ISSN: 2378-315X BBIJ

Biometrics & Biostatistics International Journal
Research Article
Volume 4 Issue 3 - 2016
On Poisson-Amarendra Distribution and Its Applications
Rama Shanker1* and Hagos Fesshaye2
1Department of Statistics, Eritrea Institute of Technology, Eritrea
2Department of Economics, College of Business and Economics, Eritrea
Received: July 19, 2016 | Published: August 27, 2016
*Corresponding author: Rama Shanker, Department of Statistics, Eritrea Institute of Technology, Asmara, Eritrea, Email:
Citation: Shanker R, Fesshaye H (2016) On Poisson-Amarendra Distribution and Its Applications. Biom Biostat Int J 4(3): 00099. DOI: 10.15406/bbij.2016.04.00099

Abstract

In this paper a simple method for obtaining moments of Poisson-Amarendra distribution (PAD) introduced by Shanker [1] has been suggested and hence the first four moments about origin and the variance has been given. The applications and the goodness of fit of the PAD have been discussed with data-sets relating to ecology, genetics and thunderstorms and the fit is compared with Poisson distribution, Poisson-Lindley distribution (PLD) introduced by Sankaran [2] and Poisson-Sujatha distribution introduced by Shanker [3] and the fit of PAD shows satisfactory fit in most of data-sets.

Keywords: Amarendra distribution; Poisson-Amarendra distribution; Poisson-Lindley distribution; Poisson-Sujatha distribution; Moments; Estimation of parameter; Goodness of fit

Introduction

The Poisson-Amarendra distribution (PAD) defined by its probability mass function (p.m.f.)

P( X=x )= θ 4 θ 3 + θ 2 +2θ+6 x 3 +( θ+7 ) x 2 +( θ 2 +5θ+15 )x+( θ 3 +4 θ 2 +7θ+10 ) ( θ+1 ) x+4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiuam aabmaabaGaamiwaiabg2da9iaadIhaaiaawIcacaGLPaaacqGH9aqp daWcaaqaaiabeI7aXnaaCaaabeqcfasaaiaaisdaaaaajuaGbaGaeq iUde3aaWbaaeqajuaibaGaaG4maaaajuaGcqGHRaWkcqaH4oqCdaah aaqabKqbGeaacaaIYaaaaKqbakabgUcaRiaaikdacqaH4oqCcqGHRa WkcaaI2aaaamaalaaabaGaamiEamaaCaaabeqcfasaaiaaiodaaaqc faOaey4kaSYaaeWaaeaacqaH4oqCcqGHRaWkcaaI3aaacaGLOaGaay zkaaGaamiEamaaCaaabeqcfasaaiaaikdaaaqcfaOaey4kaSYaaeWa aeaacqaH4oqCdaahaaqabKqbGeaacaaIYaaaaKqbakabgUcaRiaaiw dacqaH4oqCcqGHRaWkcaaIXaGaaGynaaGaayjkaiaawMcaaiaadIha cqGHRaWkdaqadaqaaiabeI7aXnaaCaaabeqcfasaaiaaiodaaaqcfa Oaey4kaSIaaGinaiabeI7aXnaaCaaabeqcfasaaiaaikdaaaqcfaOa ey4kaSIaaG4naiabeI7aXjabgUcaRiaaigdacaaIWaaacaGLOaGaay zkaaaabaWaaeWaaeaacqaH4oqCcqGHRaWkcaaIXaaacaGLOaGaayzk aaWaaWbaaeqajuaibaGaamiEaiabgUcaRiaaisdaaaaaaKqbakaayk W7aaa@7E4F@

                                 

;x=0,1,2,...,θ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaai4oai aadIhacqGH9aqpcaaIWaGaaiilaiaaigdacaGGSaGaaGOmaiaacYca caGGUaGaaiOlaiaac6cacaGGSaGaeqiUdeNaeyOpa4JaaGimaaaa@43C5@ (1.1)

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

f( λ;θ )= θ 4 θ 3 + θ 2 +2θ+6 ( 1+λ+ λ 2 + λ 3 ) e θλ ;λ>0,θ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aabmaabaGaeq4UdWMaai4oaiabeI7aXbGaayjkaiaawMcaaiabg2da 9maalaaabaGaeqiUde3aaWbaaeqajuaibaGaaGinaaaaaKqbagaacq aH4oqCdaahaaqabKqbGeaacaaIZaaaaKqbakabgUcaRiabeI7aXnaa CaaabeqcfasaaiaaikdaaaqcfaOaey4kaSIaaGOmaiabeI7aXjabgU caRiaaiAdaaaWaaeWaaeaacaaIXaGaey4kaSIaeq4UdWMaey4kaSIa eq4UdW2aaWbaaeqajuaibaGaaGOmaaaajuaGcqGHRaWkcqaH7oaBda ahaaqabKqbGeaacaaIZaaaaaqcfaOaayjkaiaawMcaaiaadwgadaah aaqabKqbGeaacqGHsislcqaH4oqCcaaMc8Uaeq4UdWgaaKqbakaayk W7caaMc8UaaGPaVlaaykW7caGG7aGaeq4UdWMaeyOpa4JaaGimaiaa cYcacaaMc8UaaGPaVlabeI7aXjabg6da+iaaicdaaaa@7486@               (1.2)

We have

P( X=x )= 0 e λ λ x x! θ 4 θ 3 + θ 2 +2θ+6 ( 1+λ+ λ 2 + λ 3 ) e θλ dλ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiuam aabmaabaGaamiwaiabg2da9iaadIhaaiaawIcacaGLPaaacqGH9aqp daWdXbqaamaalaaabaGaamyzamaaCaaabeqcfasaaiabgkHiTiabeU 7aSbaajuaGcqaH7oaBdaahaaqabKqbGeaacaWG4baaaaqcfayaaiaa dIhacaGGHaaaaaqcfasaaiaaicdaaeaacqGHEisPaKqbakabgUIiYd GaeyyXIC9aaSaaaeaacqaH4oqCdaahaaqabKqbGeaacaaI0aaaaaqc fayaaiabeI7aXnaaCaaabeqcfasaaiaaiodaaaqcfaOaey4kaSIaeq iUde3aaWbaaeqajuaibaGaaGOmaaaajuaGcqGHRaWkcaaIYaGaeqiU deNaey4kaSIaaGOnaaaadaqadaqaaiaaigdacqGHRaWkcqaH7oaBcq GHRaWkcqaH7oaBdaahaaqabKqbGeaacaaIYaaaaKqbakabgUcaRiab eU7aSnaaCaaabeqcfasaaiaaiodaaaaajuaGcaGLOaGaayzkaaGaam yzamaaCaaabeqcfasaaiabgkHiTiabeI7aXjaaykW7cqaH7oaBaaqc faOaamizaiabeU7aSbaa@7551@                     (1.3)

