ISSN: 2378-315X BBIJ

Biometrics & Biostatistics International Journal
Research Article
Volume 2 Issue 4 - 2015
On Poisson-Lindley Distribution and Its Applications to Biological Sciences
Rama Shanker1* and Hagos Fesshaye2
1Department of Statistics, Eritrea Institute of Technology, Eritrea
2Department of Economics, College of Business and Economics, Eritrea
Received: April 13, 2015 | Published: April 27, 2015
*Corresponding author: Rama Shanker, Department of Statistics, Eritrea Institute of Technology, Eritrea, Email:
Citation: Shanker R, Fesshaye H (2014) On Poisson-Lindley Distribution and Its Applications to Biological Sciences. Biom Biostat Int J 2(4): 00036. DOI: 10.15406/bbij.2015.02.00036

Abstract

A general expression for the r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGYb aaaa@377C@ th factorial moment of Poisson-Lindley distribution has been obtained and hence its first four moments about origin has been obtained. The distribution has been fitted to some data-sets relating to ecology and genetics to test its goodness of fit and the fit shows that it can be an important tool for modeling biological science data.

Keywords: Lindley distribution, Poisson-Lindley distribution, moments, compounding, estimation of parameters, goodness of fit.

Introduction

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

P( X=x )= θ 2 ( x+θ+2 ) ( θ+1 ) x+3 ; MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGqb qcfa4aaeWaaOqaaKqzGeGaamiwaiabg2da9iaadIhaaOGaayjkaiaa wMcaaKqzGeGaaGjbVlabg2da9iaaysW7caaMc8Ecfa4aaSaaaOqaaK qzGeGaeqiUdexcfa4aaWbaaSqabKqaGeaajugWaiaaikdaaaqcLbsa caaMc8Ecfa4aaeWaaOqaaKqzGeGaamiEaiabgUcaRiaaykW7cqaH4o qCcqGHRaWkcaaMc8UaaGOmaaGccaGLOaGaayzkaaaabaqcfa4aaeWa aOqaaKqzGeGaeqiUdeNaey4kaSIaaGPaVlaaigdaaOGaayjkaiaawM caaKqbaoaaCaaaleqajeaibaqcLbmacaWG4bGaey4kaSIaaGPaVlaa iodaaaaaaKqzGeGaaGjbVlaaykW7caaMc8UaaGPaVlaacUdaaaa@6B55@ x = 0, 1, 2,…, > 0. (1.1)

has been introduced by Sankaran (1970) to model count data. The distribution arises from the Poisson distribution when its parameter λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH7o aBaaa@3839@ follows Lindley (1958) 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=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGMb GaaGPaVNqbaoaabmaakeaajugibiabeU7aSjaacYcacaaMc8UaeqiU dehakiaawIcacaGLPaaajugibiaaysW7cqGH9aqpcaaMe8UaaGPaVN qbaoaalaaakeaajugibiabeI7aXLqbaoaaCaaaleqajeaibaqcLbma caaIYaaaaaGcbaqcLbsacqaH4oqCcqGHRaWkcaaMc8UaaGymaaaaca aMe8Ecfa4aaeWaaOqaaKqzGeGaaGymaiabgUcaRiaaykW7cqaH7oaB aOGaayjkaiaawMcaaKqzGeGaaGPaVlaaykW7caWGLbqcfa4aaWbaaS qabKqaGeaajugWaiabgkHiTiabeI7aXjaaykW7cqaH7oaBaaaaaa@686E@ x>0,θ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG4b GaeyOpa4JaaGimaiaaykW7caGGSaGaeqiUdeNaeyOpa4JaaGimaaaa @3EF7@ (1.2)

 We have

P( X=x )= MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGqb qcfa4aaeWaaOqaaKqzGeGaamiwaiabg2da9iaadIhaaOGaayjkaiaa wMcaaKqzGeGaeyypa0daaa@3E89@ 0 P ( x|λ )f( λ;θ )dλ= 0 e λ λ x x! θ 2 θ+1 ( 1+λ ) e θλ dλ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aa8qCaO qaaKqzGeGaamiuaaWcbaqcLbsacaaIWaaaleaajugibiabg6HiLcGa ey4kIipajuaGdaqadaGcbaqcLbsacaWG4bGaaiiFaiabeU7aSbGcca GLOaGaayzkaaqcLbsacaaMc8UaamOzaKqbaoaabmaakeaajugibiab eU7aSjaacUdacqaH4oqCaOGaayjkaiaawMcaaKqzGeGaamizaiabeU 7aSjabg2da9KqbaoaapehakeaajuaGdaWcaaGcbaqcLbsacaWGLbqc fa4aaWbaaSqabKqaGeaajugWaiabgkHiTiabeU7aSbaajugibiabeU 7aSLqbaoaaCaaaleqajeaibaqcLbmacaWG4baaaaGcbaqcLbsacaWG 4bGaaiyiaaaaaSqaaKqzGeGaaGimaaWcbaqcLbsacqGHEisPaiabgU IiYdGaeyyXICDcfa4aaSaaaOqaaKqzGeGaeqiUdexcfa4aaWbaaSqa bKqaGeaajugWaiaaikdaaaaakeaajugibiabeI7aXjabgUcaRiaaig daaaqcfa4aaeWaaOqaaKqzGeGaaGymaiabgUcaRiabeU7aSbGccaGL OaGaayzkaaqcLbsacaWGLbqcfa4aaWbaaSqabKqaGeaajugWaiabgk HiTiabeI7aXjabeU7aSbaajugibiaadsgacqaH7oaBaaa@83DE@ (1.3)

= θ 2 ( θ+1 )x! 0 e ( θ+1 )λ ( λ x + λ x+1 )dλ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGH9a qpjuaGdaWcaaGcbaqcLbsacqaH4oqCjuaGdaahaaWcbeqcKfaG=haa jugWaiaaikdaaaaakeaajuaGdaqadaGcbaqcLbsacqaH4oqCcqGHRa WkcaaIXaaakiaawIcacaGLPaaajugibiaaykW7caWG4bGaaiyiaaaa juaGdaWdXbGcbaqcLbsacaWGLbqcfa4aaWbaaSqabKazba4=baqcLb macqGHsisllmaabmaajqwaa+FaaKqzadGaeqiUdeNaey4kaSIaaGym aaqcKfaG=laawIcacaGLPaaajugWaiabeU7aSbaaaSqaaKqzGeGaaG imaaWcbaqcLbsacqGHEisPaiabgUIiYdqcfa4aaeWaaOqaaKqzGeGa eq4UdWwcfa4aaWbaaSqabKazba4=baqcLbmacaWG4baaaKqzGeGaey 4kaSIaeq4UdWwcfa4aaWbaaSqabKazba4=baqcLbmacaWG4bGaey4k aSIaaGymaaaaaOGaayjkaiaawMcaaKqzGeGaamizaiabeU7aSbaa@7834@

