ISSN: 2378-315X BBIJ

Biometrics & Biostatistics International Journal
Research Article
Volume 3 Issue 4 - 2016
On Poisson-Sujatha Distribution and its Applications to Model Count Data from Biological Sciences
Rama Shanker1*, Hagos Fesshaye2
Department of Statistics, Eritrea Institute of Technology, Eritrea
Received: January 29, 2016 | Published: March 10, 2016
*Corresponding author: Rama Shanker, Department of Statistics, Eritrea Institute of Technology, Asmara, Eritrea, Email:
Citation: Shanker R, Fesshaye H (2016) On Poisson-Sujatha Distribution and its Applications to Model Count Data from Biological Sciences. Biom Biostat Int J 3(4): 00069. DOI: 10.15406/bbij.2016.03.00069

Abstract

In this paper a simple method for finding moments of Poisson-Sujatha distribution (PSD) introduced by Shanker [1] has been suggested and hence the first four moments about origin and the variance has been given. The PSD has been fitted to the same data-sets relating to ecology and genetics to which earlier Shanker & Hagos [2] has fitted Poisson-Lindley distribution (PLD) introduced by Sankaran [3] and Poisson-distribution (PD) and the goodness of fit of PSD shows satisfactory fit in majority of data-sets.

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

Introduction

The Poisson-Sujatha distribution (PSD) having probability mass function

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@   (1.1)

has been introduced by Shanker [1] for modeling count data-sets. The PSD arises from Poisson distribution when its parameter follows Sujatha distribution introduced by Shanker [4] having probability density function

f( λ;θ )= θ 3 θ 2 +θ+2 ( 1+λ+ λ 2 ) e θλ ;λ>0,θ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aabmaabaGaeq4UdWMaai4oaiabeI7aXbGaayjkaiaawMcaaiabg2da 9maalaaabaGaeqiUde3aaWbaaeqajuaibaGaaG4maaaaaKqbagaacq aH4oqCdaahaaqabKqbGeaacaaIYaaaaKqbakabgUcaRiabeI7aXjab gUcaRiaaikdaaaWaaeWaaeaacaaIXaGaey4kaSIaeq4UdWMaey4kaS Iaeq4UdW2aaWbaaeqajuaibaGaaGOmaaaaaKqbakaawIcacaGLPaaa caWGLbWaaWbaaeqajuaibaGaeyOeI0IaeqiUdeNaaGPaVlabeU7aSb aajuaGcaaMc8UaaGPaVlaaykW7caaMc8Uaai4oaiabeU7aSjabg6da +iaaicdacaGGSaGaaGPaVlaaykW7cqaH4oqCcqGH+aGpcaaIWaaaaa@6B61@   (1.2)

We have

P( X=x )= 0 e λ λ x x! θ 3 θ 2 +θ+2 ( 1+λ+ λ 2 ) e θλ dλ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiuam aabmaabaGaamiwaiabg2da9iaadIhaaiaawIcacaGLPaaacqGH9aqp daWdXbqaamaalaaabaGaamyzamaaCaaabeqcfasaaiabgkHiTiabeU 7aSbaajuaGcqaH7oaBdaahaaqabKqbGeaacaWG4baaaaqcfayaaiaa dIhacaGGHaaaaaqcfasaaiaaicdaaeaacqGHEisPaKqbakabgUIiYd GaeyyXIC9aaSaaaeaacqaH4oqCdaahaaqabKqbGeaacaaIZaaaaaqc fayaaiabeI7aXnaaCaaabeqcfasaaiaaikdaaaqcfaOaey4kaSIaeq iUdeNaey4kaSIaaGOmaaaadaqadaqaaiaaigdacqGHRaWkcqaH7oaB cqGHRaWkcqaH7oaBdaahaaqabKqbGeaacaaIYaaaaaqcfaOaayjkai aawMcaaiaadwgadaahaaqabKqbGeaacqGHsislcqaH4oqCcaaMc8Ua eq4UdWgaaKqbakaadsgacqaH7oaBaaa@6C2C@    (1.3)