= θ 4 ( θ 3 + θ 2 +2θ+6 )x! 0 λ x ( 1+λ+ λ 2 + λ 3 ) e ( θ+1 )λ dλ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyypa0 ZaaSaaaeaacqaH4oqCdaahaaqabKqbGeaacaaI0aaaaaqcfayaamaa bmaabaGaeqiUde3aaWbaaeqajuaibaGaaG4maaaajuaGcqGHRaWkcq aH4oqCdaahaaqabKqbGeaacaaIYaaaaKqbakabgUcaRiaaikdacqaH 4oqCcqGHRaWkcaaI2aaacaGLOaGaayzkaaGaaGPaVlaadIhacaGGHa aaamaapehabaGaeq4UdW2aaWbaaeqajuaibaGaamiEaaaaaeaacaaI WaaabaGaeyOhIukajuaGcqGHRiI8amaabmaabaGaaGymaiabgUcaRi abeU7aSjabgUcaRiabeU7aSnaaCaaabeqcfasaaiaaikdaaaqcfaOa ey4kaSIaeq4UdW2aaWbaaeqajuaibaGaaG4maaaaaKqbakaawIcaca GLPaaacaWGLbWaaWbaaeqajuaibaGaeyOeI0scfa4aaeWaaKqbGeaa cqaH4oqCcqGHRaWkcaaIXaaacaGLOaGaayzkaaGaaGPaVlabeU7aSb aajuaGcaWGKbGaeq4UdWgaaa@6F8A@

= θ 4 θ 3 + θ 2 +2θ+6 x 3 +( θ+7 ) x 2 +( θ 2 +5θ+15 )x+( θ 3 +4 θ 2 +7θ+10 ) ( θ+1 ) x+4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyypa0 ZaaSaaaeaacqaH4oqCdaahaaqabKqbGeaacaaI0aaaaaqcfayaaiab eI7aXnaaCaaabeqcfasaaiaaiodaaaqcfaOaey4kaSIaeqiUde3aaW baaeqajuaibaGaaGOmaaaajuaGcqGHRaWkcaaIYaGaeqiUdeNaey4k aSIaaGOnaaaacqGHflY1daWcaaqaaiaadIhadaahaaqabKqbGeaaca aIZaaaaKqbakabgUcaRmaabmaabaGaeqiUdeNaey4kaSIaaG4naaGa ayjkaiaawMcaaiaadIhadaahaaqabKqbGeaacaaIYaaaaKqbakabgU caRmaabmaabaGaeqiUde3aaWbaaeqajuaibaGaaGOmaaaajuaGcqGH RaWkcaaI1aGaeqiUdeNaey4kaSIaaGymaiaaiwdaaiaawIcacaGLPa aacaWG4bGaey4kaSYaaeWaaeaacqaH4oqCdaahaaqabKqbGeaacaaI ZaaaaKqbakabgUcaRiaaisdacqaH4oqCdaahaaqabKqbGeaacaaIYa aaaKqbakabgUcaRiaaiEdacqaH4oqCcqGHRaWkcaaIXaGaaGimaaGa ayjkaiaawMcaaaqaamaabmaabaGaeqiUdeNaey4kaSIaaGymaaGaay jkaiaawMcaamaaCaaabeqcfasaaiaadIhacqGHRaWkcaaI0aaaaaaa juaGcaaMc8UaaGPaVdaa@7CE6@ .                                   

                                                                    

;x=0,1,2,...,θ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaai4oai aadIhacqGH9aqpcaaIWaGaaiilaiaaigdacaGGSaGaaGOmaiaacYca caGGUaGaaiOlaiaac6cacaGGSaGaeqiUdeNaeyOpa4JaaGimaaaa@43C5@ (1.4)

which is the Poisson-Amarendra distribution (PAD).

 It has been shown by Shanker [4] that Amarendra distribution is a four component mixture of exponential ( θ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacqaH4oqCaiaawIcacaGLPaaaaaa@39C3@ distribution, a gamma ( 2,θ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacaaIYaGaaiilaiabeI7aXbGaayjkaiaawMcaaaaa@3B2F@ distribution, a gamma ( 3,θ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacaaIZaGaaiilaiabeI7aXbGaayjkaiaawMcaaaaa@3B30@  distribution  and a gamma ( 4,θ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacaaI0aGaaiilaiabeI7aXbGaayjkaiaawMcaaaaa@3B31@ distribution with their mixing proportions θ 3 θ 3 + θ 2 +2θ+6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacqaH4oqCdaahaaqabKqbGeaacaaIZaaaaaqcfayaaiabeI7aXnaa CaaabeqcfasaaiaaiodaaaqcfaOaey4kaSIaeqiUde3aaWbaaeqaju aibaGaaGOmaaaajuaGcqGHRaWkcaaIYaGaeqiUdeNaey4kaSIaaGOn aaaaaaa@465E@ , θ 2 θ 3 + θ 2 +2θ+6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacqaH4oqCdaahaaqabKqbGeaacaaIYaaaaaqcfayaaiabeI7aXnaa CaaabeqcfasaaiaaiodaaaqcfaOaey4kaSIaeqiUde3aaWbaaeqaju aibaGaaGOmaaaajuaGcqGHRaWkcaaIYaGaeqiUdeNaey4kaSIaaGOn aaaaaaa@465D@ , 2θ θ 3 + θ 2 +2θ+6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacaaIYaGaeqiUdehabaGaeqiUde3aaWbaaeqajuaibaGaaG4maaaa juaGcqGHRaWkcqaH4oqCdaahaaqabKqbGeaacaaIYaaaaKqbakabgU caRiaaikdacqaH4oqCcqGHRaWkcaaI2aaaaaaa@457F@ , and 6 θ 3 + θ 2 +2θ+6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacaaI2aaabaGaeqiUde3aaWbaaeqajuaibaGaaG4maaaajuaGcqGH RaWkcqaH4oqCdaahaaqabKqbGeaacaaIYaaaaKqbakabgUcaRiaaik dacqaH4oqCcqGHRaWkcaaI2aaaaaaa@43CD@  respectively. Shanker [4] has discussed its various mathematical and statistical properties including its shape for different values of its parameter, moment generating function, moments, skewness, kurtosis, hazard rate function, mean residual life function, stochastic orderings, mean deviations, Bonferroni and Lorenz curves, amongst others. Further, Shanker [4] has also discussed the estimation of its parameter using maximum likelihood estimation and method of moments along with applications for modeling lifetime data and observed that it gives much closer fit than Akash, Shanker and Sujatha distributions introduced by Shanker [5-7], Lindley [8] and exponential distributions. It would be worth mentioning that Shanker [5-7] has proposed  Akash, Shanker and Sujatha, distributions along with their various mathematical and statistical properties to model lifetime data arising from engineering and biomedical sciences and showed that these distributions provide much closer fit than Lindley and exponential distributions.

The Poisson-Lindley distribution (PLD) defined by its p.m.f.

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 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiUdehaaa@37AC@  > 0. (1.5)

has been introduced by Sankaran [2] to model count data. The PLD is a Poisson mixture of Lindley [8] 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 [8] distribution with its p.d.f.

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 & Hagos [9] has done detailed study about applications of Poisson-Lindley distribution for modeling count data from biological sciences and showed that it gives better fit than Poisson-distribution. Shanker et al. [10] discussed the comparative study of zero-truncated Poisson and Poisson-Lindley distributions and observed that in majority of data sets zero-truncated Poisson-Lindley distribution gives better fit.

Shanker [11] obtained Poisson-Sujatha distribution (PSD) having p.m.f.

