ISSN: 2378-315X BBIJ

Biometrics & Biostatistics International Journal
Research Article
Volume 3 Issue 1 - 2016
Poisson Area-Biased Lindley Distribution and its Applications on Biological Data
Shakila Bashir* and Mujahid Rasul
Department of Statistics, Forman Christian College, Pakistan
Received:December 07, 2015 | Published: January 13, 2016
*Corresponding author: Shakila Bashir, Assistant Professor, Department of Statistics, Forman Christian College (A Chartered University) Ferozepur Road Lahore (54600), Pakistan, Tel: +92 (42) 9923 1581; Email: ,
Citation: Bashir S, Rasul M (2016) Poisson Area-Biased Lindley Distribution and its Applications on Biological Data. Biom Biostat Int J 3(1): 00058. DOI: 10.15406/bbij.2016.03.00058

Abstract

The purpose of this paper is to introduce a discrete distribution named Poisson-area-biased Lindley distribution and its applications on biological data. Poisson area-biased Lindley distribution is introduced with some of its basic properties including moments, coefficient of skewness and kurtosis are discussed. The method of moments and maximum likelihood estimation of the parameters of Poisson area-biased Lindley distribution are investigated. It is found that the parameter estimated by method of moments is positively biased, consistent and asymptotically normal. Application of the model to some biological data sets is compared with Poisson distribution.

Keywords: PABLD; PD; PLD; Area-biased; MOM; MLE; Factorial moments

Introduction

Lindley [1] introduced a single parameter distribution named as Lindley distribution with probability distribution function (pdf)

f( x;θ )= θ 2 θ+1 ( 1+x ) e θx , x>0,θ>0. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGMb qcfa4aaeWaaOqaaKqzGeGaamiEaiaacUdacqaH4oqCaOGaayjkaiaa wMcaaKqzGeGaeyypa0tcfa4aaSaaaOqaaKqzGeGaeqiUdexcfa4aaW baaSqabeaajugWaiaaikdaaaaakeaajugibiabeI7aXjabgUcaRiaa igdaaaqcfa4aaeWaaOqaaKqzGeGaaGymaiabgUcaRiaadIhaaOGaay jkaiaawMcaaKqzGeGaamyzaKqbaoaaCaaaleqabaqcLbmacqGHsisl cqaH4oqCcaWG4baaaKqzGeGaaiilauaabeqabmaaaOqaaaqaaaqaaK qzGeGaamiEaiabg6da+iaaicdacaGGSaGaeqiUdeNaeyOpa4JaaGim aiaac6caaaaaaa@5E85@ (1.1)

The pdf (1.1) is the mixture of exponential ( θ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeGaeqiUdehakiaawIcacaGLPaaaaaa@3A66@ and gamma ( 2,θ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeGaaGOmaiaacYcacqaH4oqCaOGaayjkaiaawMcaaaaa@3BD2@ distributions. The cumulative distribution function (cdf) of the Lindley distribution is

F( x )=1 θ+1+θx θ+1 e θx , x>0,θ>0. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGgb qcfa4aaeWaaOqaaKqzGeGaamiEaaGccaGLOaGaayzkaaqcLbsacqGH 9aqpcaaIXaGaeyOeI0scfa4aaSaaaOqaaKqzGeGaeqiUdeNaey4kaS IaaGymaiabgUcaRiabeI7aXjaadIhaaOqaaKqzGeGaeqiUdeNaey4k aSIaaGymaaaacaWGLbqcfa4aaWbaaSqabeaajugWaiabgkHiTiabeI 7aXjaadIhaaaqcLbsacaGGSaqbaeqabeWaaaGcbaaabaaabaqcLbsa caWG4bGaeyOpa4JaaGimaiaacYcacqaH4oqCcqGH+aGpcaaIWaGaai Olaaaaaaa@5A42@  (1.2)

The first two moments of the Lindley distribution are

μ 1 = θ+2 θ( θ+1 ) , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacuaH8o qBgaqbaKqbaoaaBaaaleaajugWaiaaigdaaSqabaqcLbsacqGH9aqp juaGdaWcaaGcbaqcLbsacqaH4oqCcqGHRaWkcaaIYaaakeaajugibi abeI7aXLqbaoaabmaakeaajugibiabeI7aXjabgUcaRiaaigdaaOGa ayjkaiaawMcaaaaajugibiaacYcaaaa@4AB0@   μ 2 = 2( θ+3 ) θ 2 ( θ+1 ) . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacuaH8o qBgaqbaKqbaoaaBaaaleaajugWaiaaikdaaSqabaqcLbsacqGH9aqp juaGdaWcaaGcbaqcLbsacaaIYaqcfa4aaeWaaOqaaKqzGeGaeqiUde Naey4kaSIaaG4maaGccaGLOaGaayzkaaaabaqcLbsacqaH4oqCjuaG daahaaWcbeqaaKqzadGaaGOmaaaajuaGdaqadaGcbaqcLbsacqaH4o qCcqGHRaWkcaaIXaaakiaawIcacaGLPaaaaaqcLbsacaGGUaaaaa@50C5@

Sankaran [2] introduced the Lindley mixture of Poisson distribution named Poisson-Lindley distribution with the following pdf

f( x;θ )= θ 2 ( x+θ+2 ) ( θ+1 ) . , x=0,1,2,......., θ>0. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGMb qcfa4aaeWaaOqaaKqzGeGaamiEaiaacUdacqaH4oqCaOGaayjkaiaa wMcaaKqzGeGaeyypa0tcfa4aaSaaaOqaaKqzGeGaeqiUdexcfa4aaW baaSqabeaajugWaiaaikdaaaqcfa4aaeWaaOqaaKqzGeGaamiEaiab gUcaRiabeI7aXjabgUcaRiaaikdaaOGaayjkaiaawMcaaaqaaKqbao aabmaakeaajugibiabeI7aXjabgUcaRiaaigdaaOGaayjkaiaawMca aKqbaoaaCaaaleqabaqcLbmacaGGUaaaaaaajugibiaacYcafaqabe qaeaaaaOqaaaqaaaqaaKqzGeGaamiEaiabg2da9iaaicdacaGGSaGa aGymaiaacYcacaaIYaGaaiilaiaac6cacaGGUaGaaiOlaiaac6caca GGUaGaaiOlaiaac6cacaGGSaaakeaajugibiabeI7aXjabg6da+iaa icdacaGGUaaaaaaa@67D6@ (1.3)

The pdf (1.3) is applied to count data and arises from Poisson distribution when its parameter λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH7o aBaaa@3839@ follows a Lindley distribution. Ghitany & Al-Mutairi [3] discussed various properties of the Lindley distribution. Ghitany & Al-Mutairi [3] introduced size-biased Poisson Lindley distribution with applications. They considered the size biased form of the Poisson-Lindley distribution. Ghitany & Al-Mutairi [4] discussed estimation methods for the discrete Poisson-Lindley distribution. Srivastava & Adhikari [5] introduced a size-biased Poisson-Lindley distribution which is obtained by considering the size-biased form of the Poisson distribution with Lindley distribution without its size-biased form. Adhikari & Srivastava [6] proposed a Poisson size-biased Lindley distribution which is obtained by computing Poisson distribution without its size-biased form with size-biased Lindley distribution. Shanker & Fesshaye [7] discussed Poisson-Lindley distribution with several of its properties including factorial moments and parameter estimation. They applied the Poisson-Lindley distribution on ecology and genetics data sets and showed that it can be an important tool for modeling biological science data.