= θ 3 ( θ 2 +θ+2 )x! 0 λ x ( 1+λ+ λ 2 ) e ( θ+1 )λ dλ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyypa0 ZaaSaaaeaacqaH4oqCdaahaaqabKqbGeaacaaIZaaaaaqcfayaamaa bmaabaGaeqiUde3aaWbaaeqajuaibaGaaGOmaaaajuaGcqGHRaWkcq aH4oqCcqGHRaWkcaaIYaaacaGLOaGaayzkaaGaaGPaVlaadIhacaGG HaaaamaapehabaGaeq4UdW2aaWbaaeqajuaibaGaamiEaaaaaeaaca aIWaaabaGaeyOhIukajuaGcqGHRiI8amaabmaabaGaaGymaiabgUca RiabeU7aSjabgUcaRiabeU7aSnaaCaaabeqcfasaaiaaikdaaaaaju aGcaGLOaGaayzkaaGaamyzamaaCaaabeqcfasaaiabgkHiTKqbaoaa bmaajuaibaGaeqiUdeNaey4kaSIaaGymaaGaayjkaiaawMcaaiaayk W7cqaH7oaBaaqcfaOaamizaiabeU7aSbaa@6665@

= 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=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyypa0 ZaaSaaaeaacqaH4oqCdaahaaqabKqbGeaacaaIZaaaaaqcfayaaiab eI7aXnaaCaaabeqcfasaaiaaikdaaaqcfaOaey4kaSIaeqiUdeNaey 4kaSIaaGOmaaaacqGHflY1daWcaaqaaiaadIhadaahaaqabKqbGeaa caaIYaaaaKqbakabgUcaRmaabmaabaGaeqiUdeNaey4kaSIaaGinaa GaayjkaiaawMcaaiaadIhacqGHRaWkdaqadaqaaiabeI7aXnaaCaaa beqcfasaaiaaikdaaaqcfaOaey4kaSIaaG4maiabeI7aXjabgUcaRi aaisdaaiaawIcacaGLPaaaaeaadaqadaqaaiabeI7aXjabgUcaRiaa igdaaiaawIcacaGLPaaadaahaaqabKqbGeaacaWG4bGaey4kaSIaaG 4maaaaaaqcfaOaaGPaVlaaykW7caGG7aGaamiEaiabg2da9iaaicda caGGSaGaaGymaiaacYcacaaIYaGaaiilaiaac6cacaGGUaGaaiOlai aacYcacqaH4oqCcqGH+aGpcaaIWaaaaa@717D@   (1.4)

Which is the Poisson-Sujatha distribution (PSD).

Shanker [4] has shown that the Sujatha distribution (1.2) is a three component mixture of an exponential (θ) distribution, a gamma (2,θ) distribution, and a gamma (3,θ) distributionwith their mixing proportions θ 2 θ 2 +θ+2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacqaH4oqCdaahaaqabKqbGeaacaaIYaaaaaqcfayaaiabeI7aXnaa CaaabeqcfasaaiaaikdaaaqcfaOaey4kaSIaeqiUdeNaey4kaSIaaG Omaaaaaaa@416A@ , θ θ 2 +θ+2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacqaH4oqCaeaacqaH4oqCdaahaaqabKqbGeaacaaIYaaaaKqbakab gUcaRiabeI7aXjabgUcaRiaaikdaaaaaaa@3FD0@ and 2 θ 2 +θ+2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacaaIYaaabaGaeqiUde3aaWbaaeqajuaibaGaaGOmaaaajuaGcqGH RaWkcqaH4oqCcqGHRaWkcaaIYaaaaaaa@3ED6@ respectively. Shanker [4] has discussed its various mathematical and statistical properties including its shape, moment generating function, moments, skewness, kurtosis, hazard rate function, mean residual life function, stochastic orderings, mean deviations, distribution of order statistics, Bonferroni and Lorenz curves, Renyi entropy measure, stress-strength reliability , amongst others along with the estimation of the parameter and applications for modeling lifetime data.

Shanker [1] has detailed study about various mathematical and statistical properties of PSD including moment generating function, coefficient of variation, skewness, kurtosis, over-dispersion, hazard rate and unimodality along with the estimation of the parameter and applications. Shanker & Hagos [5,6] have obtained size-biased Poisson-Sujatha distribution (SBPSD) and zero-truncated Poisson-Sujatha distribution(ZTPSD) and discussed their statistical properties, estimation of the parameter and applications. Further, Shanker & Hagos [7] have detailed study about zero-truncation of Poisson, Poisson-Lindley and Poisson-Sujatha distributions and their applications.

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

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

has been introduced by Sankaran [3] to model count data. The distribution arises from the Poisson distribution when its parameter follows Lindley [8] distribution with its probability density function

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

In this paper a simple method for finding moments of Poisson-Sujatha distribution (PSD) 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 PSD so far. The PSD has been fitted to the same data-sets relating to ecology and genetics to which Shanker & Hagos [2] has fitted Poisson-Lindley distribution (PLD) introduced by Sankaran [3] and Poisson-distribution (PD) and the goodness of fit of PSD shows satisfactory fit in majority of data-sets.