P( X=x )= θ 3 θ 2 +θ+2 x 2 +( θ+4 )x+( θ 2 +3θ+4 ) ( 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 iUde3aaWbaaeqajuaibaGaaGOmaaaajuaGcqGHRaWkcqaH4oqCcqGH RaWkcaaIYaaaaiabgwSixpaalaaabaGaamiEamaaCaaabeqcfasaai aaikdaaaqcfaOaey4kaSYaaeWaaeaacqaH4oqCcqGHRaWkcaaI0aaa caGLOaGaayzkaaGaamiEaiabgUcaRmaabmaabaGaeqiUde3aaWbaae qajuaibaGaaGOmaaaajuaGcqGHRaWkcaaIZaGaeqiUdeNaey4kaSIa aGinaaGaayjkaiaawMcaaaqaamaabmaabaGaeqiUdeNaey4kaSIaaG ymaaGaayjkaiaawMcaamaaCaaabeqcfasaaiaadIhacqGHRaWkcaaI ZaaaaaaajuaGcaaMc8UaaGPaVlaacUdacaWG4bGaeyypa0JaaGimai aacYcacaaIXaGaaiilaiaaikdacaGGSaGaaiOlaiaac6cacaGGUaGa aiilaiabeI7aXjabg6da+iaaicdaaaa@76BB@ (1.7)

by compounding Poisson distribution with Sujatha distribution, introduced by Shanker [7] having p.d.f.

f( x;θ )= θ 3 θ 2 +θ+2 ( 1+x+ x 2 ) e θx ;x>0,θ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aabmaabaGaamiEaiaacUdacqaH4oqCaiaawIcacaGLPaaacqGH9aqp daWcaaqaaiabeI7aXnaaCaaabeqcfasaaiaaiodaaaaajuaGbaGaeq iUde3aaWbaaeqajuaibaGaaGOmaaaajuaGcqGHRaWkcqaH4oqCcqGH RaWkcaaIYaaaamaabmaabaGaaGymaiabgUcaRiaadIhacqGHRaWkca WG4bWaaWbaaeqajuaibaGaaGOmaaaaaKqbakaawIcacaGLPaaacaWG LbWaaWbaaeqajuaibaGaeyOeI0IaeqiUdeNaamiEaaaajuaGcaaMc8 UaaGPaVlaaykW7caaMc8Uaai4oaiaadIhacqGH+aGpcaaIWaGaaiil aiaaykW7caaMc8UaeqiUdeNaeyOpa4JaaGimaaaa@6643@                              (1.8)

Sujatha distribution introduced by Shanker [7] is a better model than exponential and Lindley distributions for modeling lifetime data from biomedical science and engineering. Further, Shanker & Hagos [11] has detailed study about applications of Poisson-Sujatha distribution (PSD) for modeling count data from biological science and observed that it gives better fit than Poisson-Lindley (PLD) and Poisson-distribution. Shanker & Hagos [12,13] have obtained the size-biased Poisson-Sujatha distribution (SBPSD) and zero-truncated Poisson-Sujatha distribution (ZTPSD) and discussed their various mathematical and statistical properties, estimation of their parameter and applications. Further, Shanker & Hagos [14] have detailed study regarding applications of zero-truncated Poisson (ZTPD), zero-truncated Poisson-Lindley distribution (ZTPLD), and zero-truncated Poisson-Sujatha distribution (ZTPSD) for modeling data-sets excluding zero counts from demography and biological sciences and concluded that in majority of data-sets ZTPSD gives better fit than ZTPD and ZTPLD.

In this paper a simple method for obtaining moments of Poisson-Amarendra distribution (PAD) introduced by Shanker [1] has been suggested and hence the first four 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.  The PAD has been fitted to some data-sets relating to ecology ,genetics and thunderstorms and its goodness of fit has been compared with Poisson distribution (PD), Poisson-Lindley distribution (PLD), Poisson-Sujatha distribution (PSD).

Moments of Pad

Using (1.3), the r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOCaa aa@377B@ th moment about origin of PAD (1.1) can be obtained as

μ r =E[ E( X r |λ ) ]= θ 4 θ 3 + θ 2 +2θ+6 0 [ x=0 x r e λ λ x x! ] ( 1+λ+ λ 2 + λ 3 ) e θλ dλ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaKqbGeaacaWGYbaajuaGbeaadaahaaqabKqbGeaacWaGGBOm GikaaKqbakabg2da9iaadweadaWadaqaaiaadweadaqadaqaaiaadI fadaahaaqabKqbGeaacaWGYbaaaKqbakaacYhacqaH7oaBaiaawIca caGLPaaaaiaawUfacaGLDbaacqGH9aqpdaWcaaqaaiabeI7aXnaaCa aabeqcfasaaiaaisdaaaaajuaGbaGaeqiUde3aaWbaaeqajuaibaGa aG4maaaajuaGcqGHRaWkcqaH4oqCdaahaaqabKqbGeaacaaIYaaaaK qbakabgUcaRiaaikdacqaH4oqCcqGHRaWkcaaI2aaaamaapehabaWa amWaaeaadaaeWbqaaiaadIhadaahaaqabeaacaWGYbaaamaalaaaba GaamyzamaaCaaabeqcfasaaiabgkHiTiabeU7aSbaajuaGcqaH7oaB daahaaqabKqbGeaacaWG4baaaaqcfayaaiaadIhacaGGHaaaaaqcfa saaiaadIhacqGH9aqpcaaIWaaabaGaeyOhIukajuaGcqGHris5aaGa ay5waiaaw2faaaqcfasaaiaaicdaaeaacqGHEisPaKqbakabgUIiYd WaaeWaaeaacaaIXaGaey4kaSIaeq4UdWMaey4kaSIaeq4UdW2aaWba aeqajuaibaGaaGOmaaaajuaGcqGHRaWkcqaH7oaBdaahaaqabKqbGe aacaaIZaaaaaqcfaOaayjkaiaawMcaaiaadwgadaahaaqabKqbGeaa cqGHsislcqaH4oqCcaaMc8Uaeq4UdWgaaKqbakaadsgacqaH7oaBaa a@8BB0@ (2.1)

clearly that the expression under the bracket in (2.1) is the r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOCaa aa@377B@ 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 mean of the Poisson distribution, the mean of the PAD (1.1) can be obtained as

μ 1 = θ 4 θ 3 + θ 2 +2θ+6 0 λ( 1+λ+ λ 2 + λ 3 ) e θλ dλ= θ 3 +2 θ 2 +6θ+24 θ( θ 3 + θ 2 +2θ+6 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaKqbGeaacaaIXaaajuaGbeaadaahaaqabKqbGeaacWaGGBOm GikaaKqbakabg2da9maalaaabaGaeqiUde3aaWbaaeqajuaibaGaaG inaaaaaKqbagaacqaH4oqCdaahaaqabKqbGeaacaaIZaaaaKqbakab gUcaRiabeI7aXnaaCaaabeqcfasaaiaaikdaaaqcfaOaey4kaSIaaG OmaiabeI7aXjabgUcaRiaaiAdaaaWaa8qCaeaacqaH7oaBdaqadaqa aiaaigdacqGHRaWkcqaH7oaBcqGHRaWkcqaH7oaBdaahaaqabKqbGe aacaaIYaaaaKqbakabgUcaRiabeU7aSnaaCaaabeqcfasaaiaaioda aaaajuaGcaGLOaGaayzkaaGaaGPaVdqcfasaaiaaicdaaeaacqGHEi sPaKqbakabgUIiYdGaamyzamaaCaaabeqcfasaaiabgkHiTiabeI7a XjaaykW7cqaH7oaBaaqcfaOaamizaiabeU7aSjabg2da9maalaaaba GaeqiUde3aaWbaaeqajuaibaGaaG4maaaajuaGcqGHRaWkcaaIYaGa eqiUde3aaWbaaeqajuaibaGaaGOmaaaajuaGcqGHRaWkcaaI2aGaeq iUdeNaey4kaSIaaGOmaiaaisdaaeaacqaH4oqCdaqadaqaaiabeI7a XnaaCaaabeqcfasaaiaaiodaaaqcfaOaey4kaSIaeqiUde3aaWbaae qajuaibaGaaGOmaaaajuaGcqGHRaWkcaaIYaGaeqiUdeNaey4kaSIa aGOnaaGaayjkaiaawMcaaaaaaaa@8D33@