= θ 2 θ+1 [ 1 ( θ+1 ) x+1 + x+1 ( θ+1 ) x+2 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGH9a qpjuaGdaWcaaGcbaqcLbsacqaH4oqCjuaGdaahaaWcbeqcKfaG=haa jugWaiaaikdaaaaakeaajugibiabeI7aXjabgUcaRiaaigdaaaqcfa 4aamWaaOqaaKqbaoaalaaakeaajugibiaaigdaaOqaaKqbaoaabmaa keaajugibiabeI7aXjabgUcaRiaaigdaaOGaayjkaiaawMcaaKqbao aaCaaaleqajqwaa+FaaKqzadGaamiEaiabgUcaRiaaigdaaaaaaKqz GeGaey4kaSscfa4aaSaaaOqaaKqzGeGaamiEaiabgUcaRiaaigdaaO qaaKqbaoaabmaakeaajugibiabeI7aXjabgUcaRiaaigdaaOGaayjk aiaawMcaaKqbaoaaCaaaleqajqwaa+FaaKqzadGaamiEaiabgUcaRi aaikdaaaaaaaGccaGLBbGaayzxaaaaaa@65CD@

= θ 2 ( x+θ+2 ) ( θ+1 ) x+3 ;θ>0,x=0,1,2,3,... MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGH9a qpjuaGdaWcaaGcbaqcLbsacqaH4oqCjuaGdaahaaWcbeqcbasaaKqz adGaaGOmaaaajuaGdaqadaGcbaqcLbsacaWG4bGaey4kaSIaeqiUde Naey4kaSIaaGOmaaGccaGLOaGaayzkaaaabaqcfa4aaeWaaOqaaKqz GeGaeqiUdeNaey4kaSIaaGymaaGccaGLOaGaayzkaaqcfa4aaWbaaS qabKqaGeaajugWaiaadIhacqGHRaWkcaaIZaaaaaaajugibiaaykW7 caaMc8Uaai4oaiaaykW7caaMc8UaaGPaVlabeI7aXjabg6da+iaaic dacaGGSaGaaGPaVlaaysW7caaMe8UaamiEaiabg2da9iaaicdacaGG SaGaaGymaiaacYcacaaIYaGaaiilaiaaiodacaGGSaGaaiOlaiaac6 cacaGGUaaaaa@6B89@

which is the Poisson-Lindley distribution (PLD).

The PLD has been extensively studied by Sankaran [1] and Ghitany and Mutairi [2] and they have discussed its various properties. The PLD has been generalized by many researchers. Shanker & Mishra [3] obtained a two parameter Poisson-Lindley distribution by compounding Poisson distribution with a two parameter Lindley distribution introduced by Shanker & Mishra [4]. A quasi Poisson-Lindley distribution has been introduced by Shanker & Mishra [5] by compounding Poisson distribution with a quasi Lindley distribution introduced by Shanker & Mishra [6]. Shanker et al. [7] obtained a discrete two parameter Poisson-Lindley distribution by mixing Poisson distribution with a two parameter Lindley distribution for modeling waiting and survival time’s data introduced by Shanker et al. [8]. Further, Shanker & Tekie [9] obtained a new quasi Poisson-Lindley distribution by compounding Poisson distribution with a new quasi Lindley distribution introduced by Shanker & Amanuel [10].

In this paper, a general expression for the th factorial moment of PLD has been obtained and hence its first four moments about origin has also been obtained. It seems that not much work has been done on the applications of PLD. The PLD has been fitted to some data sets in ecology and genetics along with Poisson distribution and it has been found that PLD is more flexible for analyzing different types of count data than Poisson distribution.

Moments of Poisson-Lindley Distribution

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

μ ( r ) =E[ E( X ( r ) |λ ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGafqiVd0 MbauaalmaaBaaajqwaG9FaaSWaaeWaaKazba2=baqcLbmacaWGYbaa jqwaG9VaayjkaiaawMcaaaqcbasabaqcLbmacaaMc8EcLbsacqGH9a qpcaaMe8UaaGPaVlaadweacaaMc8Ecfa4aamWaaOqaaKqzGeGaaGPa VlaadweacaaMc8Ecfa4aaeWaaOqaaKqzGeGaamiwaSWaaWbaaKqaGe qabaWcdaqadaqcKfay=haajugWaiaadkhaaKqaGiaawIcacaGLPaaa aaqcLbsacaGG8bGaeq4UdWgakiaawIcacaGLPaaaaiaawUfacaGLDb aaaaa@5F47@ (2.1)

 where

X ( r ) =X( X1 )( X2 )...( Xr+1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGyb WcdaahaaqcKfaG=hqabaWcdaqadaqcKfaG=haajugWaiaadkhaaKaz ba4=caGLOaGaayzkaaaaaKqzGeGaeyypa0JaamiwaKqbaoaabmaake aajugibiaadIfacqGHsislcaaIXaaakiaawIcacaGLPaaajuaGdaqa daGcbaqcLbsacaWGybGaeyOeI0IaaGOmaaGccaGLOaGaayzkaaqcLb sacaaMc8UaaiOlaiaac6cacaGGUaGaaGPaVNqbaoaabmaakeaajugi biaadIfacqGHsislcaWGYbGaey4kaSIaaGymaaGccaGLOaGaayzkaa aaaa@5AD3@