Moments of Poisson-Sujatha Distribution

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 PSD (1.1) can be obtained as

     

  r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOCaa aa@377B@ = θ 3 θ 2 +θ+2 0 [ x=0 x r e λ λ x x! ] ( 1+λ+ λ 2 ) e θλ dλ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyypa0 ZaaSaaaeaacqaH4oqCdaahaaqabKqbGeaacaaIZaaaaaqcfayaaiab eI7aXnaaCaaabeqcfasaaiaaikdaaaqcfaOaey4kaSIaeqiUdeNaey 4kaSIaaGOmaaaadaWdXbqaamaadmaabaWaaabCaeaacaWG4bWaaWba aeqajuaibaGaamOCaaaajuaGdaWcaaqaaiaadwgadaahaaqabKqbGe aacqGHsislcqaH7oaBaaqcfaOaeq4UdW2aaWbaaeqajuaibaGaamiE aaaaaKqbagaacaWG4bGaaiyiaaaaaKqbGeaacaWG4bGaeyypa0JaaG imaaqaaiabg6HiLcqcfaOaeyyeIuoaaiaawUfacaGLDbaaaKqbGeaa caaIWaaabaGaeyOhIukajuaGcqGHRiI8amaabmaabaGaaGymaiabgU caRiabeU7aSjabgUcaRiabeU7aSnaaCaaabeqcfasaaiaaikdaaaaa juaGcaGLOaGaayzkaaGaamyzamaaCaaabeqcfasaaiabgkHiTiabeI 7aXjaaykW7cqaH7oaBaaqcfaOaamizaiabeU7aSbaa@707C@     (2.1)

Clearly 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 first moment about origin of the Poisson distribution, the first moment about origin of the PSD (1.1) can be obtained as

                 

  μ 2 = θ 3 θ 2 +θ+2 0 ( λ 2 +λ )( 1+λ+ λ 2 ) e θλ dλ= θ 3 +4 θ 2 +12θ+24 θ 2 ( θ 2 +θ+2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaKqbGeaacaaIYaaajuaGbeaadaahaaqabeaacWaGGBOmGika aiabg2da9maalaaabaGaeqiUde3aaWbaaeqajuaibaGaaG4maaaaaK qbagaacqaH4oqCdaahaaqabKqbGeaacaaIYaaaaKqbakabgUcaRiab eI7aXjabgUcaRiaaikdaaaWaa8qCaeaadaqadaqaaiabeU7aSnaaCa aabeqaaiaaikdaaaGaey4kaSIaeq4UdWgacaGLOaGaayzkaaWaaeWa aeaacaaIXaGaey4kaSIaeq4UdWMaey4kaSIaeq4UdW2aaWbaaeqaba GaaGOmaaaaaiaawIcacaGLPaaaaeaacaaIWaaabaGaeyOhIukacqGH RiI8aiaaykW7caWGLbWaaWbaaeqabaGaeyOeI0IaeqiUdeNaaGPaVl abeU7aSbaacaWGKbGaeq4UdWMaeyypa0ZaaSaaaeaacqaH4oqCdaah aaqabeaacaaIZaaaaiabgUcaRiaaisdacqaH4oqCdaahaaqabeaaca aIYaaaaiabgUcaRiaaigdacaaIYaGaeqiUdeNaey4kaSIaaGOmaiaa isdaaeaacqaH4oqCdaahaaqabeaacaaIYaaaamaabmaabaGaeqiUde 3aaWbaaeqabaGaaGOmaaaacqGHRaWkcqaH4oqCcqGHRaWkcaaIYaaa caGLOaGaayzkaaaaaaaa@808C@                           (2.2)

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

    

      μ 2 = θ 3 θ 2 +θ+2 0 ( λ 2 +λ )( 1+λ+ λ 2 ) e θλ dλ= θ 3 +4 θ 2 +12θ+24 θ 2 ( θ 2 +θ+2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaKqbGeaacaaIYaaajuaGbeaadaahaaqabeaacWaGGBOmGika aiabg2da9maalaaabaGaeqiUde3aaWbaaeqajuaibaGaaG4maaaaaK qbagaacqaH4oqCdaahaaqabKqbGeaacaaIYaaaaKqbakabgUcaRiab eI7aXjabgUcaRiaaikdaaaWaa8qCaeaadaqadaqaaiabeU7aSnaaCa aabeqaaiaaikdaaaGaey4kaSIaeq4UdWgacaGLOaGaayzkaaWaaeWa aeaacaaIXaGaey4kaSIaeq4UdWMaey4kaSIaeq4UdW2aaWbaaeqaba GaaGOmaaaaaiaawIcacaGLPaaaaeaacaaIWaaabaGaeyOhIukacqGH RiI8aiaaykW7caWGLbWaaWbaaeqabaGaeyOeI0IaeqiUdeNaaGPaVl abeU7aSbaacaWGKbGaeq4UdWMaeyypa0ZaaSaaaeaacqaH4oqCdaah aaqabeaacaaIZaaaaiabgUcaRiaaisdacqaH4oqCdaahaaqabeaaca aIYaaaaiabgUcaRiaaigdacaaIYaGaeqiUdeNaey4kaSIaaGOmaiaa isdaaeaacqaH4oqCdaahaaqabeaacaaIYaaaamaabmaabaGaeqiUde 3aaWbaaeqabaGaaGOmaaaacqGHRaWkcqaH4oqCcqGHRaWkcaaIYaaa caGLOaGaayzkaaaaaaaa@808C@                (2.3)