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 4 θ 3 + θ 2 +2θ+6 0 ( λ 2 +λ )( 1+λ+ λ 2 + λ 3 ) e θλ dλ= θ 4 +4 θ 3 +12 θ 2 +48θ+120 θ 2 ( θ 3 + θ 2 +2θ+6 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaKqbGeaacaaIYaaajuaGbeaadaahaaqabKqbGeaacWaGGBOm GikaaKqbakabg2da9maalaaabaGaeqiUde3aaWbaaeqajuaibaGaaG inaaaaaKqbagaacqaH4oqCdaahaaqabKqbGeaacaaIZaaaaKqbakab gUcaRiabeI7aXnaaCaaabeqcfasaaiaaikdaaaqcfaOaey4kaSIaaG OmaiabeI7aXjabgUcaRiaaiAdaaaWaa8qCaeaadaqadaqaaiabeU7a SnaaCaaabeqcfasaaiaaikdaaaqcfaOaey4kaSIaeq4UdWgacaGLOa GaayzkaaWaaeWaaeaacaaIXaGaey4kaSIaeq4UdWMaey4kaSIaeq4U dW2aaWbaaeqajuaibaGaaGOmaaaajuaGcqGHRaWkcqaH7oaBdaahaa qabKqbGeaacaaIZaaaaaqcfaOaayjkaiaawMcaaiaaykW7aKqbGeaa caaIWaaabaGaeyOhIukajuaGcqGHRiI8aiaadwgadaahaaqabKqbGe aacqGHsislcqaH4oqCcaaMc8Uaeq4UdWgaaKqbakaadsgacqaH7oaB cqGH9aqpdaWcaaqaaiabeI7aXnaaCaaabeqcfasaaiaaisdaaaqcfa Oaey4kaSIaaGinaiabeI7aXnaaCaaabeqcfasaaiaaiodaaaqcfaOa ey4kaSIaaGymaiaaikdacqaH4oqCdaahaaqabKqbGeaacaaIYaaaaK qbakabgUcaRiaaisdacaaI4aGaeqiUdeNaey4kaSIaaGymaiaaikda caaIWaaabaGaeqiUde3aaWbaaeqajuaibaGaaGOmaaaajuaGdaqada qaaiabeI7aXnaaCaaabeqcfasaaiaaiodaaaqcfaOaey4kaSIaeqiU de3aaWbaaeqajuaibaGaaGOmaaaajuaGcqGHRaWkcaaIYaGaeqiUde Naey4kaSIaaGOnaaGaayjkaiaawMcaaaaaaaa@9BAB@  

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 PAD (1.1) are obtained as

μ 3 = θ 5 +8 θ 4 +30 θ 3 +120 θ 2 +480θ+720 θ 3 ( θ 3 + θ 2 +2θ+6 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaKqbGeaacaaIZaaajuaGbeaadaahaaqabKqbGeaacWaGGBOm GikaaKqbakabg2da9maalaaabaGaeqiUde3aaWbaaeqajuaibaGaaG ynaaaajuaGcqGHRaWkcaaI4aGaeqiUde3aaWbaaeqajuaibaGaaGin aaaajuaGcqGHRaWkcaaIZaGaaGimaiabeI7aXnaaCaaabeqcfasaai aaiodaaaqcfaOaey4kaSIaaGymaiaaikdacaaIWaGaeqiUde3aaWba aeqajuaibaGaaGOmaaaajuaGcqGHRaWkcaaI0aGaaGioaiaaicdacq aH4oqCcqGHRaWkcaaI3aGaaGOmaiaaicdaaeaacqaH4oqCdaahaaqa bKqbGeaacaaIZaaaaKqbaoaabmaabaGaeqiUde3aaWbaaeqajuaiba GaaG4maaaajuaGcqGHRaWkcqaH4oqCdaahaaqabKqbGeaacaaIYaaa aKqbakabgUcaRiaaikdacqaH4oqCcqGHRaWkcaaI2aaacaGLOaGaay zkaaaaaaaa@6C44@                                                                     

μ 4 = θ 6 +16 θ 5 +84 θ 4 +360 θ 3 +1680 θ 2 +5040θ+5040 θ 4 ( θ 3 + θ 2 +2θ+6 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaKqbGeaacaaI0aaajuaGbeaadaahaaqabKqbGeaacWaGGBOm GikaaKqbakabg2da9maalaaabaGaeqiUde3aaWbaaeqajuaibaGaaG OnaaaajuaGcqGHRaWkcaaIXaGaaGOnaiabeI7aXnaaCaaabeqcfasa aiaaiwdaaaqcfaOaey4kaSIaaGioaiaaisdacqaH4oqCdaahaaqabK qbGeaacaaI0aaaaKqbakabgUcaRiaaiodacaaI2aGaaGimaiabeI7a XnaaCaaabeqcfasaaiaaiodaaaqcfaOaey4kaSIaaGymaiaaiAdaca aI4aGaaGimaiabeI7aXnaaCaaabeqcfasaaiaaikdaaaqcfaOaey4k aSIaaGynaiaaicdacaaI0aGaaGimaiabeI7aXjabgUcaRiaaiwdaca aIWaGaaGinaiaaicdaaeaacqaH4oqCdaahaaqabKqbGeaacaaI0aaa aKqbaoaabmaabaGaeqiUde3aaWbaaeqajuaibaGaaG4maaaajuaGcq GHRaWkcqaH4oqCdaahaaqabKqbGeaacaaIYaaaaKqbakabgUcaRiaa ikdacqaH4oqCcqGHRaWkcaaI2aaacaGLOaGaayzkaaaaaaaa@75AC@

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

μ 2 = θ 7 +4 θ 6 +14 θ 5 +58 θ 4 +144 θ 3 +156 θ 2 +240θ+144 θ 2 ( θ 3 + θ 2 +2θ+6 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaKqbGeaacaaIYaaajuaGbeaacqGH9aqpdaWcaaqaaiabeI7a XnaaCaaabeqcfasaaiaaiEdaaaqcfaOaey4kaSIaaGinaiabeI7aXn aaCaaabeqcfasaaiaaiAdaaaqcfaOaey4kaSIaaGymaiaaisdacqaH 4oqCdaahaaqabKqbGeaacaaI1aaaaKqbakabgUcaRiaaiwdacaaI4a GaeqiUde3aaWbaaeqajuaibaGaaGinaaaajuaGcqGHRaWkcaaIXaGa aGinaiaaisdacqaH4oqCdaahaaqabKqbGeaacaaIZaaaaKqbakabgU caRiaaigdacaaI1aGaaGOnaiabeI7aXnaaCaaabeqcfasaaiaaikda aaqcfaOaey4kaSIaaGOmaiaaisdacaaIWaGaeqiUdeNaey4kaSIaaG ymaiaaisdacaaI0aaabaGaeqiUde3aaWbaaeqajuaibaGaaGOmaaaa juaGdaqadaqaaiabeI7aXnaaCaaabeqcfasaaiaaiodaaaqcfaOaey 4kaSIaeqiUde3aaWbaaeqajuaibaGaaGOmaaaajuaGcqGHRaWkcaaI YaGaeqiUdeNaey4kaSIaaGOnaaGaayjkaiaawMcaamaaCaaabeqcfa saaiaaikdaaaaaaaaa@75AE@