From (1.3), we thus have

μ ( r ) = 0 [ x=0 x ( r ) e λ λ x x! ] θ 2 θ+1 ( 1+λ ) e θλ dλ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGafqiVd0 MbauaalmaaBaaajeaibaWcdaqadaqcbasaaKqzadGaamOCaaqcbaIa ayjkaiaawMcaaaqabaqcLbsacaaMe8Uaeyypa0JaaGjbVlaaykW7ju aGdaWdXbGcbaqcLbsacaaMc8Ecfa4aamWaaOqaaKqbaoaaqahakeaa jugibiaaykW7caWG4bWcdaahaaqcbasabeaalmaabmaajeaibaqcLb macaWGYbaajeaicaGLOaGaayzkaaaaaKqzGeGaaGjbVNqbaoaalaaa keaajugibiaadwgajuaGdaahaaWcbeqcbasaaKqzadGaeyOeI0IaaG PaVlabeU7aSbaajugibiaaysW7cqaH7oaBjuaGdaahaaWcbeqcbasa aKqzadGaamiEaaaaaOqaaKqzGeGaamiEaiaacgcaaaaaleaajugibi aadIhacqGH9aqpcaaMc8UaaGimaaWcbaqcLbsacqGHEisPaiabggHi LdaakiaawUfacaGLDbaaaSqaaKqzGeGaaGimaaWcbaqcLbsacqGHEi sPaiabgUIiYdGaaGjbVNqbaoaalaaakeaajugibiabeI7aXLqbaoaa CaaaleqajeaibaqcLbmacaaIYaaaaaGcbaqcLbsacqaH4oqCcqGHRa WkcaaMc8UaaGymaaaacaaMc8Ecfa4aaeWaaOqaaKqzGeGaaGymaiab gUcaRiaaykW7cqaH7oaBcaaMc8oakiaawIcacaGLPaaajugibiaayk W7caWGLbqcfa4aaWbaaSqabKqaGeaajugWaiabgkHiTiaaykW7cqaH 4oqCcaaMc8Uaeq4UdWgaaKqzGeGaaGjbVlaadsgacqaH7oaBaaa@9B39@

= 0 [ λ r x=r e λ λ xr ( xr )! ] θ 2 θ+1 ( 1+λ ) e θλ dλ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeyypa0 JaaGjbVNqbaoaapehakeaajugibiaaykW7juaGdaWadaGcbaqcLbsa cqaH7oaBjuaGdaahaaWcbeqcbasaaKqzadGaamOCaaaajuaGdaaeWb GcbaqcLbsacaaMe8Ecfa4aaSaaaOqaaKqzGeGaamyzaKqbaoaaCaaa leqajeaibaqcLbmacqGHsislcqaH7oaBaaqcLbsacaaMe8Uaeq4UdW wcfa4aaWbaaSqabKqaGeaajugWaiaadIhacqGHsislcaWGYbaaaaGc baqcfa4aaeWaaOqaaKqzGeGaamiEaiabgkHiTiaadkhaaOGaayjkai aawMcaaKqzGeGaaiyiaaaaaKqaGeaajugWaiaadIhacqGH9aqpcaaM c8UaamOCaaqcbasaaKqzadGaeyOhIukajugibiabggHiLdaakiaawU facaGLDbaaaKqaGeaajugWaiaaicdaaKqaGeaajugWaiabg6HiLcqc LbsacqGHRiI8aiaaysW7juaGdaWcaaGcbaqcLbsacqaH4oqCjuaGda ahaaWcbeqcbasaaKqzadGaaGOmaaaaaOqaaKqzGeGaeqiUdeNaey4k aSIaaGPaVlaaigdaaaGaaGPaVNqbaoaabmaakeaajugibiaaigdacq GHRaWkcaaMc8Uaeq4UdWMaaGPaVdGccaGLOaGaayzkaaqcLbsacaaM c8UaamyzaKqbaoaaCaaaleqajeaibaqcLbmacqGHsislcaaMc8Uaeq iUdeNaaGPaVlabeU7aSbaajugibiaaysW7caWGKbGaeq4UdWgaaa@9889@

Taking x+r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgU caRiaadkhaaaa@38CC@ in place of x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaaaa@36F3@ , we get

μ ( r ) = 0 λ r [ x=0 e λ λ x x! ] θ 2 θ+1 ( 1+λ ) e θλ dλ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGafqiVd0 MbauaajuaGdaWgaaWcbaqcfa4aaeWaaSqaaKqzGeGaamOCaaWccaGL OaGaayzkaaaabeaajugibiabg2da9KqbaoaapehakeaajugibiabeU 7aSLqbaoaaCaaaleqajeaibaqcLbmacaWGYbaaaKqzGeGaaGPaVNqb aoaadmaakeaajuaGdaaeWbGcbaqcLbsacaaMe8Ecfa4aaSaaaOqaaK qzGeGaamyzaKqbaoaaCaaaleqajeaibaqcLbmacqGHsislcaaMc8Ua eq4UdWgaaKqzGeGaaGjbVlabeU7aSLqbaoaaCaaaleqajeaibaqcLb macaWG4baaaaGcbaqcLbsacaWG4bGaaiyiaaaaaKqaGeaajugWaiaa dIhacqGH9aqpcaaMc8UaaGimaaqcbasaaKqzadGaeyOhIukajugibi abggHiLdaakiaawUfacaGLDbaaaKqaGeaajugWaiaaicdaaKqaGeaa jugWaiabg6HiLcqcLbsacqGHRiI8aiaaysW7juaGdaWcaaGcbaqcLb sacqaH4oqCjuaGdaahaaWcbeqcbasaaKqzadGaaGOmaaaaaOqaaKqz GeGaeqiUdeNaey4kaSIaaGPaVlaaigdaaaGaaGPaVNqbaoaabmaake aajugibiaaigdacqGHRaWkcaaMc8Uaeq4UdWMaaGPaVdGccaGLOaGa ayzkaaqcLbsacaaMc8UaamyzaKqbaoaaCaaaleqajeaibaqcLbmacq GHsislcaaMc8UaeqiUdeNaaGPaVlabeU7aSbaajugibiaaysW7caWG KbGaeq4UdWgaaa@9886@