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

          

  μ 3 = θ 4 +8 θ 3 +30 θ 2 +96θ+120 θ 3 ( θ 2 +θ+2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaKqbGeaacaaIZaaajuaGbeaadaahaaqabeaacWaGGBOmGika aiabg2da9maalaaabaGaeqiUde3aaWbaaeqajuaibaGaaGinaaaaju aGcqGHRaWkcaaI4aGaeqiUde3aaWbaaeqajuaibaGaaG4maaaajuaG cqGHRaWkcaaIZaGaaGimaiabeI7aXnaaCaaabeqcfasaaiaaikdaaa qcfaOaey4kaSIaaGyoaiaaiAdacqaH4oqCcqGHRaWkcaaIXaGaaGOm aiaaicdaaeaacqaH4oqCdaahaaqabKqbGeaacaaIZaaaaKqbaoaabm aabaGaeqiUde3aaWbaaeqajuaibaGaaGOmaaaajuaGcqGHRaWkcqaH 4oqCcqGHRaWkcaaIYaaacaGLOaGaayzkaaaaaaaa@5F72@                                                                     (2.4)

            μ 4 = θ 5 +16 θ 4 +84 θ 3 +336 θ 2 +840θ+720 θ 4 ( θ 2 +θ+2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaKqbGeaacaaI0aaajuaGbeaadaahaaqabeaacWaGGBOmGika aiabg2da9maalaaabaGaeqiUde3aaWbaaeqajuaibaGaaGynaaaaju aGcqGHRaWkcaaIXaGaaGOnaiabeI7aXnaaCaaabeqcfasaaiaaisda aaqcfaOaey4kaSIaaGioaiaaisdacqaH4oqCdaahaaqabKqbGeaaca aIZaaaaKqbakabgUcaRiaaiodacaaIZaGaaGOnaiabeI7aXnaaCaaa beqcfasaaiaaikdaaaqcfaOaey4kaSIaaGioaiaaisdacaaIWaGaeq iUdeNaey4kaSIaaG4naiaaikdacaaIWaaabaGaeqiUde3aaWbaaeqa juaibaGaaGinaaaajuaGdaqadaqaaiabeI7aXnaaCaaabeqcfasaai aaikdaaaqcfaOaey4kaSIaeqiUdeNaey4kaSIaaGOmaaGaayjkaiaa wMcaaaaaaaa@6762@                                                (2.5)

Thus the variance of the PSD (1.1) can be obtained as

μ 2 = θ 5 +4 θ 4 +14 θ 3 +28 θ 2 +24θ+12 θ 2 ( θ 2 +θ+2 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaKqbGeaacaaIYaaajuaGbeaacqGH9aqpdaWcaaqaaiabeI7a XnaaCaaabeqcfasaaiaaiwdaaaqcfaOaey4kaSIaaGinaiabeI7aXn aaCaaabeqcfasaaiaaisdaaaqcfaOaey4kaSIaaGymaiaaisdacqaH 4oqCdaahaaqabKqbGeaacaaIZaaaaKqbakabgUcaRiaaikdacaaI4a GaeqiUde3aaWbaaeqajuaibaGaaGOmaaaajuaGcqGHRaWkcaaIYaGa aGinaiabeI7aXjabgUcaRiaaigdacaaIYaaabaGaeqiUde3aaWbaae qajuaibaGaaGOmaaaajuaGdaqadaqaaiabeI7aXnaaCaaabeqcfasa aiaaikdaaaqcfaOaey4kaSIaeqiUdeNaey4kaSIaaGOmaaGaayjkai aawMcaamaaCaaabeqcfasaaiaaikdaaaaaaaaa@6260@                                    (2.6)

Shanker [1] has shown that the PSD is always over-dispersed, has increasing hazard rate and unimodal. Further, Shanker [1] has also shown that the graphs of coefficient of variation, skewness, and kurtosis of PSD are increasing for increasing values of the parameter.