Parameter Estimation

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=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOBaa aa@3777@ from the PAD (1.1) and let f x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBa aaleaacaWG4baabeaaaaa@380A@ 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=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamitaaaa@36C7@ of the PAD (1.1) is given by

L= ( θ 4 θ 3 + θ 2 +2θ+6 ) n 1 ( θ+1 ) x=1 k f x ( x+4 ) x=1 k [ x 3 +( θ+7 ) x 2 +( θ 2 +5θ+15 )x +( θ 3 +4 θ 2 +7θ+10 ) ] f x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamitai abg2da9maabmaabaWaaSaaaeaacqaH4oqCdaahaaqabKqbGeaacaaI 0aaaaaqcfayaaiabeI7aXnaaCaaabeqcfasaaiaaiodaaaqcfaOaey 4kaSIaeqiUde3aaWbaaeqajuaibaGaaGOmaaaajuaGcqGHRaWkcaaI YaGaeqiUdeNaey4kaSIaaGOnaaaaaiaawIcacaGLPaaadaahaaqabK qbGeaacaWGUbaaaKqbaoaalaaabaGaaGymaaqaamaabmaabaGaeqiU deNaey4kaSIaaGymaaGaayjkaiaawMcaamaaCaaabeqaamaaqahaba GaamOzamaaBaaajuaibaGaamiEaaqcfayabaWaaeWaaeaacaWG4bGa ey4kaSIaaGinaaGaayjkaiaawMcaaaqcfasaaiaadIhacqGH9aqpca aIXaaabaGaam4AaaqcfaOaeyyeIuoaaaaaamaarahabaWaamWaaqaa beqaaiaadIhadaahaaqabKqbGeaacaaIZaaaaKqbakabgUcaRmaabm aabaGaeqiUdeNaey4kaSIaaG4naaGaayjkaiaawMcaaiaadIhadaah aaqabKqbGeaacaaIYaaaaKqbakabgUcaRmaabmaabaGaeqiUde3aaW baaeqajuaibaGaaGOmaaaajuaGcqGHRaWkcaaI1aGaeqiUdeNaey4k aSIaaGymaiaaiwdaaiaawIcacaGLPaaacaWG4baabaGaey4kaSYaae WaaeaacqaH4oqCdaahaaqabKqbGeaacaaIZaaaaKqbakabgUcaRiaa isdacqaH4oqCdaahaaqabKqbGeaacaaIYaaaaKqbakabgUcaRiaaiE dacqaH4oqCcqGHRaWkcaaIXaGaaGimaaGaayjkaiaawMcaaaaacaGL BbGaayzxaaWaaWbaaeqajuaibaGaamOzaKqbaoaaBaaajuaibaGaam iEaaqabaaaaaqaaiaadIhacqGH9aqpcaaIXaaabaGaam4AaaqcfaOa ey4dIunaaaa@91F5@

The log likelihood function is thus obtained as

logL=nlog( θ 4 θ 3 + θ 2 +2θ+6 ) x=1 k f x ( x+4 ) log( θ+1 )+ x=1 k f x log[ x 3 +( θ+7 ) x 2 +( θ 2 +5θ+15 )x+( θ 3 +4 θ 2 +7θ+10 ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaciiBai aac+gacaGGNbGaamitaiabg2da9iaad6gaciGGSbGaai4BaiaacEga daqadaqaamaalaaabaGaeqiUde3aaWbaaeqajuaibaGaaGinaaaaaK qbagaacqaH4oqCdaahaaqabKqbGeaacaaIZaaaaKqbakabgUcaRiab eI7aXnaaCaaabeqcfasaaiaaikdaaaqcfaOaey4kaSIaaGOmaiabeI 7aXjabgUcaRiaaiAdaaaaacaGLOaGaayzkaaGaeyOeI0YaaabCaeaa caWGMbWaaSbaaKqbGeaacaWG4baajuaGbeaadaqadaqaaiaadIhacq GHRaWkcaaI0aaacaGLOaGaayzkaaaajuaibaGaamiEaiabg2da9iaa igdaaeaacaWGRbaajuaGcqGHris5aiGacYgacaGGVbGaai4zamaabm aabaGaeqiUdeNaey4kaSIaaGymaaGaayjkaiaawMcaaiabgUcaRmaa qahabaGaamOzamaaBaaajuaibaGaamiEaaqcfayabaGaciiBaiaac+ gacaGGNbWaamWaaeaacaWG4bWaaWbaaeqajuaibaGaaG4maaaajuaG cqGHRaWkdaqadaqaaiabeI7aXjabgUcaRiaaiEdaaiaawIcacaGLPa aacaWG4bWaaWbaaeqajuaibaGaaGOmaaaajuaGcqGHRaWkdaqadaqa aiabeI7aXnaaCaaabeqcfasaaiaaikdaaaqcfaOaey4kaSIaaGynai abeI7aXjabgUcaRiaaigdacaaI1aaacaGLOaGaayzkaaGaamiEaiab gUcaRmaabmaabaGaeqiUde3aaWbaaeqajuaibaGaaG4maaaajuaGcq GHRaWkcaaI0aGaeqiUde3aaWbaaeqajuaibaGaaGOmaaaajuaGcqGH RaWkcaaI3aGaeqiUdeNaey4kaSIaaGymaiaaicdaaiaawIcacaGLPa aaaiaawUfacaGLDbaaaKqbGeaacaWG4bGaeyypa0JaaGymaaqaaiaa dUgaaKqbakabggHiLdaaaa@9D21@

The first derivative of the log likelihood function is given by

dlogL dθ = 4n θ n( 3 θ 2 +2θ+2 ) ( θ 3 + θ 2 +2θ+6 ) n( x ¯ +4 ) θ+1 + x=1 k [ x 2 +( 2θ+5 )x+( 3 θ 2 +8θ+7 ) ] f x [ x 3 +( θ+7 ) x 2 +( θ 2 +5θ+15 )x+( θ 3 +4 θ 2 +7θ+10 ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacaWGKbGaciiBaiaac+gacaGGNbGaamitaaqaaiaadsgacqaH4oqC aaGaeyypa0ZaaSaaaeaacaaI0aGaamOBaaqaaiabeI7aXbaacqGHsi sldaWcaaqaaiaad6gadaqadaqaaiaaiodacqaH4oqCdaahaaqabKqb GeaacaaIYaaaaKqbakabgUcaRiaaikdacqaH4oqCcqGHRaWkcaaIYa aacaGLOaGaayzkaaaabaWaaeWaaeaacqaH4oqCdaahaaqabKqbGeaa caaIZaaaaKqbakabgUcaRiabeI7aXnaaCaaabeqcfasaaiaaikdaaa qcfaOaey4kaSIaaGOmaiabeI7aXjabgUcaRiaaiAdaaiaawIcacaGL PaaaaaGaeyOeI0YaaSaaaeaacaWGUbWaaeWaaeaaceWG4bGbaebacq GHRaWkcaaI0aaacaGLOaGaayzkaaaabaGaeqiUdeNaey4kaSIaaGym aaaacaaMc8UaaGPaVlaaykW7cqGHRaWkdaaeWbqaamaalaaabaWaam WaaeaacaWG4bWaaWbaaeqajuaibaGaaGOmaaaajuaGcqGHRaWkdaqa daqaaiaaikdacqaH4oqCcqGHRaWkcaaI1aaacaGLOaGaayzkaaGaam iEaiabgUcaRmaabmaabaGaaG4maiabeI7aXnaaCaaabeqcfasaaiaa ikdaaaqcfaOaey4kaSIaaGioaiabeI7aXjabgUcaRiaaiEdaaiaawI cacaGLPaaaaiaawUfacaGLDbaacaWGMbWaaSbaaKqbGeaacaWG4baa juaGbeaaaeaadaWadaqaaiaadIhadaahaaqabKqbGeaacaaIZaaaaK qbakabgUcaRmaabmaabaGaeqiUdeNaey4kaSIaaG4naaGaayjkaiaa wMcaaiaadIhadaahaaqabKqbGeaacaaIYaaaaKqbakabgUcaRmaabm aabaGaeqiUde3aaWbaaeqajuaibaGaaGOmaaaajuaGcqGHRaWkcaaI 1aGaeqiUdeNaey4kaSIaaGymaiaaiwdaaiaawIcacaGLPaaacaWG4b Gaey4kaSYaaeWaaeaacqaH4oqCdaahaaqabKqbGeaacaaIZaaaaKqb akabgUcaRiaaisdacqaH4oqCdaahaaqabKqbGeaacaaIYaaaaKqbak abgUcaRiaaiEdacqaH4oqCcqGHRaWkcaaIXaGaaGimaaGaayjkaiaa wMcaaaGaay5waiaaw2faaaaaaKqbGeaacaWG4bGaeyypa0JaaGymaa qaaiaadUgaaKqbakabggHiLdaaaa@B646@