Rao [8] introduced the distributions that are used in situations when the recorded observations do not have an equal probability of selection and do not have the original distribution. The distributions used to handle such situations are called weighted distributions. Suppose that the original distribution comes from a distribution with pdf f 0 ( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGMb qcfa4aaSbaaSqaaKqzadGaaGimaaWcbeaajuaGdaqadaGcbaqcLbsa caWG4baakiaawIcacaGLPaaaaaa@3DD4@  and the observations is recorded to a probability re-weighted by a weight function w( x )>0, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG3b qcfa4aaeWaaOqaaKqzGeGaamiEaaGccaGLOaGaayzkaaqcLbsacqGH +aGpcaaIWaGaaiilaaaa@3E39@ then the weighted distribution is defined as

f( x )= w( x ) E[ w( X ) ] f 0 ( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGMb qcfa4aaeWaaOqaaKqzGeGaamiEaaGccaGLOaGaayzkaaqcLbsacqGH 9aqpjuaGdaWcaaGcbaqcLbsacaWG3bqcfa4aaeWaaOqaaKqzGeGaam iEaaGccaGLOaGaayzkaaaabaqcLbsacaWGfbqcfa4aamWaaOqaaKqz GeGaam4DaKqbaoaabmaakeaajugibiaadIfaaOGaayjkaiaawMcaaa Gaay5waiaaw2faaaaajugibiaadAgajuaGdaWgaaWcbaqcLbmacaaI WaaaleqaaKqbaoaabmaakeaajugibiaadIhaaOGaayjkaiaawMcaaa aa@5389@ (1.4)

The weighted distribution with w( x )=x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG3b qcfa4aaeWaaOqaaKqzGeGaamiEaaGccaGLOaGaayzkaaqcLbsacqGH 9aqpcaWG4baaaa@3DCA@  is called size-biased/length-biased distributions and w( x )= x 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4Dam aabmaabaGaamiEaaGaayjkaiaawMcaaiabg2da9iaadIhadaahaaqa bKqbGeaacaaIYaaaaaaa@3D15@ is called area-biased distribution. Patil & Ord [9] discussed size-biased sampling and related form-invariant weighted distributions. Patil & Rao [10] discussed some models leading to weighted distributions and showed applications of weighted distributions in many real sampling problems. Mir & Ahmad [11] introduced size-biased form of some discrete distributions with their applications.

In this paper we consider the Poisson area-biased Lindley distribution (PABLD) which is obtained by considering Poisson distribution without its area-biased form with area-biased Lindley distribution (ABLD).

Poisson Area-Biased Lindley Distribution

The Poisson area-biased Lindley distribution (PABLD) arises from the Poisson distribution with pdf

f( x;λ )= e λ λ x x! , x=0,1,2,...... λ>0, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aabmaabaGaamiEaiaacUdacqaH7oaBaiaawIcacaGLPaaacqGH9aqp daWcaaqaaiaadwgadaahaaqabKqbGeaacqGHsislcqaH7oaBaaqcfa Oaeq4UdW2aaWbaaeqajuaibaGaamiEaaaaaKqbagaacaWG4bGaaiyi aaaacaGGSaqbaeqabeabaaaabaaabaaabaGaamiEaiabg2da9iaaic dacaGGSaGaaGymaiaacYcacaaIYaGaaiilaiaac6cacaGGUaGaaiOl aiaac6cacaGGUaGaaiOlaaqaaiabeU7aSjabg6da+iaaicdacaGGSa aaaaaa@566F@  (2.1)

when its parameter λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4UdW gaaa@3838@ follows the area-biased Lindley distribution (ABLD) in (2.1) with pdf

f( x;θ )= θ 4 2( θ+3 ) x 2 ( 1+x ) e θx , x>0,θ>0. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aabmaabaGaamiEaiaacUdacqaH4oqCaiaawIcacaGLPaaacqGH9aqp daWcaaqaaiabeI7aXnaaCaaabeqcfasaaiaaisdaaaaajuaGbaGaaG OmamaabmaabaGaeqiUdeNaey4kaSIaaG4maaGaayjkaiaawMcaaaaa caWG4bWaaWbaaeqajuaibaGaaGOmaaaajuaGdaqadaqaaiaaigdacq GHRaWkcaWG4baacaGLOaGaayzkaaGaamyzamaaCaaabeqcfasaaiab gkHiTiabeI7aXjaadIhaaaqcfaOaaiilauaabeqabmaaaeaaaeaaae aacaWG4bGaeyOpa4JaaGimaiaacYcacqaH4oqCcqGH+aGpcaaIWaGa aiOlaaaaaaa@5AE6@  (2.3)

So

0 f( x;λ )f( λ;θ ) dλ= θ 4 2( θ+3 )x! 0 e λ( θ+1 ) ( λ x+2 + λ x+3 ) dλ. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aa8qCae aacaWGMbWaaeWaaeaacaWG4bGaai4oaiabeU7aSbGaayjkaiaawMca aiaadAgadaqadaqaaiabeU7aSjaacUdacqaH4oqCaiaawIcacaGLPa aaaeaacaaIWaaabaGaeyOhIukacqGHRiI8aiaadsgacqaH7oaBcqGH 9aqpdaWcaaqaaiabeI7aXnaaCaaabeqcfasaaiaaisdaaaaajuaGba GaaGOmamaabmaabaGaeqiUdeNaey4kaSIaaG4maaGaayjkaiaawMca aiaadIhacaGGHaaaamaapehabaGaamyzamaaCaaabeqcfasaaiabgk HiTiabeU7aSLqbaoaabmaajuaibaGaeqiUdeNaey4kaSIaaGymaaGa ayjkaiaawMcaaaaajuaGdaqadaqaaiabeU7aSnaaCaaabeqcfasaai aadIhacqGHRaWkcaaIYaaaaKqbakabgUcaRiabeU7aSnaaCaaabeqc fasaaiaadIhacqGHRaWkcaaIZaaaaaqcfaOaayjkaiaawMcaaaqaai aaicdaaeaacqGHEisPaiabgUIiYdGaamizaiabeU7aSjaac6caaaa@7454@

After simplifying it the pdf of PABLD is obtained

f( x;θ )= ( θ θ+1 ) 4 ( x+1 )( x+2 )( θ+x+4 ) 2( θ+3 ) ( θ+1 ) x , x=0,1,2,..... θ>0. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aabmaabaGaamiEaiaacUdacqaH4oqCaiaawIcacaGLPaaacqGH9aqp daqadaqaamaalaaabaGaeqiUdehabaGaeqiUdeNaey4kaSIaaGymaa aaaiaawIcacaGLPaaadaahaaqabKqbGeaacaaI0aaaaKqbaoaalaaa baWaaeWaaeaacaWG4bGaey4kaSIaaGymaaGaayjkaiaawMcaamaabm aabaGaamiEaiabgUcaRiaaikdaaiaawIcacaGLPaaadaqadaqaaiab eI7aXjabgUcaRiaadIhacqGHRaWkcaaI0aaacaGLOaGaayzkaaaaba GaaGOmamaabmaabaGaeqiUdeNaey4kaSIaaG4maaGaayjkaiaawMca amaabmaabaGaeqiUdeNaey4kaSIaaGymaaGaayjkaiaawMcaamaaCa aabeqcfasaaiaadIhaaaaaaKqbakaacYcafaqabeqaeaaaaeaaaeaa aeaacaWG4bGaeyypa0JaaGimaiaacYcacaaIXaGaaiilaiaaikdaca GGSaGaaiOlaiaac6cacaGGUaGaaiOlaiaac6caaeaacqaH4oqCcqGH +aGpcaaIWaGaaiOlaaaaaaa@6FBD@  (2.4)

Properties of the poisson-area-biased-lindley distribution

The factorial moments of the PABLD in (2.1)

μ ( r ) =E[ E( x ( r ) /λ ) ], MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiVd0 MbauaadaWgaaqaamaabmaajuaibaGaamOCaaqcfaOaayjkaiaawMca aaqabaGaeyypa0JaamyramaadmaabaGaamyramaabmaabaWaaSGbae aacaWG4bWaaWbaaeqabaWcdaqadaqcfasaaKqzadGaamOCaaqcfaOa ayjkaiaawMcaaaaaaeaacqaH7oaBaaaacaGLOaGaayzkaaaacaGLBb GaayzxaaGaaiilaaaa@49C6@

μ ( r ) = ( θ+r+3 )( r+2 )! 2( θ+3 ) θ r . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiVd0 MbauaadaWgaaqaamaabmaajuaibaGaamOCaaqcfaOaayjkaiaawMca aaqabaGaeyypa0ZaaSaaaeaadaqadaqaaiabeI7aXjabgUcaRiaadk hacqGHRaWkcaaIZaaacaGLOaGaayzkaaWaaeWaaeaacaWGYbGaey4k aSIaaGOmaaGaayjkaiaawMcaaiaacgcaaeaacaaIYaWaaeWaaeaacq aH4oqCcqGHRaWkcaaIZaaacaGLOaGaayzkaaGaeqiUde3aaWbaaeqa juaibaGaamOCaaaaaaqcfaOaaiOlaaaa@520A@ (2.5)