Estimation of the Parameter

Maximum likelihood estimate (MLE) of the parameter: Let ( x 1 , x 2 ,..., x n ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacaWG4bWaaSbaaKqbGeaacaaIXaaajuaGbeaacaGGSaGaamiEamaa BaaajuaibaGaaGOmaaqcfayabaGaaiilaiaac6cacaGGUaGaaiOlai aacYcacaWG4bWaaSbaaKqbGeaacaWGUbaajuaGbeaaaiaawIcacaGL Paaaaaa@442B@ be a random sample of size n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaaaa@36E9@  from the PSD (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=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Aaaaa@36E6@  is the largest observed value having non-zero frequency. The likelihood function L MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamitaa aa@3755@ of the PSD (1.1) is given by

L= ( θ 3 θ 2 +θ+2 ) n 1 ( θ+1 ) x=1 k f x ( x+3 ) x=1 k [ x 2 +( θ+4 )x+( θ 2 +3θ+4 ) ] f x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamitai abg2da9maabmaabaWaaSaaaeaacqaH4oqCdaahaaqabKqbGeaacaaI ZaaaaaqcfayaaiabeI7aXnaaCaaabeqcfasaaiaaikdaaaqcfaOaey 4kaSIaeqiUdeNaey4kaSIaaGOmaaaaaiaawIcacaGLPaaadaahaaqa bKqbGeaacaWGUbaaaKqbaoaalaaabaGaaGymaaqaamaabmaabaGaeq iUdeNaey4kaSIaaGymaaGaayjkaiaawMcaamaaCaaabeqaamaaqaha baGaamOzamaaBaaajuaibaGaamiEaaqcfayabaWaaeWaaeaacaWG4b Gaey4kaSIaaG4maaGaayjkaiaawMcaaaqcfasaaiaadIhacqGH9aqp caaIXaaabaGaam4AaaqcfaOaeyyeIuoaaaaaamaarahabaWaamWaae aacaWG4bWaaWbaaeqajuaibaGaaGOmaaaajuaGcqGHRaWkdaqadaqa aiabeI7aXjabgUcaRiaaisdaaiaawIcacaGLPaaacaWG4bGaey4kaS YaaeWaaeaacqaH4oqCdaahaaqabKqbGeaacaaIYaaaaKqbakabgUca RiaaiodacqaH4oqCcqGHRaWkcaaI0aaacaGLOaGaayzkaaaacaGLBb GaayzxaaWaaWbaaeqajuaibaGaamOzaKqbaoaaBaaajuaibaGaamiE aaqabaaaaaqaaiaadIhacqGH9aqpcaaIXaaabaGaam4AaaqcfaOaey 4dIunaaaa@7944@

The log likelihood function is thus obtained as

logL=nlog( θ 3 θ 2 +θ+2 ) x=1 k f x ( x+3 ) log( θ+1 )+ x=1 k f x log[ x 2 +( θ+4 )x+( θ 2 +3θ+4 ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaciiBai aac+gacaGGNbGaamitaiabg2da9iaad6gaciGGSbGaai4BaiaacEga daqadaqaamaalaaabaGaeqiUde3aaWbaaeqajuaibaGaaG4maaaaaK qbagaacqaH4oqCdaahaaqabKqbGeaacaaIYaaaaKqbakabgUcaRiab eI7aXjabgUcaRiaaikdaaaaacaGLOaGaayzkaaGaeyOeI0YaaabCae aacaWGMbWaaSbaaKqbGeaacaWG4baajuaGbeaadaqadaqaaiaadIha cqGHRaWkcaaIZaaacaGLOaGaayzkaaaajuaibaGaamiEaiabg2da9i aaigdaaeaacaWGRbaajuaGcqGHris5aiGacYgacaGGVbGaai4zamaa bmaabaGaeqiUdeNaey4kaSIaaGymaaGaayjkaiaawMcaaiabgUcaRm aaqahabaGaamOzamaaBaaajuaibaGaamiEaaqcfayabaGaciiBaiaa c+gacaGGNbWaamWaaeaacaWG4bWaaWbaaeqajuaibaGaaGOmaaaaju aGcqGHRaWkdaqadaqaaiabeI7aXjabgUcaRiaaisdaaiaawIcacaGL PaaacaWG4bGaey4kaSYaaeWaaeaacqaH4oqCdaahaaqabKqbGeaaca aIYaaaaKqbakabgUcaRiaaiodacqaH4oqCcqGHRaWkcaaI0aaacaGL OaGaayzkaaaacaGLBbGaayzxaaaajuaibaGaamiEaiabg2da9iaaig daaeaacaWGRbaajuaGcqGHris5aaaa@8477@