The expression within the bracket is clearly unity and hence we have

μ ( r ) = θ 2 θ+1 0 λ r ( 1+λ ) e θλ dλ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGafqiVd0 MbauaalmaaBaaajeaibaWcdaqadaqcbasaaKqzadGaamOCaaqcbaIa ayjkaiaawMcaaaqabaqcLbsacqGH9aqpjuaGdaWcaaGcbaqcLbsacq aH4oqCjuaGdaahaaWcbeqcbasaaKqzadGaaGOmaaaaaOqaaKqzGeGa eqiUdeNaey4kaSIaaGPaVlaaigdaaaqcfa4aa8qCaOqaaKqzGeGaeq 4UdWwcfa4aaWbaaSqabKqaGeaajugWaiaadkhaaaaajeaibaqcLbma caaIWaaajeaibaqcLbmacqGHEisPaKqzGeGaey4kIipacaaMe8UaaG PaVNqbaoaabmaakeaajugibiaaigdacqGHRaWkcaaMc8Uaeq4UdWMa aGPaVdGccaGLOaGaayzkaaqcLbsacaaMc8UaamyzaKqbaoaaCaaale qajeaibaqcLbmacqGHsislcaaMc8UaeqiUdeNaaGPaVlabeU7aSbaa jugibiaaysW7caWGKbGaeq4UdWgaaa@747D@

Using gamma integral and a little algebraic simplification, we get finally a general expression for the th factorial moment of PLD as

μ ( r ) = r!( θ+r+1 ) θ r ( θ+1 ) ;r=1,2,3,.... MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGafqiVd0 MbauaalmaaBaaajeaibaWcdaqadaqcbasaaKqzadGaamOCaaqcbaIa ayjkaiaawMcaaaqabaqcLbsacqGH9aqpjuaGdaWcaaGcbaqcLbsaca WGYbGaaiyiaKqbaoaabmaakeaajugibiabeI7aXjabgUcaRiaadkha cqGHRaWkcaaIXaaakiaawIcacaGLPaaaaeaajugibiabeI7aXLqbao aaCaaaleqajeaibaqcLbmacaWGYbaaaKqbaoaabmaakeaajugibiab eI7aXjabgUcaRiaaigdaaOGaayjkaiaawMcaaaaajugibiaaykW7ca aMc8Uaai4oaiaadkhacqGH9aqpcaaIXaGaaiilaiaaikdacaGGSaGa aG4maiaacYcacaGGUaGaaiOlaiaac6cacaGGUaaaaa@61B6@ (2.2)

Substituting r=1,2,3,and4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGYb Gaeyypa0JaaGymaiaacYcacaaIYaGaaiilaiaaiodacaGGSaGaaGPa VlaabggacaqGUbGaaeizaiaaykW7caaMc8UaaGinaaaa@44E1@  in (2.2), first four factorial moments can be obtained and using the relationship between factorial moments and moments about origin, the first four moment about origin of the PLD (1.1) are given by

μ 1 = MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGafqiVd0 MbauaalmaaBaaajeaibaqcLbmacaaIXaaajeaibeaajugibiabg2da 9aaa@3C3A@ θ+2 θ( θ+1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaake aajugibiabeI7aXjabgUcaRiaaikdaaOqaaKqzGeGaeqiUdexcfa4a aeWaaOqaaKqzGeGaeqiUdeNaey4kaSIaaGymaaGccaGLOaGaayzkaa aaaaaa@42D2@ (2.3)

μ 2 = MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGafqiVd0 MbauaajuaGdaWgaaqcKfay=haajugWaiaaikdaaKqaGeqaaKqzGeGa eyypa0daaa@3E61@ θ + 2 θ ( θ + 1 ) + 2 ( θ + 3 ) θ 2 ( θ + 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaake aajugibiabeI7aXjabgUcaRiaaikdaaOqaaKqzGeGaeqiUdexcfa4a aeWaaOqaaKqzGeGaeqiUdeNaey4kaSIaaGymaaGccaGLOaGaayzkaa aaaKqzGeGaey4kaSscfa4aaSaaaOqaaKqzGeGaaGOmaKqbaoaabmaa keaajugibiabeI7aXjabgUcaRiaaiodaaOGaayjkaiaawMcaaaqaaK qzGeGaeqiUdexcfa4aaWbaaSqabKqaGeaajugWaiaaikdaaaqcfa4a aeWaaOqaaKqzGeGaeqiUdeNaey4kaSIaaGymaaGccaGLOaGaayzkaa aaaaaa@5766@ (2.5) μ 3 = θ+2 θ( θ+1 ) + 6( θ+3 ) θ 2 ( θ+1 ) + 6( θ+4 ) θ 3 ( θ+1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGafqiVd0 MbauaajuaGdaWgaaqcbasaaKqzadGaaG4maaWcbeaajugibiabg2da 9KqbaoaalaaakeaajugibiabeI7aXjabgUcaRiaaikdaaOqaaKqzGe GaeqiUdexcfa4aaeWaaOqaaKqzGeGaeqiUdeNaey4kaSIaaGymaaGc caGLOaGaayzkaaaaaKqzGeGaey4kaSscfa4aaSaaaOqaaKqzGeGaaG OnaKqbaoaabmaakeaajugibiabeI7aXjabgUcaRiaaiodaaOGaayjk aiaawMcaaaqaaKqzGeGaeqiUdexcfa4aaWbaaSqabKqaGeaajugWai aaikdaaaqcfa4aaeWaaOqaaKqzGeGaeqiUdeNaey4kaSIaaGymaaGc caGLOaGaayzkaaaaaKqzGeGaey4kaSscfa4aaSaaaOqaaKqzGeGaaG OnaKqbaoaabmaakeaajugibiabeI7aXjabgUcaRiaaisdaaOGaayjk aiaawMcaaaqaaKqzGeGaeqiUdexcfa4aaWbaaSqabKqaGeaajugWai aaiodaaaqcfa4aaeWaaOqaaKqzGeGaeqiUdeNaey4kaSIaaGymaaGc caGLOaGaayzkaaaaaaaa@72B9@ (2.4)