For r=1,2,3&4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOCai abg2da9iaaigdacaGGSaGaaGOmaiaacYcacaaIZaGaaiOjaiaaisda aaa@3D7D@ in (2.5), the first four factorial moments of the PABLD are

μ ( 1 ) = 3( θ+4 ) θ( θ+3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiVd0 MbauaadaWgaaqcfasaaKqbaoaabmaajuaibaGaaGymaaGaayjkaiaa wMcaaaqcfayabaGaeyypa0ZaaSaaaeaacaaIZaWaaeWaaeaacqaH4o qCcqGHRaWkcaaI0aaacaGLOaGaayzkaaaabaGaeqiUde3aaeWaaeaa cqaH4oqCcqGHRaWkcaaIZaaacaGLOaGaayzkaaaaaaaa@4969@ , μ ( 2 ) = 12( θ+5 ) θ 2 ( θ+3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiVd0 MbauaadaWgaaqcfasaaKqbaoaabmaajuaibaGaaGOmaaGaayjkaiaa wMcaaaqcfayabaGaeyypa0ZaaSaaaeaacaaIXaGaaGOmamaabmaaba GaeqiUdeNaey4kaSIaaGynaaGaayjkaiaawMcaaaqaaiabeI7aXnaa CaaabeqcfasaaKqzadGaaGOmaaaajuaGdaqadaqaaiabeI7aXjabgU caRiaaiodaaiaawIcacaGLPaaaaaaaaa@4CED@ , μ ( 3 ) = 60( θ+6 ) θ 3 ( θ+3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiVd0 MbauaadaWgaaqcfasaaKqbaoaabmaajuaibaGaaG4maaGaayjkaiaa wMcaaaqcfayabaGaeyypa0ZaaSaaaeaacaaI2aGaaGimamaabmaaba GaeqiUdeNaey4kaSIaaGOnaaGaayjkaiaawMcaaaqaaiabeI7aXnaa Caaabeqcfasaaiaaiodaaaqcfa4aaeWaaeaacqaH4oqCcqGHRaWkca aIZaaacaGLOaGaayzkaaaaaaaa@4BC5@ , μ ( 4 ) = 360( θ+7 ) θ 4 ( θ+3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiVd0 MbauaadaWgaaqcfasaaKqbaoaabmaajuaibaGaaGinaaGaayjkaiaa wMcaaaqcfayabaGaeyypa0ZaaSaaaeaacaaIZaGaaGOnaiaaicdada qadaqaaiabeI7aXjabgUcaRiaaiEdaaiaawIcacaGLPaaaaeaacqaH 4oqCdaahaaqabKqbGeaacaaI0aaaaKqbaoaabmaabaGaeqiUdeNaey 4kaSIaaG4maaGaayjkaiaawMcaaaaaaaa@4C85@ (2.6)

Since the first four raw moments of the PABLD are

μ 1 = 3( θ+4 ) θ( θ+3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiVd0 MbauaadaWgaaqcfasaaiaaigdaaKqbagqaaiabg2da9maalaaabaGa aG4mamaabmaabaGaeqiUdeNaey4kaSIaaGinaaGaayjkaiaawMcaaa qaaiabeI7aXnaabmaabaGaeqiUdeNaey4kaSIaaG4maaGaayjkaiaa wMcaaaaaaaa@4724@ , μ 2 = 3( θ 2 +8θ+30 ) θ 2 ( θ+3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiVd0 MbauaadaWgaaqcfasaaiaaikdaaKqbagqaaiabg2da9maalaaabaGa aG4mamaabmaabaGaeqiUde3aaWbaaeqajuaibaGaaGOmaaaajuaGcq GHRaWkcaaI4aGaeqiUdeNaey4kaSIaaG4maiaaicdaaiaawIcacaGL PaaaaeaacqaH4oqCdaahaaqabKqbGeaacaaIYaaaaKqbaoaabmaaba GaeqiUdeNaey4kaSIaaG4maaGaayjkaiaawMcaaaaaaaa@4E6C@ (2.7)

μ 3 = 3( θ 3 +16 θ 2 +80θ+120 ) θ 3 ( θ+3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiVd0 MbauaadaWgaaqcfasaaiaaiodaaKqbagqaaiabg2da9maalaaabaGa aG4mamaabmaabaGaeqiUde3aaWbaaeqajuaibaGaaG4maaaajuaGcq GHRaWkcaaIXaGaaGOnaiabeI7aXnaaCaaabeqcfasaaiaaikdaaaqc faOaey4kaSIaaGioaiaaicdacqaH4oqCcqGHRaWkcaaIXaGaaGOmai aaicdaaiaawIcacaGLPaaaaeaacqaH4oqCdaahaaqabKqbGeaacaaI ZaaaaKqbaoaabmaabaGaeqiUdeNaey4kaSIaaG4maaGaayjkaiaawM caaaaaaaa@5590@ , μ 4 = 3( θ 4 +32 θ 3 +260 θ 2 +840θ+840 ) θ 4 ( θ+3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiVd0 MbauaadaWgaaqcfasaaiaaisdaaKqbagqaaiabg2da9maalaaabaGa aG4mamaabmaabaGaeqiUde3aaWbaaeqajuaibaGaaGinaaaajuaGcq GHRaWkcaaIZaGaaGOmaiabeI7aXnaaCaaabeqcfasaaiaaiodaaaqc faOaey4kaSIaaGOmaiaaiAdacaaIWaGaeqiUde3aaWbaaeqajuaiba GaaGOmaaaajuaGcqGHRaWkcaaI4aGaaGinaiaaicdacqaH4oqCcqGH RaWkcaaI4aGaaGinaiaaicdaaiaawIcacaGLPaaaaeaacqaH4oqCda ahaaqabKqbGeaacaaI0aaaaKqbaoaabmaabaGaeqiUdeNaey4kaSIa aG4maaGaayjkaiaawMcaaaaaaaa@5CC1@ (2.8)

The mean moments of PABLD are

μ 2 = σ 2 = 3( θ 3 +8 θ 2 +30θ+42 ) θ 2 ( θ+3 ) 2 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaKqbGeaacaaIYaaajuaGbeaacqGH9aqpcqaHdpWCdaahaaqa bKqbGeaacaaIYaaaaKqbakabg2da9maalaaabaGaaG4mamaabmaaba GaeqiUde3aaWbaaeqajuaibaGaaG4maaaajuaGcqGHRaWkcaaI4aGa eqiUde3aaWbaaeqajuaibaGaaGOmaaaajuaGcqGHRaWkcaaIZaGaaG imaiabeI7aXjabgUcaRiaaisdacaaIYaaacaGLOaGaayzkaaaabaGa eqiUde3aaWbaaeqajuaibaGaaGOmaaaajuaGdaqadaqaaiabeI7aXj abgUcaRiaaiodaaiaawIcacaGLPaaadaahaaqabKqbGeaacaaIYaaa aaaajuaGcaGGUaaaaa@5ABC@  (2.9)

μ 3 = 3( θ 5 +10 θ 4 +14 θ 3 +36 θ 2 2160θ+2664 ) θ 3 ( θ+3 ) 3 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaKqbGeaacaaIZaaajuaGbeaacqGH9aqpdaWcaaqaaiaaioda daqadaqaaiabeI7aXnaaCaaabeqcfasaaiaaiwdaaaqcfaOaey4kaS IaaGymaiaaicdacqaH4oqCdaahaaqabKqbGeaacaaI0aaaaKqbakab gUcaRiaaigdacaaI0aGaeqiUde3aaWbaaeqajqwba+FaaKqzadGaaG 4maaaajuaGcqGHRaWkcaaIZaGaaGOnaiabeI7aXnaaCaaabeqcfasa aiaaikdaaaqcfaOaeyOeI0IaaGOmaiaaigdacaaI2aGaaGimaiabeI 7aXjabgUcaRiaaikdacaaI2aGaaGOnaiaaisdaaiaawIcacaGLPaaa aeaacqaH4oqCdaahaaqabKqbGeaacaaIZaaaaKqbaoaabmaabaGaeq iUdeNaey4kaSIaaG4maaGaayjkaiaawMcaamaaCaaabeqcfasaaiaa iodaaaaaaKqbakaac6caaaa@6864@  (2.10)