The first derivative of the log likelihood function is given by

dlogL dθ = n( θ 2 +2θ+6 ) θ( θ 2 +θ+2 ) n( x ¯ +3 ) θ+1 + x=1 k [ x+( 2θ+3 ) ] f x [ x 2 +( θ+4 )x+( θ 2 +3θ+4 ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacaWGKbGaciiBaiaac+gacaGGNbGaamitaaqaaiaadsgacqaH4oqC aaGaeyypa0ZaaSaaaeaacaWGUbWaaeWaaeaacqaH4oqCdaahaaqabK qbGeaacaaIYaaaaKqbakabgUcaRiaaikdacqaH4oqCcqGHRaWkcaaI 2aaacaGLOaGaayzkaaaabaGaeqiUde3aaeWaaeaacqaH4oqCdaahaa qabKqbGeaacaaIYaaaaKqbakabgUcaRiabeI7aXjabgUcaRiaaikda aiaawIcacaGLPaaaaaGaeyOeI0YaaSaaaeaacaWGUbWaaeWaaeaace WG4bGbaebacqGHRaWkcaaIZaaacaGLOaGaayzkaaaabaGaeqiUdeNa ey4kaSIaaGymaaaacqGHRaWkdaaeWbqaamaalaaabaWaamWaaeaaca WG4bGaey4kaSYaaeWaaeaacaaIYaGaeqiUdeNaey4kaSIaaG4maaGa ayjkaiaawMcaaaGaay5waiaaw2faaiaadAgadaWgaaqcfasaaiaadI haaKqbagqaaaqaamaadmaabaGaamiEamaaCaaabeqcfasaaiaaikda aaqcfaOaey4kaSYaaeWaaeaacqaH4oqCcqGHRaWkcaaI0aaacaGLOa GaayzkaaGaamiEaiabgUcaRmaabmaabaGaeqiUde3aaWbaaeqajuai baGaaGOmaaaajuaGcqGHRaWkcaaIZaGaeqiUdeNaey4kaSIaaGinaa GaayjkaiaawMcaaaGaay5waiaaw2faaaaaaKqbGeaacaWG4bGaeyyp a0JaaGymaaqaaiaadUgaaKqbakabggHiLdaaaa@8787@

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

The maximum likelihood estimate (MLE), θ ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaaaaa@384A@  of θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde haaa@383A@  of PSD (1.1) is the solution of the equation  dlogL dθ =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacaWGKbGaciiBaiaac+gacaGGNbGaamitaaqaaiaadsgacqaH4oqC aaGaeyypa0JaaGimaaaa@3F7D@  and is given by the solution of the following non-linear equation

n( θ 2 +2θ+6 ) θ( θ 2 +θ+2 ) n( x ¯ +3 ) θ+1 + x=1 k [ x+( 2θ+3 ) ] f x [ x 2 +( θ+4 )x+( θ 2 +3θ+4 ) ] =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaGUajuaGda Wcaaqaaiaad6gadaqadaqaaiabeI7aXnaaCaaabeqcfasaaiaaikda aaqcfaOaey4kaSIaaGOmaiabeI7aXjabgUcaRiaaiAdaaiaawIcaca GLPaaaaeaacqaH4oqCdaqadaqaaiabeI7aXnaaCaaabeqcfasaaiaa ikdaaaqcfaOaey4kaSIaeqiUdeNaey4kaSIaaGOmaaGaayjkaiaawM caaaaacqGHsisldaWcaaqaaiaad6gadaqadaqaaiqadIhagaqeaiab gUcaRiaaiodaaiaawIcacaGLPaaaaeaacqaH4oqCcqGHRaWkcaaIXa aaaiabgUcaRmaaqahabaWaaSaaaeaadaWadaqaaiaadIhacqGHRaWk daqadaqaaiaaikdacqaH4oqCcqGHRaWkcaaIZaaacaGLOaGaayzkaa aacaGLBbGaayzxaaGaamOzamaaBaaajuaibaGaamiEaaqcfayabaaa baWaamWaaeaacaWG4bWaaWbaaeqajuaibaGaaGOmaaaajuaGcqGHRa WkdaqadaqaaiabeI7aXjabgUcaRiaaisdaaiaawIcacaGLPaaacaWG 4bGaey4kaSYaaeWaaeaacqaH4oqCdaahaaqabKqbGeaacaaIYaaaaK qbakabgUcaRiaaiodacqaH4oqCcqGHRaWkcaaI0aaacaGLOaGaayzk aaaacaGLBbGaayzxaaaaaaqcfasaaiaadIhacqGH9aqpcaaIXaaaba Gaam4AaaqcfaOaeyyeIuoacqGH9aqpcaaIWaaaaa@81F5@       

This non-linear equation can be solved by any numerical iteration methods such as Newton- Raphson, Bisection method, Regula–Falsi method etc.