μ 4 = θ+2 θ( θ+1 ) + 14( θ+3 ) θ 2 ( θ+1 ) + 36( θ+4 ) θ 3 ( θ+1 ) + 24( θ+5 ) θ 4 ( θ+1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGafqiVd0 MbauaajuaGdaWgaaqcbasaaKqzadGaaGinaaWcbeaajugibiabg2da 9KqbaoaalaaakeaajugibiabeI7aXjabgUcaRiaaikdaaOqaaKqzGe GaeqiUdexcfa4aaeWaaOqaaKqzGeGaeqiUdeNaey4kaSIaaGymaaGc caGLOaGaayzkaaaaaKqzGeGaey4kaSscfa4aaSaaaOqaaKqzGeGaaG ymaiaaisdajuaGdaqadaGcbaqcLbsacqaH4oqCcqGHRaWkcaaIZaaa kiaawIcacaGLPaaaaeaajugibiabeI7aXLqbaoaaCaaaleqajeaiba qcLbmacaaIYaaaaKqbaoaabmaakeaajugibiabeI7aXjabgUcaRiaa igdaaOGaayjkaiaawMcaaaaajugibiabgUcaRKqbaoaalaaakeaaju gibiaaiodacaaI2aqcfa4aaeWaaOqaaKqzGeGaeqiUdeNaey4kaSIa aGinaaGccaGLOaGaayzkaaaabaqcLbsacqaH4oqCjuaGdaahaaWcbe qcbasaaKqzadGaaG4maaaajuaGdaqadaGcbaqcLbsacqaH4oqCcqGH RaWkcaaIXaaakiaawIcacaGLPaaaaaqcLbsacqGHRaWkjuaGdaWcaa GcbaqcLbsacaaIYaGaaGinaKqbaoaabmaakeaajugibiabeI7aXjab gUcaRiaaiwdaaOGaayjkaiaawMcaaaqaaKqzGeGaeqiUdexcfa4aaW baaSqabKqaGeaajugWaiaaisdaaaqcfa4aaeWaaOqaaKqzGeGaeqiU deNaey4kaSIaaGymaaGccaGLOaGaayzkaaaaaaaa@8986@ (2.6)

Ghitany et al. [2] discussed the estimation methods for the PLD (1.1) and its applications.

Estimation Of Parameters

Maximum Likelihood (ML) Estimates: Let x 1 , x 2 , , x n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG4b WcdaWgaaqcbasaaKqzadGaaGymaaqcbasabaqcLbsacaGGSaGaamiE aKqbaoaaBaaajeaibaqcLbmacaaIYaaaleqaaKqzGeGaaiilaiablA ciljaacYcacaWG4bqcfa4aaSbaaKqaGeaajugWaiaad6gaaSqabaaa aa@4608@ be a random sample of size n from the PLD (1.1). Let f x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGMb qcfa4aaSbaaKqaGeaajugWaiaadIhaaSqabaaaaa@3A7F@ be the observed frequency in the sample corresponding to X = x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGyb Gaeyypa0JaamiEaaaa@3965@ ( x = 1 , 2 , 3 , ... , k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG4b Gaeyypa0JaaGymaiaacYcacaaIYaGaaiilaiaaiodacaGGSaGaaiOl aiaac6cacaGGUaGaaiilaiaadUgaaaa@4082@ ) such that x = 1 k f x = n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaabCaO qaaKqzGeGaamOzaKqbaoaaBaaajeaibaqcLbmacaWG4baaleqaaaqc basaaKqzadGaamiEaiabg2da9iaaigdaaKqaGeaajugWaiaadUgaaK qzGeGaeyyeIuoacqGH9aqpcaWGUbaaaa@4627@ , where k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Aaaaa@36E6@ is the largest observed value having non-zero frequency. The likelihood function, L MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamitaaaa@36C7@ , of the PLD (1.1) is given by

L= θ 2n 1 ( θ+1 ) x=1 k f x ( x+3 ) x=1 k ( x+θ+2 ) f x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGmb Gaeyypa0JaeqiUdexcfa4aaWbaaSqabKqaGeaajugWaiaaikdacaWG UbaaaKqbaoaalaaakeaajugibiaaigdaaOqaaKqbaoaabmaakeaaju gibiabeI7aXjabgUcaRiaaigdaaOGaayjkaiaawMcaaKqbaoaaCaaa leqabaqcfa4aaabCaSqaaKqzGeGaamOzaKqbaoaaBaaajiaibaqcLb macaWG4baameqaaKqbaoaabmaaleaajugibiaadIhacqGHRaWkcaaI ZaaaliaawIcacaGLPaaaaKGaGeaajugWaiaadIhacqGH9aqpcaaIXa aajiaibaqcLbmacaWGRbaajugibiabggHiLdaaaaaacaaMc8UaaGPa VNqbaoaaxadakeaajugibiabg+GivdqcbasaaKqzadGaamiEaiabg2 da9iaaigdaaKqaGeaajugWaiaadUgaaaqcfa4aaeWaaOqaaKqzGeGa amiEaiabgUcaRiabeI7aXjabgUcaRiaaikdaaOGaayjkaiaawMcaaK qbaoaaCaaaleqabaqcLbsacaWGMbqcfa4aaSbaaKGaGeaajugWaiaa dIhaaWqabaaaaaaa@73DA@

The log likelihood function is given by

LogL=2nlogθ x=1 k f x ( x+3 ) log( θ+1 )+ x=1 k f x log( x+θ+2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaqGmb Gaae4BaiaabEgacaWGmbGaeyypa0JaaGOmaiaad6gacaaMc8UaciiB aiaac+gacaGGNbGaeqiUdeNaeyOeI0scfa4aaabCaOqaaKqzGeGaam OzaKqbaoaaBaaajeaibaqcLbmacaWG4baaleqaaKqbaoaabmaakeaa jugibiaadIhacqGHRaWkcaaIZaaakiaawIcacaGLPaaaaKqaGeaaju gWaiaadIhacqGH9aqpcaaIXaaajeaibaqcLbmacaWGRbaajugibiab ggHiLdGaciiBaiaac+gacaGGNbqcfa4aaeWaaOqaaKqzGeGaeqiUde Naey4kaSIaaGymaaGccaGLOaGaayzkaaqcLbsacqGHRaWkjuaGdaae WbGcbaqcLbsacaWGMbqcfa4aaSbaaKqaGeaajugWaiaadIhaaSqaba aajeaibaqcLbmacaWG4bGaeyypa0JaaGymaaqcbasaaKqzadGaam4A aaqcLbsacqGHris5aiGacYgacaGGVbGaai4zaKqbaoaabmaakeaaju gibiaadIhacqGHRaWkcqaH4oqCcqGHRaWkcaaIYaaakiaawIcacaGL Paaaaaa@7A85@