μ 4 = 3( θ 7 +20 θ 6 +2 θ 5 +61122 θ 4 366276 θ 3 548280 θ 2 +19224θ+41688 ) θ 4 ( θ+3 ) 4 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaKqbGeaacaaI0aaajuaGbeaacqGH9aqpdaWcaaqaaiaaioda daqadaqaaiabeI7aXnaaCaaabeqcfasaaiaaiEdaaaqcfaOaey4kaS IaaGOmaiaaicdacqaH4oqCdaahaaqabKqbGeaacaaI2aaaaKqbakab gUcaRiaaikdacqaH4oqCdaahaaqabKqbGeaacaaI1aaaaKqbakabgU caRiaaiAdacaaIXaGaaGymaiaaikdacaaIYaGaeqiUde3aaWbaaeqa juaibaGaaGinaaaajuaGcqGHsislcaaIZaGaaGOnaiaaiAdacaaIYa GaaG4naiaaiAdacqaH4oqCdaahaaqabKqbGeaacaaIZaaaaKqbakab gkHiTiaaiwdacaaI0aGaaGioaiaaikdacaaI4aGaaGimaiabeI7aXn aaCaaabeqcfasaaiaaikdaaaqcfaOaey4kaSIaaGymaiaaiMdacaaI YaGaaGOmaiaaisdacqaH4oqCcqGHRaWkcaaI0aGaaGymaiaaiAdaca aI4aGaaGioaaGaayjkaiaawMcaaaqaaiabeI7aXnaaCaaabeqcfasa aiaaisdaaaqcfa4aaeWaaeaacqaH4oqCcqGHRaWkcaaIZaaacaGLOa GaayzkaaWaaWbaaeqajuaibaGaaGinaaaaaaqcfaOaaiOlaaaa@79DA@  (2.11)

The coefficient of skewness and kurtosis of the PABLD are

γ 1 = β 1 = ( θ 5 +10 θ 4 +14 θ 3 +36 θ 2 2160θ2664 ) 3 ( θ 3 +8 θ 2 +30θ+42 ) 3 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHZo WzjuaGdaWgaaqcfasaaKqzadGaaGymaaqcfayabaqcLbsacqGH9aqp juaGdaGcaaqaaKqzGeGaeqOSdiwcfa4aaSbaaKqbGeaajugWaiaaig daaKqbagqaaaqabaqcLbsacqGH9aqpjuaGdaWcaaqaamaabmaabaqc LbsacqaH4oqCjuaGdaahaaqabKqbGeaajugWaiaaiwdaaaqcLbsacq GHRaWkcaaIXaGaaGimaiabeI7aXLqbaoaaCaaabeqcfasaaKqzadGa aGinaaaajugibiabgUcaRiaaigdacaaI0aGaeqiUdexcfa4aaWbaae qajuaibaqcLbmacaaIZaaaaKqzGeGaey4kaSIaaG4maiaaiAdacqaH 4oqCjuaGdaahaaqabKqbGeaajugWaiaaikdaaaqcLbsacqGHsislca aIYaGaaGymaiaaiAdacaaIWaGaeqiUdeNaeyOeI0IaaGOmaiaaiAda caaI2aGaaGinaaqcfaOaayjkaiaawMcaaaqaamaakaaabaqcLbsaca aIZaqcfa4aaeWaaeaajugibiabeI7aXLqbaoaaCaaabeqcfasaaKqz adGaaG4maaaajugibiabgUcaRiaaiIdacqaH4oqCjuaGdaahaaqabK qbGeaajugWaiaaikdaaaqcLbsacqGHRaWkcaaIZaGaaGimaiabeI7a XjabgUcaRiaaisdacaaIYaaajuaGcaGLOaGaayzkaaWaaWbaaeqaju aibaqcLbmacaaIZaaaaaqcfayabaaaaKqzGeGaaiOlaaaa@879A@  (2.12)

β 2 = ( θ 7 +20 θ 6 +2 θ 5 +61122 θ 4 366276 θ 3 548280 θ 2 +19224θ+41688 ) 3 ( θ 3 +8 θ 2 +30θ+42 ) 2 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOSdi 2aaSbaaKqbGeaacaaIYaaajuaGbeaacqGH9aqpdaWcaaqaamaabmaa baGaeqiUde3aaWbaaeqajuaibaGaaG4naaaajuaGcqGHRaWkcaaIYa GaaGimaiabeI7aXnaaCaaabeqcfasaaiaaiAdaaaqcfaOaey4kaSIa aGOmaiabeI7aXnaaCaaabeqcfasaaiaaiwdaaaqcfaOaey4kaSIaaG OnaiaaigdacaaIXaGaaGOmaiaaikdacqaH4oqCdaahaaqabKqbGeaa caaI0aaaaKqbakabgkHiTiaaiodacaaI2aGaaGOnaiaaikdacaaI3a GaaGOnaiabeI7aXnaaCaaabeqcfasaaiaaiodaaaqcfaOaeyOeI0Ia aGynaiaaisdacaaI4aGaaGOmaiaaiIdacaaIWaGaeqiUde3aaWbaae qajuaibaGaaGOmaaaajuaGcqGHRaWkcaaIXaGaaGyoaiaaikdacaaI YaGaaGinaiabeI7aXjabgUcaRiaaisdacaaIXaGaaGOnaiaaiIdaca aI4aaacaGLOaGaayzkaaaabaGaaG4mamaabmaabaGaeqiUde3aaWba aeqajuaibaGaaG4maaaajuaGcqGHRaWkcaaI4aGaeqiUde3aaWbaae qajuaibaGaaGOmaaaajuaGcqGHRaWkcaaIZaGaaGimaiabeI7aXjab gUcaRiaaisdacaaIYaaacaGLOaGaayzkaaWaaWbaaeqajuaibaGaaG OmaaaaaaqcfaOaaiOlaaaa@81CA@  (2.13)

For the PABLD, from (2.12) and (2.13) it can be seen that ( γ 1 , β 2 )( 5.65,7.88 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacqaHZoWzdaWgaaqcfasaaiaaigdaaKqbagqaaiaacYcacqaHYoGy daWgaaqcfasaaiaaikdaaKqbagqaaaGaayjkaiaawMcaaiabgkziUo aabmaabaGaeyOeI0IaaGynaiaac6cacaaI2aGaaGynaiaacYcacaaI 3aGaaiOlaiaaiIdacaaI4aaacaGLOaGaayzkaaaaaa@4A30@  as θ0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde NaeyOKH4QaaGimaaaa@3AE1@ , the model is negatively skewed and leptokurtic.

Some more properties of the PABLD are

f( x+1;θ ) f( x;θ ) = ( x+3 )( θ+x+5 ) ( θ+1 )( x+1 )( θ+x+4 ) . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacaWGMbWaaeWaaeaacaWG4bGaey4kaSIaaGymaiaacUdacqaH4oqC aiaawIcacaGLPaaaaeaacaWGMbWaaeWaaeaacaWG4bGaai4oaiabeI 7aXbGaayjkaiaawMcaaaaacqGH9aqpdaWcaaqaamaabmaabaGaamiE aiabgUcaRiaaiodaaiaawIcacaGLPaaadaqadaqaaiabeI7aXjabgU caRiaadIhacqGHRaWkcaaI1aaacaGLOaGaayzkaaaabaWaaeWaaeaa cqaH4oqCcqGHRaWkcaaIXaaacaGLOaGaayzkaaWaaeWaaeaacaWG4b Gaey4kaSIaaGymaaGaayjkaiaawMcaamaabmaabaGaeqiUdeNaey4k aSIaamiEaiabgUcaRiaaisdaaiaawIcacaGLPaaaaaGaaiOlaaaa@6066@

f( x+1;θ ) f( x;θ ) = ( 1+ 3 x )( θ+ 1 x +5 ) ( θ+1 )( 1+ 1 x )( θ+ 1 x +4 ) . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacaWGMbWaaeWaaeaacaWG4bGaey4kaSIaaGymaiaacUdacqaH4oqC aiaawIcacaGLPaaaaeaacaWGMbWaaeWaaeaacaWG4bGaai4oaiabeI 7aXbGaayjkaiaawMcaaaaacqGH9aqpdaWcaaqaamaabmaabaGaaGym aiabgUcaRmaalaaabaGaaG4maaqaaiaadIhaaaaacaGLOaGaayzkaa WaaeWaaeaacqaH4oqCcqGHRaWkdaWcaaqaaiaaigdaaeaacaWG4baa aiabgUcaRiaaiwdaaiaawIcacaGLPaaaaeaadaqadaqaaiabeI7aXj abgUcaRiaaigdaaiaawIcacaGLPaaadaqadaqaaiaaigdacqGHRaWk daWcaaqaaiaaigdaaeaacaWG4baaaaGaayjkaiaawMcaamaabmaaba GaeqiUdeNaey4kaSYaaSaaaeaacaaIXaaabaGaamiEaaaacqGHRaWk caaI0aaacaGLOaGaayzkaaaaaiaac6caaaa@6392@  (2.15)

The dispersion of the PABLD is defined to be

From equation (2.14) and Table 1, it can be observed that the PABLD is over-dispersed but as θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde NaeyOKH4QaeyOhIukaaa@3B98@ then μ= σ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 Maeyypa0Jaeq4Wdm3aaWbaaeqajuaibaGaaGOmaaaaaaa@3C0F@ and the PABLD is equi-dispersed. Therefore for large θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde haaa@383A@ the PABLD is equi-dispersed.