Method of moment estimate (MOME) of the parameter: Let ( x 1 , x 2 ,..., x n ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca WG4bWaaSbaaSqaaiaaigdaaeqaaOGaaiilaiaadIhadaWgaaWcbaGa aGOmaaqabaGccaGGSaGaaiOlaiaac6cacaGGUaGaaiilaiaadIhada WgaaWcbaGaamOBaaqabaaakiaawIcacaGLPaaaaaa@41A8@ be a random sample of size n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaaaa@36E9@  from the PSD (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 PSD (1.1) is the solution of the following cubic equation

                            x ¯ θ 3 +( x ¯ 1 ) θ 2 +2( x ¯ 1 )θ6=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOabmiEay aaraGaeqiUde3aaWbaaeqajuaibaGaaG4maaaajuaGcqGHRaWkdaqa daqaaiqadIhagaqeaiabgkHiTiaaigdaaiaawIcacaGLPaaacqaH4o qCdaahaaqabKqbGeaacaaIYaaaaKqbakabgUcaRiaaikdadaqadaqa aiqadIhagaqeaiabgkHiTiaaigdaaiaawIcacaGLPaaacqaH4oqCcq GHsislcaaI2aGaeyypa0JaaGimaaaa@4E69@                                        

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

Applications of Poisson-Sujatha Distribution

The Poisson distribution is a suitable statistical model for the situations where events seem to occur at random including the number of customers arriving at a service point, the number of telephone calls arriving at an exchange, the number of fatal traffic accidents per week in a given state, the number of radioactive particle emissions per unit of time, the number of meteorites that collide with a test satellite during a single orbit, the number of organisms per unit volume of some fluid, the number of defects per unit of some materials, the number of flaws per unit length of some wire, are some amongst others. Since the condition for the applications for Poisson distribution is the independence of events and the equality of mean and variance, this condition is rarely satisfied completely in biological and medical science due to the fact that the occurrences of successive events are dependent. Further, the negative binomial distribution is a possible alternative to the Poisson distribution when successive events are possibly dependent Johnson et al. [9], but for fitting negative binomial distribution (NBD) to the count data, mean should be less than the variance. In biological and medical sciences, these conditions are also not fully satisfied. Generally, the count data in biological science and medical science are either over-dispersed or under-dispersed. The main reason for selecting PLD and PSD to fit biological science data is that these two distributions are always over-dispersed and PSD has some flexibility over PLD.

Applications in ecology

Ecology is the branch of biology dealing with the relations and interactions between organisms and their environment, including other organisms. The organisms and their environment in the nature are complex, dynamic, interdependent, mutually reactive and interrelated. Ecology deals with the various principles which govern such relationship between organisms and their environment. It was Fisher et al. [10] who have firstly discussed the applications of Logarithmic series distribution (LSD) to model count data in the science of ecology. Later, Kempton [11] who fitted the generalized form of Fisher’s Logarithmic series distribution (LSD) to model insect data and concluded that it gives a superior fit as compared to ordinary Logarithmic series distribution (LSD). He also concluded that it gives better explanation for the data having exceptionally long tail. Tripathi & Gupta [12] 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 [13] have discussed applications of generalized logarithmic series distributions (GLSD) to models data in ecology. Shanker & Hagos [2] have tried to fit PLD for data relating to ecology and observed that PLD gives satisfactory fit.

In this section we have tried to fit Poisson distribution (PD), Poisson-Lindley distribution (PLD) and Poisson-Sujatha distribution (PSD) 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 (Table 1-3).

It is obvious from above tables that both PSD and PLD give much closer fit than Poisson distribution. Further, in some data-sets PSD gives much closer fit than PLD while in some data-sets PLD gives much closer fit than PSD and thus both PSD and PLD can be considered as important tools for modeling data in ecology.

Number of Yeast Cells per Square

Observed Frequency

Expected Frequency

PD

PLD

PSD

0

213

202.1

234.0

233.2

1

128

138.0

99.4

99.6

2

37

47.1

40.5

41.0

3

18

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

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

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

4

3

5

1

6

0

Total

400.0

400.0

400.0

Estimate of Parameter

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

 

 

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

10.08

11.04

10.86

d.f.