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 PAD (1.1) is the solution of dlogL dθ =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacaWGKbGaciiBaiaac+gacaGGNbGaamitaaqaaiaadsgacqaH4oqC aaGaeyypa0JaaGimaaaa@3F7D@  and is the solution of the following  non-linear equation

4n θ n( 3 θ 2 +2θ+2 ) ( θ 3 + θ 2 +2θ+6 ) n( x ¯ +4 ) θ+1 + x=1 k [ x 2 +( 2θ+5 )x+( 3 θ 2 +8θ+7 ) ] f x [ x 3 +( θ+7 ) x 2 +( θ 2 +5θ+15 )x+( θ 3 +4 θ 2 +7θ+10 ) ] =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacaaI0aGaamOBaaqaaiabeI7aXbaacqGHsisldaWcaaqaaiaad6ga daqadaqaaiaaiodacqaH4oqCdaahaaqabKqbGeaacaaIYaaaaKqbak abgUcaRiaaikdacqaH4oqCcqGHRaWkcaaIYaaacaGLOaGaayzkaaaa baWaaeWaaeaacqaH4oqCdaahaaqabKqbGeaacaaIZaaaaKqbakabgU caRiabeI7aXnaaCaaabeqcfasaaiaaikdaaaqcfaOaey4kaSIaaGOm aiabeI7aXjabgUcaRiaaiAdaaiaawIcacaGLPaaaaaGaeyOeI0YaaS aaaeaacaWGUbWaaeWaaeaaceWG4bGbaebacqGHRaWkcaaI0aaacaGL OaGaayzkaaaabaGaeqiUdeNaey4kaSIaaGymaaaacqGHRaWkdaaeWb qaamaalaaabaWaamWaaeaacaWG4bWaaWbaaeqajuaibaGaaGOmaaaa juaGcqGHRaWkdaqadaqaaiaaikdacqaH4oqCcqGHRaWkcaaI1aaaca GLOaGaayzkaaGaamiEaiabgUcaRmaabmaabaGaaG4maiabeI7aXnaa CaaabeqcfasaaiaaikdaaaqcfaOaey4kaSIaaGioaiabeI7aXjabgU caRiaaiEdaaiaawIcacaGLPaaaaiaawUfacaGLDbaacaWGMbWaaSba aKqbGeaacaWG4baajuaGbeaaaeaadaWadaqaaiaadIhadaahaaqabK qbGeaacaaIZaaaaKqbakabgUcaRmaabmaabaGaeqiUdeNaey4kaSIa aG4naaGaayjkaiaawMcaaiaadIhadaahaaqabKqbGeaacaaIYaaaaK qbakabgUcaRmaabmaabaGaeqiUde3aaWbaaeqajuaibaGaaGOmaaaa juaGcqGHRaWkcaaI1aGaeqiUdeNaey4kaSIaaGymaiaaiwdaaiaawI cacaGLPaaacaWG4bGaey4kaSYaaeWaaeaacqaH4oqCdaahaaqabKqb GeaacaaIZaaaaKqbakabgUcaRiaaisdacqaH4oqCdaahaaqabKqbGe aacaaIYaaaaKqbakabgUcaRiaaiEdacqaH4oqCcqGHRaWkcaaIXaGa aGimaaGaayjkaiaawMcaaaGaay5waiaaw2faaaaaaKqbGeaacaWG4b Gaeyypa0JaaGymaaqaaiaadUgaaKqbakabggHiLdGaeyypa0JaaGim aaaa@AB26@           

This non-linear equation can be solved using 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 the above equation for estimating the parameter θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde haaa@383A@ .

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=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOBaa aa@3777@  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 biquadratic equation

x ¯ θ 4 +( x ¯ 1 ) θ 3 +2( x ¯ 1 ) θ 2 +6( x ¯ 1 )θ24=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOabmiEay aaraGaeqiUde3aaWbaaeqajuaibaGaaGinaaaajuaGcqGHRaWkdaqa daqaaiqadIhagaqeaiabgkHiTiaaigdaaiaawIcacaGLPaaacqaH4o qCdaahaaqabKqbGeaacaaIZaaaaKqbakabgUcaRiaaikdadaqadaqa aiqadIhagaqeaiabgkHiTiaaigdaaiaawIcacaGLPaaacqaH4oqCda ahaaqabKqbGeaacaaIYaaaaKqbakabgUcaRiaaiAdadaqadaqaaiqa dIhagaqeaiabgkHiTiaaigdaaiaawIcacaGLPaaacqaH4oqCcqGHsi slcaaIYaGaaGinaiabg2da9iaaicdaaaa@585D@

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

The Poisson distribution is a suitable statistical model for the situations where events are independent and mean equals variance, which is unrealistic in most of data sets in biological science and thunderstorms. Further, the negative binomial distribution is a possible alternative to the Poisson distribution when successive events are possibly dependent Johnson et al. [15] but for fitting negative binomial distribution (NBD) to the count data, mean must be less than the variance. In biological science and thunderstorms, these conditions are not fully satisfied. Generally, the count data in biological science and thunderstorms are either over-dispersed or under-dispersed. The main reason for selecting PAD, PSD, and PLD to fit data from biological science and thunderstorms are that these distributions are always over-dispersed and PAD has some flexibility over PSD and 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. The organisms and their environment in the nature are complex, dynamic, interdependent, mutually reactive and interrelated. Ecology deals with various principles which govern such relationship between organisms and their environment. Fisher et al. [16] firstly discussed the applications of Logarithmic series distribution (LSD) to model count data in ecology. Later, Kempton [17] 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 [18] 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. Mishra & Shanker [19] have discussed applications of generalized logarithmic series distributions (GLSD) to models data in ecology. Shanker & Hagos [10] have tried to fit PLD for data relating to ecology and observed that PLD gives satisfactory fit. Shanker & Hagos [11] has discussed applications of PSD to model count data from biological science and concluded that PSD gives superior fit than PLD in majority of data.