The maximum likelihood estimate, θ ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacuaH4o qCgaqcaaaa@384B@  of θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH4o qCaaa@383B@  is the solution of the equation dlogL dθ =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaO qaaKqzGeGaamizaiGacYgacaGGVbGaai4zaiaadYeaaOqaaKqzGeGa amizaiabeI7aXbaacqGH9aqpcaaIWaaaaa@40AF@  and is given by solution of the following non-linear equation

2n θ n( x ¯ +3 ) θ+1 + x=1 k f x x+θ+2 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaO qaaKqzGeGaaGOmaiaad6gaaOqaaKqzGeGaeqiUdehaaiabgkHiTKqb aoaalaaakeaajugibiaad6gajuaGdaqadaGcbaqcLbsaceWG4bGbae bacqGHRaWkcaaIZaaakiaawIcacaGLPaaaaeaajugibiabeI7aXjab gUcaRiaaigdaaaGaey4kaSscfa4aaabCaOqaaKqbaoaalaaakeaaju gibiaadAgajuaGdaWgaaqcbasaaKqzadGaamiEaaWcbeaaaOqaaKqz GeGaamiEaiabgUcaRiabeI7aXjabgUcaRiaaikdaaaaajeaibaqcLb macaWG4bGaeyypa0JaaGymaaqcbasaaKqzadGaam4AaaqcLbsacqGH ris5aiabg2da9iaaicdaaaa@5EE0@

Where x ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaceWG4b Gbaebaaaa@379A@  is the sample mean. It has been shown by Ghitany & Mutairi [2] that the ML estimator θ ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacuaH4o qCgaqcaaaa@384B@  of θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH4o qCaaa@383B@  is consistent and asymptotically normal.

Estimates from Moments: Let x 1 , x 2 ,, x n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG4b qcfa4aaSbaaKqaGeaajugWaiaaigdaaSqabaqcLbsacaGGSaGaamiE aKqbaoaaBaaajeaibaqcLbmacaaIYaaaleqaaKqzGeGaaiilaiablA ciljaacYcacaWG4bqcfa4aaSbaaKqaGeaajugWaiaad6gaaSqabaaa aa@466C@  be a random sample of size n from the PLD (1.1). Equating the first moment about origin to the sample mean, the method of moment (MOM) estimate, θ ˜ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacuaH4o qCgaacaaaa@384A@ , of θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH4o qCaaa@383B@  is given by

θ ˜ = ( x ¯ 1 )+ ( x ¯ 1 ) 2 +8 x ¯ 2 x ¯ ; x ¯ >0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacuaH4o qCgaacaiabg2da9KqbaoaalaaakeaajugibiabgkHiTKqbaoaabmaa keaajugibiqadIhagaqeaiabgkHiTiaaigdaaOGaayjkaiaawMcaaK qzGeGaey4kaSscfa4aaOaaaOqaaKqbaoaabmaakeaajugibiqadIha gaqeaiabgkHiTiaaigdaaOGaayjkaiaawMcaaKqbaoaaCaaaleqaje aibaqcLbmacaaIYaaaaKqzGeGaey4kaSIaaGioaiqadIhagaqeaaWc beaaaOqaaKqzGeGaaGOmaiqadIhagaqeaaaacaaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caGG7aGabmiEayaaraGaeyOpa4JaaGim aaaa@5DDF@

Where x ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiEayaara aaaa@370B@ is the sample mean? It has been shown by Ghitany & Mutairi [2] that the MOM estimator θ ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacuaH4o qCgaqcaaaa@384B@  of θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcaaIaeqiUde haaa@37D5@  is positively biased, consistent and asymptotically normal.

Applications of Poisson-Lindley Distribution

The Poisson distribution is a suitable model for the situations where events seem to occur at random such as 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, etc. However, the Poisson distribution requires events to be independent- a condition which is rarely satisfied completely. In biological science and medical science, the occurrence of successive events is dependent. The negative binomial distribution is a possible alternative to the Poisson distribution when successive events are possibly dependent [11]. Further, for fitting Poisson distribution to the count data equality of mean and variance should be satisfied. Similarly, 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 not fully satisfied.

The theoretical and empirical justification for the selection of the PLD to describe biological science and medical science data is that PLD is over dispersed ( μ< σ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcaaIaeqiVd0 MaeyipaWJaeq4WdmNcdaahaaqcbasabeaajugWaiaaikdaaaaaaa@3CDC@ )

Application in Ecology

The organisms and their environment in the nature are not only complex and dynamic but also interdependent, mutually reactive and interrelated. Ecology deals with the various principles which govern such relationship between organisms and their environment. Fisher et al. [12] has discussed the applications of Logarithmic series distribution (LSD) to model count data in the science of ecology. It was Kempton [13] 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 [14] proposed another generalization of the Logarithmic series distribution (LSD) and fitted it to insect data and found that it gives better fit as compared to ordinary Logarithmic series distribution. They concluded that the distribution is flexible to describe short-tailed as well as long-tailed data. Mishra & Shanker [15] have discussed applications of generalized logarithmic series distributions (GLSD) to models data in ecology.

In this section we have tried to fit Poisson distribution and Poisson -Lindley distribution to many biological data 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 (Table 1-3) that PLD gives much closer fit than Poisson distribution and thus it can be considered as an important tool for modeling data in ecology.

Number of cells per square

Observed frequency

Expected frequency

Poisson distribution

Poisson-Lindley distribution

0
1

128
37

118.1
54.3

127.4
41.1

2
3
4
5+

18
3
1
0

12.5 1.9 0.2 0.0 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaK qzaeabaeqajuaGbaqcLbqacaaIXaGaaGOmaiaac6cacaaI1aaajuaG baqcLbqacaaIXaGaaiOlaiaaiMdaaKqbagaajugabiaaicdacaGGUa GaaGOmaaqcfayaaKqzaeGaaGimaiaac6cacaaIWaaaaKqbakaaw2ha aaaa@4601@

12.9 3.9 1.2 0.5 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaK qzaeabaeqajuaGbaqcLbqacaaIXaGaaGOmaiaac6cacaaI5aaajuaG baqcLbqacaaIZaGaaiOlaiaaiMdaaKqbagaajugabiaaigdacaGGUa GaaGOmaaqcfayaaKqzaeGaaGimaiaac6cacaaI1aaaaKqbakaaw2ha aaaa@460D@

Total

187

187.0

187.0

Estimate of parameter

 

θ ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbqacuaH4o qCgaqcaaaa@382B@ =0.459893

θ ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbqacuaH4o qCgaqcaaaa@382B@ =2.751579

χ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbqacqaHhp WyjuaGdaahaaWcbeqcbasaaKqzGcGaaGOmaaaaaaa@3ACB@

 

9.903

1.431

d.f.