2

2

2

p-value

0.0065

0.0040

0.0044

Table 1: Observed and expected number of Haemocytometer yeast cell counts per square observed by ‘Student’ [17].

Number Mites per Leaf

Observed Frequency

Expected Frequency

PD

PLD

PSD

0

38

25.3

35.8

35.3

1

17

29.1

20.7

20.9

2

10

16.7

11.4

11.6

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
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@

4

3

5

2

6

1

7+

0

Total

80

80

80

80

Estimate of Parameter

θ ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaaaaa@384A@ =1.15

θ ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaaaaa@384A@ =1.255891

θ ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaaaaa@384A@ =1.64683

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

18.27

2.47

2.52

d.f.

2

3

3

p-value

0.0001

0.4807

0.4719

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

Number of Bores per Plant

Observed Frequency

Expected Frequency

PD

PLD

PSD

0

188

169.4

194.0

193.6

1

83

109.8

79.5

79.6

2

36

35.6

31.3

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@

4

2

5

1

Total

324

324.0

324.0

324.0

Estimate of parameter

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

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

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

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

15.19

1.29

1.16

d.f.

2

2

2

p-value

0.0005

0.5247

0.5599

Table 3: Observed and expected number of European corn- borer of Mc Guire et al. [18].

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

Application 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. The segregation of chromosomes has been studied using statistical tool, mainly chi-square ( χ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4Xdm 2aaWbaaeqajuaibaGaaGOmaaaaaaa@3947@ ).  In the analysis of data observed on chemically induced chromosome aberrations in cultures of human leukocytes, Loeschke & Kohler [14] suggested the negative binomial distribution while Janardan & Schaeffer [15] suggested modified Poisson distribution. Mishra and Shanker [13] have discussed applications of generalized Logarithmic series distributions (GLSD) to model data in mortality, ecology and genetics. Shanker & Hagos [2] have detailed study on the applications of PLD to model data from genetics. Much quantitative works seem to be done in genetics but so far no works has been done on fitting of PSD to data relating to genetics. In this section an attempt has been made to fit to data relating to genetics using PSD, PLD and PD using maximum likelihood estimate. Also an attempt has been made to fit PSD, PLD, and PD to the data of Catcheside et al. [16] in Table 4-7.

It is obvious from the fitting of PSD, PLD, and PD that both PSD and PLD gives much satisfactory fit than PD while in some data-sets PSD gives much closer fit than PLD whereas PLD gives much closer fit than PSD in some data-sets. Thus both PSD and PLD can be considered as important tools for modeling data in genetics

Number of Aberrations

Observed Frequency

Expected Frequency

PD

PLD

PSD

0

268

231.3

257

257.6

1

87

126.7

93.4

93

2

26

34.7

32.8

32.7

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@

4

4

5

2

6

1

7+

3

Total

400

400

400

400

Estimate of Parameter

θ ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaaaaa@384A@ =0.5475

θ ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaaaaa@384A@ =2.380442

θ ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaaaaa@384A@ =2.829241

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

38.21

6.21

6.28

d.f.

2

3

3

p-value

0

0.1018

0.0987

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

0

413

374

405.7

406.1

1

124

177.4

133.6

132.9

2

42

42.1

42.6

42.7

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@

4

5

5

0

6

2

Total

601

601

601

601

Estimate of parameter

θ ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaaaaa@384A@ =0.47421

θ ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaaaaa@384A@ =2.685373

θ ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaaaaa@384A@ =3.125788

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

48.17

1.34

1.1

d.f.

2

3

3

p-value

0

0.7196

0.7771

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

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

Observed Frequency

Expected Frequency

PD

PLD

PSD

0

200

172.5

191.8

192

1

57

95.4

70.3

70.1

2

30

26.4

24.9

24.9

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@

4

4

5

0

6

2

Total

300

300

300

300

Estimate of parameter

θ ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaaaaa@384A@ =0.55333

θ ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaaaaa@384A@ =2.353339

θ ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaaaaa@384A@ =2.795745

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

29.68

3.91

3.81

d.f.

2

2

2

p-value

0

0.1415

0.1488

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

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

Observed Frequency

Expected Frequency

PD

PLD

PSD

0

155

127.8

158.3

157.5

1

83

109

77.2

77.5

2

33

46.5

35.9

36.4

3

14

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

16.1
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@

4

11

5

3

6

1

Total

300

300

300

300

Estimate of parameter

θ ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaaaaa@384A@ =0.853333

θ ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaaaaa@384A@ =1.617611

θ ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaaaaa@384A@ =2.034077

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

24.97

1.51

1.74

d.f.

2

3

3

p-value

0

0.6799

0.6281

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@

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