In this section an attempt has been made to fit Poisson distribution (PD), Poisson -Lindley distribution (PLD), Poisson-Sujatha distribution (PSD) and Poisson-Amarendra 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 1, PD gives better fit than PLD, PSD and PAD; in Table 2 PAD gives better fit than PD, PLD, and PSD while in Table 3, PSD gives better fit than PD, PLD and PAD.

Number of Yeast Cells Per Square

Observed Frequency

Expected Frequency

PD

PLD

PSD

PAD

0

213

202.1

234.0

233.7

233.7

1

128

138.0

99.4

99.6

98.4

2

37

47.1

40.5

41.0

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@

16.7 6.5 2.4 1.3 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaigdacaaI2aGaaiOlaiaaiEdaaeaacaaI2aGaaiOlaiaa iwdaaeaacaaIYaGaaiOlaiaaisdaaeaacaaIXaGaaiOlaiaaiodaaa GaayzFaaaaaa@411F@

4

3

5

1

6

0

Total

400.0

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 ^ =2.373052 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpcaaIYaGaaiOlaiaaiodacaaI3aGaaG4maiaaicda caaI1aGaaGOmaaaa@3F2E@

θ ^ =2.759978 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpcaaIYaGaaiOlaiaaiEdacaaI1aGaaGyoaiaaiMda caaI3aGaaGioaaaa@3F47@

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

10.08

11.04

10.86

12.01

d.f.

2

2

2

2

p-value

0.0065

0.004

0.0044

0.0025

Table 1: Observed and expected number of Haemocytometer yeast cell counts per square observed by ‘Student’ 1907.

Number Mites Per Leaf

Observed Frequency

Expected Frequency

PD

PLD

PSD

PAD

0

38

25.3

35.8

35.3

35.3

1

17

29.1

20.7

20.9

20.8

2

10

16.7

11.4

11.6

11.7

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
3.1 1.5 0.7 0.8 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaiodacaGGUaGaaGymaaqaaiaaigdacaGGUaGaaGynaaqa aiaaicdacaGGUaGaaG4naaqaaiaaicdacaGGUaGaaGioaaaacaGL9b aaaaa@405B@

6.2
3.1 1.5 0.8 0.6 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaiodacaGGUaGaaGymaaqaaiaaigdacaGGUaGaaGynaaqa aiaaicdacaGGUaGaaGioaaqaaiaaicdacaGGUaGaaGOnaaaacaGL9b aaaaa@405A@

4

3

5

2

6

1

7+

0

Total

80

80.0

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 ^ =1.264683 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpcaaIXaGaaiOlaiaaikdacaaI2aGaaGinaiaaiAda caaI4aGaaG4maaaa@3F36@

θ ^ =2.04047 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpcaaIYaGaaiOlaiaaicdacaaI0aGaaGimaiaaisda caaI3aaaaa@3E6F@

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

18.27

2.47

2.52

2.41

d.f.

2

3

3

3

p-value

0.0001

0.4807

0.4719

0.4918

Table 2: Observed and expected number of red mites on Apple leaves.

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 to model count data in genetics but so far no works has been done on fitting of PAD to 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=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Xdm2aaW baaSqabeaacaaIYaaaaaaa@3896@ ).  In the analysis of data observed on chemically induced chromosome aberrations in cultures of human leukocytes, Loeschke & Kohler [20] suggested the negative binomial distribution while Janardan & Schaeffer [21] suggested modified Poisson distribution. Mishra & Shanker [19] have discussed applications of generalized Logarithmic series distributions (GLSD) to model data in mortality, ecology and genetics. Shanker & Hagos [9] have detailed study on the applications of PLD to model data from genetics. Shanker & Hagos [11] has detailed study on modeling of count data in genetics using PSD.  In this section an attempt has been made to fit to PAD, PSD, PLD and PD to data from genetics using maximum likelihood estimate. Also an attempt has been made to fit PAD, PSD, PLD, and PD to the data of Catcheside et al.[22] in Table [5-7].

It is obvious that in Table 4 & 7, PLD gives better fit than PD, PSD and PAD; in Table 5 & 6, PAD gives better fit than PD, PLD, and PSD.

Number of Bores Per Plant

Observed Frequency

Expected Frequency

PD

PLD

PSD

PAD

0

188

169.4

194.0

193.6

194.2

1

83

109.8

79.5

79.6

78.6

2

36

35.6

31.3

31.6

31.6

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 4.5 2.6 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaigdacaaIYaGaaiOlaiaaigdaaeaacaaI0aGaaiOlaiaa iwdaaeaacaaIYaGaaiOlaiaaiAdaaaGaayzFaaaaaa@3EEA@

12.3 4.7 2.6 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaigdacaaIYaGaaiOlaiaaiodaaeaacaaI0aGaaiOlaiaa iEdaaeaacaaIYaGaaiOlaiaaiAdaaaGaayzFaaaaaa@3EEE@

4

2

5

1

Total

324

324.0

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 ^ =2.471701 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpcaaIYaGaaiOlaiaaisdacaaI3aGaaGymaiaaiEda caaIWaGaaGymaaaa@3F2E@

θ ^ =2.858180 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpcaaIYaGaaiOlaiaaiIdacaaI1aGaaGioaiaaigda caaI4aGaaGimaaaa@3F38@

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

15.19

1.29

1.16

1.4

d.f.

2

2

2

2

p-value

0.0005

0.5247

0.5599

0.4966

Table 3: Observed and expected number of European corn- borer of Mc. Guire et al. [23].

Number of Aberrations

Observed Frequency

Expected Frequency

PD

PLD

PSD

PAD

0

268

231.3

257.0

257.6

259.0

1

87

126.7

93.4

93.0

91.9

2

26

34.7

32.8

32.7

32.5

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@

11.2
3.7 1.2 0.4 0.2 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaiodacaGGUaGaaG4naaqaaiaaigdacaGGUaGaaGOmaaqa aiaaicdacaGGUaGaaGinaaqaaiaaicdacaGGUaGaaGOmaaaacaGL9b aaaaa@4055@

11.3
3.8 1.3 0.2 0.0 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaiodacaGGUaGaaGioaaqaaiaaigdacaGGUaGaaG4maaqa aiaaicdacaGGUaGaaGOmaaqaaiaaicdacaGGUaGaaGimaaaacaGL9b aaaaa@4053@

4

4

5

2

6

1

7+

3

Total

400

400.0

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 ^ =2.829241 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpcaaIYaGaaiOlaiaaiIdacaaIYaGaaGyoaiaaikda caaI0aGaaGymaaaa@3F34@

θ ^ =3.216733 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpcaaIZaGaaiOlaiaaikdacaaIXaGaaGOnaiaaiEda caaIZaGaaG4maaaa@3F31@

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

38.21

6.21

6.28

6.5

d.f.

2

3

3

3

p-value

0

0.1018

0.0987

0.0897

Table 4: 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=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 Maam4zaiaacYhacaWGRbGaam4zaaaa@3C02@ )

Observed Frequency

Expected Frequency

PD

PLD

PSD

PAD

0

413

374.0

405.7

406.1

407.5

1

124

177.4

133.6

132.9

131.2

2

42

42.1

42.6

42.7

42.5

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@

13.4
4.1 1.2 0.6 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaisdacaGGUaGaaGymaaqaaiaaigdacaGGUaGaaGOmaaqa aiaaicdacaGGUaGaaGOnaaaacaGL9baaaaa@3E29@

13.6
4.3 1.3 0.6 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaisdacaGGUaGaaG4maaqaaiaaigdacaGGUaGaaG4maaqa aiaaicdacaGGUaGaaGOnaaaacaGL9baaaaa@3E2C@

4

5

5

0

6

2

Total

601

601.0

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 ^ =3.125788 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpcaaIZaGaaiOlaiaaigdacaaIYaGaaGynaiaaiEda caaI4aGaaGioaaaa@3F3A@

θ ^ =3.492243 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpcaaIZaGaaiOlaiaaisdacaaI5aGaaGOmaiaaikda caaI0aGaaG4maaaa@3F33@

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

48.17

1.34

1.10

0.70

d.f.