 

1

1

p-value

 

0.0016

0.2316

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

Poisson distribution

Poisson-Lindley distribution

0

38

25.3

35.8

1

17

29.1

20.7

2

10

16.7

11.4

3
4
5
6
7+

9
3
2
1
0

6.4 1.8 0.4 0.2 0.1 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaK qzaeabaeqajuaGbaqcLbqacaaI2aGaaiOlaiaaisdaaKqbagaajuga biaaigdacaGGUaGaaGioaaqcfayaaKqzaeGaaGimaiaac6cacaaI0a aajuaGbaqcLbqacaaIWaGaaiOlaiaaikdaaKqbagaajugabiaaicda caGGUaGaaGymaaaajuaGcaGL9baaaaa@4871@

6.0
3.1 1.6 0.8 0.6 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaK qzaeabaeqajuaGbaqcLbqacaaIZaGaaiOlaiaaigdaaKqbagaajuga biaaigdacaGGUaGaaGOnaaqcfayaaKqzaeGaaGimaiaac6cacaaI4a aajuaGbaqcLbqacaaIWaGaaiOlaiaaiAdaaaqcfaOaayzFaaaaaa@454C@

Total

80

80.0

80.0

Estimate of parameter

 

θ ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbqacuaH4o qCgaqcaaaa@382B@ =1.15

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

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

 

18.275

2.469

d.f.

 

2

3

p-value

 

0.0001

0.4809

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

Number of bores per plant

Observed frequency

Expected frequency

Poisson distribution

Poisson-Lindley distribution

0

83

78.9

87.2

1

36

42.9

31.8

2
3
4+

14
2
1

11.7 2.1 0.4 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaK qzaeabaeqajuaGbaqcLbqacaaIXaGaaGymaiaac6cacaaI3aaajuaG baqcLbqacaaIYaGaaiOlaiaaigdaaKqbagaajugabiaaicdacaGGUa GaaGinaaaajuaGcaGL9baaaaa@42D9@

11.2 3.8 2.0 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaK qzaeabaeqajuaGbaqcLbqacaaIXaGaaGymaiaac6cacaaIYaaajuaG baqcLbqacaaIZaGaaiOlaiaaiIdaaKqbagaajugabiaaikdacaGGUa GaaGimaaaajuaGcaGL9baaaaa@42DA@

Total

136

136.0

136.0

Estimate of parameter

 

θ ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaaaaa@384A@ =0.544118

θ ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaaaaa@384A@ =2.372252

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

 

1.885

0.757

d.f.

 

1

1

p-value

 

0.1698

0.3843

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

Application in Genetics

Number of aberrations

Observed frequency

Expected frequency

Poisson distribution

Poisson-Lindley distribution

0

268

231.3

257.0

1

87

126.7

93.4

2

26

34.7

32.8

3
4
5
6
7+

9
4
2
1
3

6.3 0.8 0.1 0.1 0.1 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaK qzaeabaeqajuaGbaqcLbqacaaI2aGaaiOlaiaaiodaaKqbagaajuga biaaicdacaGGUaGaaGioaaqcfayaaKqzaeGaaGimaiaac6cacaaIXa aajuaGbaqcLbqacaaIWaGaaiOlaiaaigdaaKqbagaajugabiaaicda caGGUaGaaGymaaaajuaGcaGL9baaaaa@486B@

11.2
3.8 1.2 0.4 0.2 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaK qzaeabaeqajuaGbaqcLbqacaaIZaGaaiOlaiaaiIdaaKqbagaajuga biaaigdacaGGUaGaaGOmaaqcfayaaKqzaeGaaGimaiaac6cacaaI0a aajuaGbaqcLbqacaaIWaGaaiOlaiaaikdaaaqcfaOaayzFaaaaaa@4547@

Total

400

400.0

400.0

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=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aaaaa@37BC@ =2.380442

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

 

38.208

6.208

d.f.

 

2

3

p-value

 

0.0000

0.1019

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

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. In the field of genetics much quantitative studies seem to have been done. 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=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHhp WyjuaGdaahaaWcbeqcbasaaKqzadGaaGOmaaaaaaa@3B0B@ ). In the analysis of data observed on chemically induced chromosome aberrations in cultures of human leukocytes, Loeschke & Kohler [16] suggested the negative binomial distribution while Janardan & Schaeffer [17] suggested modified Poisson distribution. Mishra & Shanker [15] have discussed applications of generalized Logarithmic series distributions (GLSD) to model data in mortality, ecology and genetics. In this section an attempt has been made to fit to data relating to genetics using PLD and Poisson distribution using maximum likelihood estimate. Also an attempt has been made to fit Poisson distribution and PLD to the data of Catcheside et al.[19,20] in (Tables 3-7) [21].

Class/Exposure
( )

Observed frequency

Expected frequency

Poisson distribution

Poisson-Lindley distribution

0

413

374.0

405.7

1

124

177.4

133.6

2

42

42.1

42.6

3
4
5
6

15
5
0
2

6.6 0.8 0.1 0.0 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaK qzaeabaeqajuaGbaqcLbqacaaI2aGaaiOlaiaaiAdaaKqbagaajuga biaaicdacaGGUaGaaGioaaqcfayaaKqzaeGaaGimaiaac6cacaaIXa aajuaGbaqcLbqacaaIWaGaaiOlaiaaicdaaaqcfaOaayzFaaaaaa@4548@

13.3
4.1 1.2 0.5 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaK qzaeabaeqajuaGbaqcLbqacaaI0aGaaiOlaiaaigdaaKqbagaajuga biaaigdacaGGUaGaaGOmaaqcfayaaKqzaeGaaGimaiaac6cacaaI1a aaaKqbakaaw2haaaaa@421C@

Total

601

601.0

601.0

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

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

 

48.169

1.336

d.f.