 

θ

μ= σ 2 3( θ 2 +18θ+42 ) θ 2 ( θ+3 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd0Maey ypa0Jaeq4Wdm3aaWbaaSqabeaacaaIYaaaaOGaeyOeI0YaaSaaaeaa caaIZaWaaeWaaeaacqaH4oqCdaahaaWcbeqaaiaaikdaaaGccqGHRa WkcaaIXaGaaGioaiabeI7aXjabgUcaRiaaisdacaaIYaaacaGLOaGa ayzkaaaabaGaeqiUde3aaWbaaSqabeaacaaIYaaaaOWaaeWaaeaacq aH4oqCcqGHRaWkcaaIZaaacaGLOaGaayzkaaWaaWbaaSqabeaacaaI Yaaaaaaaaaa@5035@

 

θ

μ= σ 2 3( θ 2 +18θ+42 ) θ 2 ( θ+3 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd0Maey ypa0Jaeq4Wdm3aaWbaaSqabeaacaaIYaaaaOGaeyOeI0YaaSaaaeaa caaIZaWaaeWaaeaacqaH4oqCdaahaaWcbeqaaiaaikdaaaGccqGHRa WkcaaIXaGaaGioaiabeI7aXjabgUcaRiaaisdacaaIYaaacaGLOaGa ayzkaaaabaGaeqiUde3aaWbaaSqabeaacaaIYaaaaOWaaeWaaeaacq aH4oqCcqGHRaWkcaaIZaaacaGLOaGaayzkaaWaaWbaaSqabeaacaaI Yaaaaaaaaaa@5035@

0.5

σ2 — 50.20408

19

σ2 — 0.012792

1

σ2 — 11.4375

20

σ2 — 0.011371

2

σ2 — 2.46

21

σ2 — 0.010169

3

σ2 — 0.972222

22

σ2 — 0.009144

4

σ2 — 0.497449

23

σ2 — 0.008263

5

σ2 — 0.294375

24

σ2 — 0.007502

6

σ2 — 0.191358

25

σ2 — 0.006839

7

σ2 — 0.132857

26

σ2 — 0.006258

8

σ2 — 0.096849

27

σ2 — 0.005748

9

σ2 — 0.073302

28

σ2 — 0.005296

10

σ2 — 0.05716

29

σ2 — 0.004894

11

σ2 — 0.045665

30

σ2 — 0.004536

12

σ2 — 0.037222

31

σ2 — 0.004215

13

σ2 — 0.030857

32

σ2 — 0.003927

14

σ2 — 0.025952

50

σ2 — 0.00147

15

σ2 — 0.022099

100

σ2 — 0.000335

16

σ2 — 0.019023

500

σ2 — 1.23E-05

17

σ2 — 0.016531

1000

σ2 — 3.04E-06

18

σ2 — 0.014487

σ2

Table 1: The dispersion of PABLD for different values of θ.

Method of Moments

If x 1 , x 2 ,...., x n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEam aaBaaajuaibaGaaGymaaqcfayabaGaaiilaiaadIhadaWgaaqcfasa aiaaikdaaKqbagqaaiaacYcacaGGUaGaaiOlaiaac6cacaGGUaGaai ilaiaadIhadaWgaaqcfasaaiaad6gaaKqbagqaaaaa@4354@  be the random sample from PABLD with pdf (2.4), the method of moments (MOM) estimate θ ˜ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaGaaaaa@3849@ of the parameter θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde haaa@383A@  is given by

θ ˜ = 3( x ¯ 1 )+ 9 ( x ¯ 1 ) 2 +48 x ¯ 2 x ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaGaacqGH9aqpdaWcaaqaaiabgkHiTiaaiodadaqadaqaaiqadIha gaqeaiabgkHiTiaaigdaaiaawIcacaGLPaaacqGHRaWkdaGcaaqaai aaiMdadaqadaqaaiqadIhagaqeaiabgkHiTiaaigdaaiaawIcacaGL PaaadaahaaqabKqbGeaacaaIYaaaaKqbakabgUcaRiaaisdacaaI4a GabmiEayaaraaabeaaaeaacaaIYaGabmiEayaaraaaaaaa@4C2C@  (3.1)

Theorem 1: The MOM estimator 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@ is positively biased.

Proof: Let θ ˜ =ψ( x ¯ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaGaacqGH9aqpcqaHipqEdaqadaqaaiqadIhagaqeaaGaayjkaiaa wMcaaaaa@3DBB@ , where Ψ( z )= 3( z1 )+ 9 ( z1 ) 2 +48z 2z . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeuiQdK 1aaeWaaeaacaWG6baacaGLOaGaayzkaaGaeyypa0ZaaSaaaeaacqGH sislcaaIZaWaaeWaaeaacaWG6bGaeyOeI0IaaGymaaGaayjkaiaawM caaiabgUcaRmaakaaabaGaaGyoamaabmaabaGaamOEaiabgkHiTiaa igdaaiaawIcacaGLPaaadaahaaqabKqbGeaacaaIYaaaaKqbakabgU caRiaaisdacaaI4aGaamOEaaqabaaabaGaaGOmaiaadQhaaaGaaiOl aaaa@4ED8@

So,

ψ ( z )= 78z+69 z 2 +297 z 3 +108 z 4 +( 108z+405 z 2 +135 z 3 ) 9 ( z1 ) 2 +48z 4 z 4 [ 9 ( z1 ) 2 +48z ] 3/2 >0, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiYdK NbayaadaqadaqaaiaadQhaaiaawIcacaGLPaaacqGH9aqpdaWcaaqa aiaaiEdacaaI4aGaamOEaiabgUcaRiaaiAdacaaI5aGaamOEamaaCa aabeqcfasaaiaaikdaaaqcfaOaey4kaSIaaGOmaiaaiMdacaaI3aGa amOEamaaCaaabeqcfasaaiaaiodaaaqcfaOaey4kaSIaaGymaiaaic dacaaI4aGaamOEamaaCaaabeqcfasaaiaaisdaaaqcfaOaey4kaSYa aeWaaeaacaaIXaGaaGimaiaaiIdacaWG6bGaey4kaSIaaGinaiaaic dacaaI1aGaamOEamaaCaaabeqcfasaaiaaikdaaaqcfaOaey4kaSIa aGymaiaaiodacaaI1aGaamOEamaaCaaabeqcfasaaiaaiodaaaaaju aGcaGLOaGaayzkaaWaaOaaaeaacaaI5aWaaeWaaeaacaWG6bGaeyOe I0IaaGymaaGaayjkaiaawMcaamaaCaaabeqcfasaaiaaikdaaaqcfa Oaey4kaSIaaGinaiaaiIdacaWG6baabeaaaeaacaaI0aGaamOEamaa Caaabeqcfasaaiaaisdaaaqcfa4aamWaaeaacaaI5aWaaeWaaeaaca WG6bGaeyOeI0IaaGymaaGaayjkaiaawMcaamaaCaaabeqcfasaaiaa ikdaaaqcfaOaey4kaSIaaGinaiaaiIdacaWG6baacaGLBbGaayzxaa WaaWbaaeqajuaibaqcfa4aaSGbaKqbGeaacaaIZaaabaGaaGOmaaaa aaaaaKqbakabg6da+iaaicdacaGGSaaaaa@7EC5@  (3.2)