2

3

3

3

p-value

0.0000

0.7196

0.7771

0.8732

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

PAD

0

200

172.5

191.8

192.0

192.8

1

57

95.4

70.3

70.1

69.1

2

30

26.4

24.9

24.9

24.8

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@

8.6 2.9 0.9 0.6 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaiIdacaGGUaGaaGOnaaqaaiaaikdacaGGUaGaaGyoaaqa aiaaicdacaGGUaGaaGyoaaqaaiaaicdacaGGUaGaaGOnaaaacaGL9b aaaaa@406A@

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

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 ^ =2.795745 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpcaaIYaGaaiOlaiaaiEdacaaI5aGaaGynaiaaiEda caaI0aGaaGynaaaa@3F3F@

θ ^ =3.178185 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpcaaIZaGaaiOlaiaaigdacaaI3aGaaGioaiaaigda caaI4aGaaGynaaaa@3F39@

χ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Xdm2aaW baaSqabeaacaaIYaaaaaaa@3896@

29.68

3.91

3.81

3.47

d.f.

2

2

2

2

p-value

0.0000

0.1415

0.1488

0.1764

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

PAD

0

155

127.8

158.3

157.5

157.9

1

83

109.0

77.2

77.5

76.8

2

33

46.5

35.9

36.4

36.5

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@

16.4
7.1 3.0 2.1 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaiEdacaGGUaGaaGymaaqaaiaaiodacaGGUaGaaGimaaqa aiaaikdacaGGUaGaaGymaaaacaGL9baaaaa@3E29@

16.6
7.2 3.0 2.0 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaiEdacaGGUaGaaGOmaaqaaiaaiodacaGGUaGaaGimaaqa aiaaikdacaGGUaGaaGimaaaacaGL9baaaaa@3E29@

4

11

5

3

6

1

Total

300

300.0

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 ^ =2.034077 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpcaaIYaGaaiOlaiaaicdacaaIZaGaaGinaiaaicda caaI3aGaaG4naaaa@3F2F@

θ ^ =2.431509 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpcaaIYaGaaiOlaiaaisdacaaIZaGaaGymaiaaiwda caaIWaGaaGyoaaaa@3F30@

χ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Xdm2aaW baaSqabeaacaaIYaaaaaaa@3896@

24.97

1.51

1.74

1.93

d.f.

2

3

3

3

p-value

0

0.6799

0.6281

0.5871

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@ .

Applications in thunderstorms

In thunderstorm activity, the occurrence of successive thunderstorm events (THE’s) is generally a dependent process meaning that the occurrence of a THE indicates that the atmosphere is unstable and the conditions are favorable for the formation for 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. [15]. The theoretical and empirical justification for using the NBD to describe THE activity has been fully explained and discussed by Falls et al. [24]. Further, for fitting Poisson distribution to the count data equality of mean and variance must be satisfied. Similarly, for fitting NBD to the count data, mean must 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 over PSD, PLD and PD because PAD, PSD and PLD are always  over-dispersed and PAD has advantage over PSD and PLD. The thunderstorms data have been considered in Tables 8-10.

No. of Thunderstorms

Observed Frequency

Expected Frequency

PD

PLD

PSD

PAD

0

187

155.6

185.3

184.8

185.4

1

77

117.0

83.5

83.6

82.7

2

40

43.9

35.9

36.3

36.3

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

15.2
6.1 2.4 1.6 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaiAdacaGGUaGaaGymaaqaaiaaikdacaGGUaGaaGinaaqa aiaaigdacaGGUaGaaGOnaaaacaGL9baaaaa@3E2F@

15.4
6.3 2.4 1.5 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaiAdacaGGUaGaaG4maaqaaiaaikdacaGGUaGaaGinaaqa aiaaigdacaGGUaGaaGynaaaacaGL9baaaaa@3E30@

4

6

5

2

6

1

Total

330

330.0

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 ^ =2.229891 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpcaaIYaGaaiOlaiaaikdacaaIYaGaaGyoaiaaiIda caaI5aGaaGymaaaa@3F39@

θ ^ =2.625345 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpcaaIYaGaaiOlaiaaiAdacaaIYaGaaGynaiaaioda caaI0aGaaGynaaaa@3F33@

χ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Xdm2aaW baaSqabeaacaaIYaaaaaaa@3896@

31.93

1.43

1.25

1.07

d.f.

2

3

3

3

p-value

0.0000

0.6985

0.741

0.7843

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. [24].

No. of Thunderstorms

Observed Frequency

Expected Frequency

PD

PLD

PSD

PAD

0

177

142.3

177.7

176.5

176.7

1

80

124.4

88.0

88.4

87.6

2

47

54.3

41.5

42.2

42.3

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

19.2
8.5 6.2 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaiIdacaGGUaGaaGynaaqaaiaaiAdacaGGUaGaaGOmaaaa caGL9baaaaa@3C09@

19.5
8.6 6.3 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaiIdacaGGUaGaaGOnaaqaaiaaiAdacaGGUaGaaG4maaaa caGL9baaaaa@3C0B@

4

9

5

2

Total

341

341.0

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

θ ^ =1.995806 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpcaaIXaGaaiOlaiaaiMdacaaI5aGaaGynaiaaiIda caaIWaGaaGOnaaaa@3F3E@

θ ^ =2.390474 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpcaaIYaGaaiOlaiaaiodacaaI5aGaaGimaiaaisda caaI3aGaaGinaaaa@3F35@

χ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Xdm2aaW baaSqabeaacaaIYaaaaaaa@3896@

39.74

5.15

4.67

4.35

d.f.

2

3

3

3

p-value

0.0000

0.1611

0.1976

0.2261

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. [24].

No. of Thunderstorms

Observed Frequency

Expected Frequency

PD

PLD

PSD

PAD

0

185

151.8

184.8

184.1

184.7

1

89

122.9

87.2

87.5

86.6

2

30

49.7

39.3

39.8

39.8

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

17.3
7.3 5.0 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaiEdacaGGUaGaaG4maaqaaiaaiwdacaGGUaGaaGimaaaa caGL9baaaaa@3C03@

17.6
7.4 4.9 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaiEdacaGGUaGaaGinaaqaaiaaisdacaGGUaGaaGyoaaaa caGL9baaaaa@3C0C@

4

10

5

3

Total

341

341.0

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

θ ^ =2.114545 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpcaaIYaGaaiOlaiaaigdacaaIXaGaaGinaiaaiwda caaI0aGaaGynaaaa@3F2E@

θ ^ =2.511962 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpcaaIYaGaaiOlaiaaiwdacaaIXaGaaGymaiaaiMda caaI2aGaaGOmaaaa@3F32@

χ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Xdm2aaW baaSqabeaacaaIYaaaaaaa@3896@

49.49

5.03

5.06

4.83

d.f.

2

3

3

3

p-value

0.0000

0.1696

0.1674

0.1847

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. [24].

Again it is obvious from fitting of PAD to thunderstorms data that PAD gives better fit than PD, PLD, and PSD in all data .

References

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