstorms (TH’s), given an occurrence of THE, the PAD can be considered because it is always over-dispersed Tables 4.3.1, 4.3.2, 4.3.3 and 4.3.4.

No. of Thunderstorms

Observed Frequency

Expected Frequency

PD

PLD

PAD

0

187

155.6

185.3

187.9

1

77

117.0

83.5

80.2

2

40

43.9

35.9

35.3

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

15.4

4

6

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

6.6 2.7 1.9 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaiAdacaGGUaGaaGOnaaqaaiaaikdacaGGUaGaaG4naaqa aiaaigdacaGGUaGaaGyoaaaacaGL9baaaaa@3E3A@

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 ^ =2.139736 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpcaaIYaGaaiOlaiaaigdacaaIZaGaaGyoaiaaiEda caaIZaGaaGOnaaaa@3F37@

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

31.93

1.43

1.35

d.f.

2

3

3

p-value

0.0000

0.6985

0.7173

Table 4.3.1: 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. [16].

No. of Thunderstorms

Observed Frequency

Expected Frequency

PD

PLD

PAD

0

177

142.3

177.7

180.0

1

80

124.4

88.0

84.7

2

47

54.3

41.5

40.9

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

19.4

4

9

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

8.9 7.1 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaiIdacaGGUaGaaGyoaaqaaiaaiEdacaGGUaGaaGymaaaa caGL9baaaaa@3C0D@

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 ^ =1.938989 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpcaaIXaGaaiOlaiaaiMdacaaIZaGaaGioaiaaiMda caaI4aGaaGyoaaaa@3F47@

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

39.74

5.15

5.02

d.f.

2

3

3

p-value

0.0000

0.1611

0.1703

Table 4.3.2: 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. [16].

No. of Thunderstorms

Observed Frequency

Expected Frequency

PD

PLD

PAD

0

185

151.8

184.8

187.5

1

89

122.9

87.2

83.9

2

30

49.7

39.3

38.6

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

17.5

4

10

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

7.6 5.9 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaiEdacaGGUaGaaGOnaaqaaiaaiwdacaGGUaGaaGyoaaaa caGL9baaaaa@3C0F@

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 ^ =2.038417 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpcaaIYaGaaiOlaiaaicdacaaIZaGaaGioaiaaisda caaIXaGaaG4naaaa@3F31@

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

49.49

5.03

4.69

d.f.

2

3

3

p-value

0.0000

0.1696

0.196

Table 4.3.3: 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. [16].

No. of Thunderstorms

Observed Frequency

Expected Frequency

PD

PLD

PAD

0

549

449.0

547.5

555.1

1

246

364.8

259.0

249.2

2

117

148.2

116.9

114.9

3

67

40.1

51.2

52.3

4

25

8.1 1.3 0.5 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaiIdacaGGUaGaaGymaaqaaiaaigdacaGGUaGaaG4maaqa aiaaicdacaGGUaGaaGynaaaacaGL9baaaaa@3E2D@

21.9

23.2

5

7

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

10.0 7.3 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaigdacaaIWaGaaiOlaiaaicdaaeaacaaI3aGaaiOlaiaa iodaaaGaayzFaaaaaa@3CB9@

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 ^ =2.033715 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpcaaIYaGaaiOlaiaaicdacaaIZaGaaG4maiaaiEda caaIXaGaaGynaaaa@3F2D@

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

119.45

9.60

9.40

d.f.

3

4

4

p-value

0.0000

0.0477

0.0518

Table 4.3.4: 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. [16].

It is obvious from the fitting of PAD, PLD, and PD that PAD gives much closer fit than PLD and PD in all data-sets relating to thunderstorms and hence PAD can be considered as an important model for modeling thunderstorms events.

References

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  16. Catcheside DG, Lea DE, Thoday JM (1946 b) 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 (1970) Probability distributions for thunderstorm activity at Cape Kennedy, Florida. Journal of Applied Meteorology 10: 97-104.
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