Then Ψ( z ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeuiQdK 1aaeWaaeaacaWG6baacaGLOaGaayzkaaaaaa@3A9B@  is strictly convex. By using the Jensen’s inequality we have

E{ ψ( X ¯ ) }>ψ{ E( X ¯ ) }. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyram aacmaabaGaeqiYdK3aaeWaaeaaceWGybGbaebaaiaawIcacaGLPaaa aiaawUhacaGL9baacqGH+aGpcqaHipqEdaGadaqaaiaadweadaqada qaaiqadIfagaqeaaGaayjkaiaawMcaaaGaay5Eaiaaw2haaiaac6ca aaa@46CC@

Since ψ{ E( X ¯ ) }=ψ( μ )=ψ( 3( θ+4 ) θ( θ+3 ) )=θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiYdK 3aaiWaaeaacaWGfbWaaeWaaeaaceWGybGbaebaaiaawIcacaGLPaaa aiaawUhacaGL9baacqGH9aqpcqaHipqEdaqadaqaaiabeY7aTbGaay jkaiaawMcaaiabg2da9iabeI8a5naabmaabaWaaSaaaeaacaaIZaWa aeWaaeaacqaH4oqCcqGHRaWkcaaI0aaacaGLOaGaayzkaaaabaGaeq iUde3aaeWaaeaacqaH4oqCcqGHRaWkcaaIZaaacaGLOaGaayzkaaaa aaGaayjkaiaawMcaaiabg2da9iabeI7aXbaa@5737@ , therefore E( θ ˜ )>θ. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyram aabmaabaGafqiUdeNbaGaaaiaawIcacaGLPaaacqGH+aGpcqaH4oqC caGGUaaaaa@3E0C@

Theorem 2: The MOM estimator 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@  is consistent and asymptotically normal:

n ( θ ˜ θ ) d N( 0, ν 2 ( θ ) ). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaOaaae aacaWGUbaabeaadaqadaqaaiqbeI7aXzaaiaGaeyOeI0IaeqiUdeha caGLOaGaayzkaaWaa4ajaeaacaWGKbaabeGaayPKHaGaamOtamaabm aabaGaaGimaiaacYcacqaH9oGBdaahaaqabKqbGeaacaaIYaaaaKqb aoaabmaabaGaeqiUdehacaGLOaGaayzkaaaacaGLOaGaayzkaaGaai Olaaaa@4AED@

Where

ν 2 ( θ )= θ 2 ( θ+3 ) 2 ( θ 3 +8 θ 2 +30θ+42 ) 3( θ 2 +8θ+12 ) . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqyVd4 2aaWbaaeqajuaibaGaaGOmaaaajuaGdaqadaqaaiabeI7aXbGaayjk aiaawMcaaiabg2da9maalaaabaGaeqiUde3aaWbaaeqajuaibaGaaG OmaaaajuaGdaqadaqaaiabeI7aXjabgUcaRiaaiodaaiaawIcacaGL PaaadaahaaqabKqbGeaacaaIYaaaaKqbaoaabmaabaGaeqiUde3aaW baaeqajuaqbaGaaG4maaaajuaGcqGHRaWkcaaI4aGaeqiUde3aaWba aeqajuaibaGaaGOmaaaajuaGcqGHRaWkcaaIZaGaaGimaiabeI7aXj abgUcaRiaaisdacaaIYaaacaGLOaGaayzkaaaabaGaaG4mamaabmaa baGaeqiUde3aaWbaaeqajuaibaGaaGOmaaaajuaGcqGHRaWkcaaI4a GaeqiUdeNaey4kaSIaaGymaiaaikdaaiaawIcacaGLPaaaaaGaaiOl aaaa@6447@  (3.3)

Proof: -

Consistency: Since μ<, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 MaeyipaWJaeyOhIuQaaiilaaaa@3B5F@  then X ¯ P μ. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOabmiway aaraWaa4ajaeaacaWGqbaabeGaayPKHaGaeqiVd0MaaiOlaaaa@3C39@  And ψ( z ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiYdK 3aaeWaaeaacaWG6baacaGLOaGaayzkaaaaaa@3ADA@  is a continuous function at z=μ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOEai abg2da9iabeY7aTbaa@3A3F@ , then ψ( X ¯ ) P ψ( μ ), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiYdK 3aaeWaaeaaceWGybGbaebaaiaawIcacaGLPaaadaGdKaqaaiaadcfa aeqacaGLsgcacqaHipqEdaqadaqaaiabeY7aTbGaayjkaiaawMcaai aacYcaaaa@42E5@  i-e. θ ˜ P θ. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaGaadaGdKaqaaiaadcfaaeqacaGLsgcacqaH4oqCcaGGUaaaaa@3D09@

Asymptotic normality: as σ 2 < MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4Wdm 3aaWbaaeqajuaibaGaaGOmaaaajuaGcqGH8aapcqGHEisPaaa@3C56@  then by using the central limit theorem we have

n ( X ¯ μ ) d N( 0, σ 2 ). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaOaaae aacaWGUbaabeaadaqadaqaaiqadIfagaqeaiabgkHiTiabeY7aTbGa ayjkaiaawMcaamaaoqcabaGaamizaaqabiaawkziaiaad6eadaqada qaaiaaicdacaGGSaGaeq4Wdm3aaWbaaeqajuaibaGaaGOmaaaaaKqb akaawIcacaGLPaaacaGGUaaaaa@46E9@

ψ( μ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiYdK 3aaeWaaeaacqaH8oqBaiaawIcacaGLPaaaaaa@3B91@  is a differentiable function and ψ ( μ )0, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiYdK NbauaadaqadaqaaiabeY7aTbGaayjkaiaawMcaaiabgcMi5kaaicda caGGSaaaaa@3ECE@  then by using the delta-method we have

n ( ψ( X ¯ )ψ( μ ) ) d N( 0, [ ψ ( μ ) ] 2 σ 2 ). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaOaaae aacaWGUbaabeaadaqadaqaaiabeI8a5naabmaabaGabmiwayaaraaa caGLOaGaayzkaaGaeyOeI0IaeqiYdK3aaeWaaeaacqaH8oqBaiaawI cacaGLPaaaaiaawIcacaGLPaaadaGdKaqaaiaadsgaaeqacaGLsgca caWGobWaaeWaaeaacaaIWaGaaiilamaadmaabaGafqiYdKNbauaada qadaqaaiabeY7aTbGaayjkaiaawMcaaaGaay5waiaaw2faamaaCaaa beqcfasaaiaaikdaaaqcfaOaeq4Wdm3aaWbaaeqajuaibaGaaGOmaa aaaKqbakaawIcacaGLPaaacaGGUaaaaa@563C@

Finally we have ψ( X ¯ )= θ ˜ , ψ( μ )=θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiYdK 3aaeWaaeaaceWGybGbaebaaiaawIcacaGLPaaacqGH9aqpcuaH4oqC gaacauaabeqabiaaaeaacaGGSaaabaGaeqiYdK3aaeWaaeaacqaH8o qBaiaawIcacaGLPaaacqGH9aqpcqaH4oqCaaaaaa@4621@  and

ψ ( μ )= 16μ6 9 ( μ1 ) 2 +48μ 4 μ 2 9 ( μ1 ) 2 +48μ = θ 2 ( θ+3 ) 2 3( θ 2 +8θ+12 ) . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiYdK NbauaadaqadaqaaiabeY7aTbGaayjkaiaawMcaaiabg2da9maalaaa baGaeyOeI0IaaGymaiabgkHiTiaaiAdacqaH8oqBcqGHsislcaaI2a WaaOaaaeaacaaI5aWaaeWaaeaacqaH8oqBcqGHsislcaaIXaaacaGL OaGaayzkaaWaaWbaaeqajuaibaGaaGOmaaaajuaGcqGHRaWkcaaI0a GaaGioaiabeY7aTbqabaaabaGaaGinaiabeY7aTnaaCaaabeqcfasa aiaaikdaaaqcfa4aaOaaaeaacaaI5aWaaeWaaeaacqaH8oqBcqGHsi slcaaIXaaacaGLOaGaayzkaaWaaWbaaeqajuaibaGaaGOmaaaajuaG cqGHRaWkcaaI0aGaaGioaiabeY7aTbqabaaaaiabg2da9iabgkHiTm aalaaabaGaeqiUde3aaWbaaeqajuaibaGaaGOmaaaajuaGdaqadaqa aiabeI7aXjabgUcaRiaaiodaaiaawIcacaGLPaaadaahaaqabKqbGe aacaaIYaaaaaqcfayaaiaaiodadaqadaqaaiabeI7aXnaaCaaabeqc fasaaiaaikdaaaqcfaOaey4kaSIaaGioaiabeI7aXjabgUcaRiaaig dacaaIYaaacaGLOaGaayzkaaaaaiaac6caaaa@7617@  (3.4)