 

2

3

p-value

 

0.0000

0.7206

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=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbqacqaH8o qBcaWGNbGaaiiFaiaadUgacaWGNbaaaa@3BE3@

Class/Exposure
( )

Observed frequency

Expected frequency

Poisson distribution

Poisson-Lindley distribution

0

200

172.5

191.8

1

57

95.4

70.3

2

30

26.4

24.9

3
4
5
6

7
4
0
2

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=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaK qzaeabaeqajuaGbaqcLbqacaaI4aGaaiOlaiaaiAdaaKqbagaajuga biaaikdacaGGUaGaaGyoaaqcfayaaKqzaeGaaGymaiaac6cacaaIWa aajuaGbaqcLbqacaaIWaGaaiOlaiaaiwdaaaqcfaOaayzFaaaaaa@4552@

Total

300

300.0

300.0

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

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

 

29.68

3.91

d.f.

 

2

2

p-value

 

0.0000

0.1415

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=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbqacqaH8o qBcaWGNbGaaiiFaiaadUgacaWGNbaaaa@3BE3@

Class/Exposure
( )

Observed frequency

Expected frequency

Poisson distribution

Poisson-Lindley distribution

0

155

127.8

158.3

1

83

109.0

77.2

2

33

46.5

35.9

3
4
5
6

14
11
3
1

13.2 2.8 0.5 0.2 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaK qzaeabaeqajuaGbaqcLbqacaaIXaGaaG4maiaac6cacaaIYaaajuaG baqcLbqacaaIYaGaaiOlaiaaiIdaaKqbagaajugabiaaicdacaGGUa GaaGynaaqcfayaaKqzaeGaaGimaiaac6cacaaIYaaaaKqbakaaw2ha aaaa@4604@

16.1
7.1 3.1 2.3 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaK qzaeabaeqajuaGbaqcLbqacaaI3aGaaiOlaiaaigdaaKqbagaajuga biaaiodacaGGUaGaaGymaaqcfayaaKqzaeGaaGOmaiaac6cacaaIZa aaaKqbakaaw2haaaaa@4220@

Total

300

300.0

300.0

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

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

 

24.969

1.51

d.f.

 

2

3

p-value

 

0.0000

0.6799

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=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbqacqaH8o qBcaWGNbGaaiiFaiaadUgacaWGNbaaaa@3BE3@

It is obvious from above tables that PLD gives much closer fit than Poisson distribution and thus it can be considered as an important tool for modeling data in genetics.

References

  1. Sankaran M (1970) The discrete Poisson-Lindley distribution. Biometrics 26(1): 145-149.
  2. Ghitany ME, Al-Mutairi DK (2009) Estimation Methods for the discrete Poisson-Lindley distribution. Journal of Statistical Computation and Simulation 79(1): 1-9.
  3. Shanker R, Mishra A (2014) A two-parameter Poisson-Lindley distribution. International Journal of Statistics and Systems 9(1): 79-85.
  4. Shanker R, Mishra A (2013 a) A two-parameter Lindley distribution. Statistics in Transition new Series 14(1): 45-56.
  5. Shanker R, Mishra A (2015) A quasi Poisson-Lindley distribution (submitted).
  6. Shanker R, Mishra A (2013 b) A quasi Lindley distribution. African journal of Mathematics and Computer Science Research 6(4): 64-71.
  7. Shanker R, Sharma S, Shanker R (2012) A Discrete two-Parameter Poisson Lindley Distribution. Journal of Ethiopian Statistical Association 21: 15-22.
  8. Shanker R, Sharma S, Shanker, R (2013) A two-parameter Lindley distribution for modeling waiting and survival times data, Applied Mathematics 4: 363-368.
  9. Shanker R, Tekie AL (2014) A new quasi Poisson-Lindley distribution. International Journal of Statistics and Systems 9(1): 87-94.
  10. Shanker R, Amanuel AG (2013) A new quasi Lindley distribution, International Journal of Statistics and Systems 8(2): 143-156.
  11. Johnson NL, Kotz S, Kemp AW (1992) Univariate Discrete Distributions, 2nd edition, John Wiley & sons Inc.
  12. Fisher RA, Corpet AS, Williams CB (1943) The relation between the number of species and the number of individuals in a random sample of an animal population. Journal of Animal Ecology 12(1): 42-58.
  13. Kempton RA (1975) A generalized form of Fisher’s logarithmic series. Biometrika 62 (1): 29-38.
  14. Tripathi RC, Gupta RC (1985) A generalization of the log-series distribution. Comm. in Stat. (Theory and Methods) 14(8): 1779-1799.
  15. Mishra A, Shanker R (2002) Generalized logarithmic series distribution-Its nature and applications, Proceedings of the Vth International Symposium on Optimization and Statistics 28-30: 155-168.
  16. Loeschke V, Kohler W (1976) Deterministic and Stochastic models of the negative binomial distribution and the analysis of chromosomal aberrations in human leukocytes. Biometrische Zeitschrift 18: 427-451.
  17. Janardan KG, Schaeffer DJ (1977) Models for the analysis of chromosomal aberrations in human leukocytes. Biometrical Journal (8)599-612.
  18. Mc Guire JU, Brindley TA, Bancroft TA (1957) The distribution of European corn-borer larvae pyrausta in field corn. Biometrics 13: 65-78.
  19. Catcheside DG, Lea DE, Thoday JM (1946) Types of chromosome structural      change induced by the irradiation on Tradescantia microspores. Journal of Genetics 47: 113-136.
  20. Catcheside DG, Lea DE, Thoday JM (1946) The production of chromosome        structural changes in Tradescantia microspores in relation to dosage, intensity and temperature. Journal of Genetics l.47: 137-149.
  21. Lindley DV (1958) Fiducial distributions and Bayes theorem. Journal of Royal Statistical Society Ser. B Vol.20: 102-107.
© 2014-2016 MedCrave Group, All rights reserved. No part of this content may be reproduced or transmitted in any form or by any means as per the standard guidelines of fair use.
Creative Commons License Open Access by MedCrave Group is licensed under a Creative Commons Attribution 4.0 International License.
Based on a work at http://medcraveonline.com
Best viewed in Mozilla Firefox | Google Chrome | Above IE 7.0 version | Opera |Privacy Policy