The theorem 2 follow the asymptotic 100( 1α )% MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaaGymai aaicdacaaIWaWaaeWaaeaacaaIXaGaeyOeI0IaeqySdegacaGLOaGa ayzkaaGaaiyjaaaa@3E2C@  confidence interval for θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde haaa@383A@  is

θ ˜ ± z α 2 ν( θ ˜ ) n . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaGaacqGHXcqScaWG6bWaaSbaaeaadaWcaaqaaiabeg7aHbqaaiaa ikdaaaaabeaadaWcaaqaaiabe27aUnaabmaabaGafqiUdeNbaGaaai aawIcacaGLPaaaaeaadaGcaaqaaiaad6gaaeqaaaaacaGGUaaaaa@448D@  (3.5)

Maximum Likelihood Estimation

Let x 1 , x 2 ,...., x n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEam aaBaaajuaibaGaaGymaaqcfayabaGaaiilaiaadIhadaWgaaqcfasa aiaaikdaaKqbagqaaiaacYcacaGGUaGaaiOlaiaac6cacaGGUaGaai ilaiaadIhadaWgaaqcfasaaiaad6gaaKqbagqaaaaa@4354@  be the random sample on size n from PABLD with pdf (2.4), the maximum likelihood estimate (MLE) θ ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaaaaa@384A@ of the parameter θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde haaa@383A@  is the solution of the non-linear equation:

4n θ n( 4 x ¯ ) ( θ+1 ) n ( θ+3 ) + i=1 n 1 θ+ x i +4 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacaaI0aGaamOBaaqaaiabeI7aXbaacqGHsisldaWcaaqaaiaad6ga daqadaqaaiaaisdacqGHsislceWG4bGbaebaaiaawIcacaGLPaaaae aadaqadaqaaiabeI7aXjabgUcaRiaaigdaaiaawIcacaGLPaaaaaGa eyOeI0YaaSaaaeaacaWGUbaabaWaaeWaaeaacqaH4oqCcqGHRaWkca aIZaaacaGLOaGaayzkaaaaaiabgUcaRmaaqahabaWaaSaaaeaacaaI XaaabaGaeqiUdeNaey4kaSIaamiEamaaBaaajuaibaGaamyAaaqcfa yabaGaey4kaSIaaGinaaaaaKqbGeaacaWGPbGaeyypa0JaaGymaaqa aiaad6gaaKqbakabggHiLdGaeyypa0JaaGimaaaa@5CD3@  (4.1)

Applications

In this section the PABLD is applied to some biological data sets and compared with PD.

  1. Guire, et al. [12] gave data on European corn borers per plant with 0, 1, 2, 3 and 4 and counts 83, 36, 14, 2, and 1.
  2. Form Table 2, it can be seen that the PABLD gives much closer fit than the PD and PLD to the data set of number of bores per plant . Thus PABLD provides a better alternative to PD and PLD for modeling count data sets.

  3. Beall [13] gave the distribution of Pyrausta nublilalis in 1937, no of insects 0, 1, 2, 3, 4 and 5 with counts 33, 12, 6, 3, 1 and 1.
  4. Form Table 3, it can be seen that the PABLD gives better fit than the PD to the data set of number of insects. Thus PABLD provides a better alternative to PD for modeling count data sets.

  5. Juday [14] and Thomas [15] gave data on macroscopic fresh-water fauna in dredge samples from the bottom of Weber Lake.
  6. Form Table 4 it can be seen that the PABLD gives better fit than PD and PLD to the animal distribution of microcalanus nauplii. Thus PABLD provides a better alternative to PD and PLD for modeling count data sets.

  7. Archibald [16] gave data on plant populations. The distribution of representing salicornia stricta.
  8. Form Table 5, it can be seen that the PABLD gives better fit than the PD and PLD. Thus PABLD provides a better alternative to PD and PLD for modeling count data sets.

  9. Archibald [16-18] gave data on plant populations. The distribution of representing Plantago maritime.

Number of Bores Per Plant X

Observed Frequency (Oi)

Expected Frequency (Ei)

Poisson Distribution

Poisson-Lindley Distribution

Poisson- Area-Biased Lindley Distribution

0

83

78.9

87.2

82.4

1

36

42.9

31.8

38.1

2

14

11.7

11.2

11.7

3

2

2.01

3.8

2

4

1

0.4

2

0.67

Total

136

136

136

135.87

Estimation of Parameters

θ ^ =0.544118 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaecaae aacqaH4oqCaiaawkWaaiabg2da9iaaicdacaGGUaGaaGynaiaaisda caaI0aGaaGymaiaaigdacaaI4aaaaa@3FE1@

θ ^ =2.372252 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaecaae aacqaH4oqCaiaawkWaaiabg2da9iaaikdacaGGUaGaaG4maiaaiEda caaIYaGaaGOmaiaaiwdacaaIYaaaaa@3FE1@

θ ^ =6.119427 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaecaae aacqaH4oqCaiaawkWaaiabg2da9iaaiAdacaGGUaGaaGymaiaaigda caaI5aGaaGinaiaaikdacaaI3aaaaa@3FE8@

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

1.885

0.757

0.312

d.f

1

1

1

p-value

0.1698

0.3843

0.576455

Table 2: Chi-square goodness of fit test for PD, PLD and PABLD to European corn-borer data.

Number of Insects x

Observed Frequency (Oi)

Expected Frequency (Ei)

Poisson Distribution

Poisson Lindley Distribution

Poisson Area-Biased Lindley Distribution

0

33

26.45

31.48

33.18

1

12

19.84

14.16

15.98

2

6

7.44

6.09

5.09

3

3

1.86

2.5

1.34

4

1

0.35

1.04

0.32

5

1

0.05

0.42

0.07

Total

56

55.99

55.73

55.98

Estimation of Parameters

θ ˜ =0.75 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaGaacqGH9aqpcaaIWaGaaiOlaiaaiEdacaaI1aaaaa@3C3B@

θ ˜ =1.808 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaGaacqGH9aqpcaaIXaGaaiOlaiaaiIdacaaIWaGaaGioaaaa@3CFA@

θ ˜ =5.859 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaGaacqGH9aqpcaaI1aGaaiOlaiaaiIdacaaI1aGaaGyoaaaa@3D04@

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

4.89

0.484

3.56

d.f

1

1

1

p-value

0.026977

0.00001

0.059131

Table 3: Chi-square goodness of fit test for PD, PLD and PABLD to distribution of Pyrausta nublilalis in 1937.

Individuals Per Unit

Microcalanus

Observed Frequency (Oi)

Expected Frequency (Ei)

Poisson Distribution

Poisson Lindley Distribution

Poisson Area-Biased Lindley Distribution

0

0

0.01

7.156

1.294

1

2

0.098

8.743

3.402

2

4

0.468

9.632

5.76

3

3

1.498

10.009

7.928

4

5

3.595

10.014

9.643

5

8

6.903

9.757

10.791

6

16

11.045

9.324

11.37

7

13

15.147

8.777

11.446

8

12

18.177

8.164

11.116

9

13

19.388

7.521

10.487

10

15

18.613

6.873

9.66

11

15

16.244

6.239

8.721

12

9

12.995

5.631

7.739

13

9

9.596

5.057

6.767

14

7

6.58

4.522

5.842

15

4

4.211

4.028

4.986

16

4

2.527

3.575

4.213

17

6

1.427

3.164

3.528

18

2

0.761

2.793

2.931

19

0

0.385

2.459

2.417

20

2

0.185

2.16

1.981

21

1

0.084

1.894

1.613

22

0

0.037

1.658

1.306

Total

150

149.97

149.7

150

Estimation of Parameters

θ ˜ =9.6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaGaacqGH9aqpcaaI5aGaaiOlaiaaiAdaaaa@3B84@

θ ˜ =0.192 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaGaacqGH9aqpcaaIWaGaaiOlaiaaigdacaaI5aGaaGOmaaaa@3CF5@

θ ˜ =0.404296 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaGaacqGH9aqpcaaIWaGaaiOlaiaaisdacaaIWaGaaGinaiaaikda caaI5aGaaGOnaaaa@3F30@

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

30.39206

62.992

20.02153

d.f

10

13

12

p-value

0.000739

0.00001

0.06669

Table 4: Chi-square goodness of fit test for PD, PLD and PABLD to animal distribution of microcalanus nauplii.

Plants Per Quadrant

Salicornia

Observed Frequency

Expected Frequency (Ei)

(Oi)

Poisson Distribution

Poisson Lindley Distribution

Poisson Area-Biased Lindley Distribution

0

4

0.127

7.874

2.277

1

3

0.843

8.939

5.267

2

8

2.804

9.199

7.861

3

13

6.216

8.947

9.553

4

11

10.333

8.389

10.265

5

9

13.743

7.665

10.156

6

8

15.232

6.871

9.465

7

10

14.471

6.069

8.43

8

3

12.029

5.299

7.245

9

3

8.888

4.582

6.05

10

8

5.91

3.931

4.934

11

3

3.573

3.35

3.943

12

4

1.98

2.839

3.099

13

4

1.013

2.394

2.399

14

0

0.481

2.01

1.834

15

3

0.213

1.681

1.387

16

0

0.089

1.402

1.038

17

0

0.035

1.165

0.77

18

1

0.013

0.966

0.566

19

0

0.004

0.799

0.414

20

3

0.001

0.659

0.3

Total

98

97.99

98

97.25275

Estimation of  Parameters

θ ˜ =6.65 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaGaacqGH9aqpcaaI2aGaaiOlaiaaiAdacaaI1aaaaa@3C40@

θ ˜ =0.269 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaGaacqGH9aqpcaaIWaGaaiOlaiaaikdacaaI2aGaaGyoaaaa@3CFA@

θ ˜ =0.577238 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaGaacqGH9aqpcaaIWaGaaiOlaiaaiwdacaaI3aGaaG4naiaaikda caaIZaGaaGioaaaa@3F37@

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

65.55225

13.01986

7.381047

d.f

7

8

8

p-value

0.00001

0.111198

0.496138

Table 5: Chi-square goodness of fit test for PD, PLD and PABLD to distribution of quadrant, representing salicornia stricta.

From Table 6 it is concluded that the PABLD gives better fit than the PD and almost equally good fit as PLD distribution to the distribution of Plantago maritime. Therefore the PABLD is better alternative to PD and PLD to model discrete data sets.

Plants per Quadrant

Plantago

Observed Frequency

Expected Frequency (Ei)

Poisson Distribution

Poisson Lindley Distribution

Poisson Area-Biased Lindley Distribution

0

12

0.6409

11.471

4.273

1

8

3.2367

12.166

8.868

2

9

8.1727

11.749

11.897

3

13

13.7574

10.746

13.009

4

6

17.3687

9.484

12.59

5

8

17.5424

8.163

11.223

6

11

14.7648

6.895

9.428

7

7

10.652

5.741

7.571

8

8

6.7239

4.725

5.868

9

7

3.7729

3.853

4.42

10

3

1.9053

3.117

3.251

11

4

0.8747

2.505

2.344

12

1

0.3681

2.002

1.662

13

1

0.143

1.592

1.161

14

0

0.0516

1.261

0.801

15

0

0.0174

0.995

0.547

16

1

0.0055

0.782

0.369

17

0

0.0016

0.613

0.247

18

0

0.0005

0.48

0.164

19

1

0.0001

0.374

0.108

20

0

0.00003

0.291

0.071

Total

100

99.999

99.89

99.8709

Estimation of Parameters

θ ˜ =5.05 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaGaacqGH9aqpcaaI1aGaaiOlaiaaicdacaaI1aaaaa@3C39@

θ ˜ =0.345 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaGaacqGH9aqpcaaIWaGaaiOlaiaaiodacaaI0aGaaGynaaaa@3CF5@

θ ˜ =0.752375 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaGaacqGH9aqpcaaIWaGaaiOlaiaaiEdacaaI1aGaaGOmaiaaioda caaI3aGaaGynaaaa@3F34@

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

55.48343

7.084

10.2781

d.f

6

7

7

p-value

0.00001

0.420187

0.173359

Table 6: Chi-square goodness of fit test for PD, PLD and PABLD to distribution of quadrant, representing Plantago maritima.

Note: The highlighted expected frequencies from Table 2-6 are the pooled frequencies that are less than 5, so the degrees of freedom are calculated according to them.

From Table 2-7, it is observed that the PABLD gives better fit than PD and PLD to the some biological count data sets. PD is a discrete distribution with parameter λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4UdW gaaa@3838@ . Lindley distribution is a continuous life time distribution and PLD is the mixture of Poisson and Lindley distributions with parameter θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde haaa@383A@ . The proposed model named PABLD is obtained by the mixture of the Poisson distribution and the area biased form of the Lindley distribution. The area biased distribution is a type of the weighted distribution with weight w( x )= x 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4Dam aabmaabaGaamiEaaGaayjkaiaawMcaaiabg2da9iaadIhadaahaaqa beaacaaIYaaaaaaa@3CE7@ , due to mixture of PD and LD with this weight, the proposed model is showing applications better than PD and PLD to biological data sets. Mostly the applications of the weighted distributions to the data relating biology can be found in Patil & Rao [10].

f. Interval Estimation: By using equation (3.5) the parameter θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde haaa@383A@ of PABLD is estimated by the interval estimation for the Biological data sets. The estimated interval for θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde haaa@383A@ of PABLD by the interval estimation is closer to the estimated value by MOM.

Table

Data Sets

95 % C. I

II

Number of bores per plant

(5.989827, 6.249026)

III

Number of insects

(5.562813, 6.155574)

IV

Microcalanus

(0.39898, 0.40902)

V

Salicornia

(0.568854, 0.591146)

VI

Plantago

(0.738042, 0.766708)

Table 7: The asymptotic 95% confidence intervals (C.I) for θ of PABLD.

Conclusion

The Poisson area-biased Lindley distribution (PABLD) is discrete distribution that is obtained by mixture of the Poisson distribution and area-biased Lindley distribution. Some important properties of the PABLD are derived. From Figure 1 it can be seen that the PABLD is positively skewed moreover it can be seen that as θ0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde NaeyOKH4QaaGimaaaa@3AE1@ , ( γ 1 , β 2 )( 5.65,7.88 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacqaHZoWzdaWgaaqcfasaaiaaigdaaKqbagqaaiaacYcacqaHYoGy daWgaaqcfasaaiaaikdaaKqbagqaaaGaayjkaiaawMcaaiabgkziUo aabmaabaGaeyOeI0IaaGynaiaac6cacaaI2aGaaGynaiaacYcacaaI 3aGaaiOlaiaaiIdacaaI4aaacaGLOaGaayzkaaaaaa@4A30@  and the PABLD is negatively skewed and leptokurtic. Furthermore it is found that the PABLD is over-dispersed but as θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde NaeyOKH4QaeyOhIukaaa@3B98@ the PABLD is equi-dispersed. The parameter of the PABLD is estimated by the method of moments (MOM) and it is proved that the 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@  is positively biased, consistent and asymptotically normal. In section 4, the proposed model PABLD is applied to some biological data sets and compared with PD and PLD. It is observed that the PABLD gives better approach to the given data sets. Therefore it is concluded that PABLD is a better alternative to PD and PLD and it has useful applications in real life biological data sets. The asymptotic 95% MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaaGyoai aaiwdacaGGLaaaaa@38AF@  confidence interval (C.I) for θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde haaa@383A@ of PABLD is also found on these data sets and it is observed that the estimated interval for θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde haaa@383A@ of PABLD by the interval estimation is closer to the estimated value obtained by MOM.

Figure 1: Plots of the pdf of PABLD for θ = 0.5, θ = 1, θ = 2, θ = 8

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