ISSN: 2378-315X BBIJ

Biometrics & Biostatistics International Journal
Review Article
Volume 4 Issue 6 - 2016
Tests of Hypotheses for the Parameters of a Bivariate Geometric Distribution
Fitrat Hossain1 and Munni Begum2*
1Department of Mathematics, Marquette University, USA
2Department of Mathematical Sciences, Ball State University, USA
Received: September 17, 2016 | Published: November 07, 2016
*Corresponding author: Munni Begum, Department of Mathematical Sciences, Ball State University, Muncie, IN 47306, USA, Email:
Citation: Hossain F, Begum M (2016) Tests of Hypotheses for the Parameters of a Bivariate Geometric Distribution. Biom Biostat Int J 4(6): 00112. DOI: 10.15406/bbij.2016.04.00112

Abstract

A bivariate geometric distribution is an extension to a univariate geo­metric distribution where the occurrence of three different types of events is considered. Many statisticians have studied and given different forms of a bivariate geometric distribution. In this paper, we considered the form given by Phatak & Sreehari [1]. We estimated the parameters of this distribution under three different models using maximum likelihood estimation (mle) and derived deviances as the goodness of fit statistics for testing the parameters and deviance difference for comparing two models. Using simulated data we found that the deviance measure works well to test a reduced model against a full model.

Keywords: Bivariate Geometric Distribution, Deviance, Deviance difference.

Introduction

Many situations in real world cannot be described by a single variable. Simul­taneous occurrence of multiple events warrants multivariate distributions. For instance, univariate geometric distribution can represent occurrence of failure of one component of a system. However, to study systems with several com­ponents that may have different types of failures, such as twin engines of an airplane or the paired organ in a human body, bivariate geometric distributions are suitable. Bivariate geometric distribution has increasingly important roles in various fields, including reliability and survival analysis. There are different forms of a bivariate geometric distribution. Phatak & Sreehari [1] pro­vided a form of the bivariate geometric distribution which is considered here. They introduced a form of probability mass function which take into considera­tion of three different types of events. There are other forms which can be seen in Nair & Nair [2], Hawkes [3], Arnold et al. [4] and Sreehari & Vasudeva [5]. Basu & Dhar [6] proposed a bivariate geometric model which is analogous to bivariate exponential model developed by Marshal & Olkin [7]. Characterization results are developed by Sun & Basu [8], Sreehari [9], and Sreehari & Vasudeva [5].

Omey & Minkova [10] considered the bivariate geometric distribution with negative correlation coefficient and analyzed some properties, probability generating function, probability mass function, moments and tail probabilities. Krishna & Pundir [11], studied the plausibility of a bivariate geometric distribution as a reliability model. They derived the maximum likelihood esti­mators and Bayes estimators of the parameters and various reliability charac­teristics. They also compared these estimators using Monte-Carlo simulation.

In this paper, the parameters of a saturated model, reduced model and gen­eralized linear model (glm) for a bivariate geometric distribution are estimated using the maximum likelihood method. We also derived deviances as the good­ness of fit statistics for testing parameters corresponding to these models and deviance difference to compare two related models in order to determine which model fits the data well. Rest of the paper is organized as follows: section 2 describes the univariate geometric distribution, section 3 presents the bivariate geometric distribution, section 4 presents hypothesis testing, section 5 discusses a numerical example with simulated data and section 6 has the conclusion.

Univariate Geometric Distribution

The probability mass function (pmf) of a random variable Y which follows a geometric distribution with probability of success p can be written as,

P( Y=y )=p ( 1p ) y ,y = 0,1,2,...;  0<p<1,  0<q=1p<1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadcfada qadaqaaiaadMfacqGH9aqpcaWG5baacaGLOaGaayzkaaGaeyypa0Ja amiCamaabmaabaGaaGymaiabgkHiTiaadchaaiaawIcacaGLPaaada ahaaqabKqbGeaacaWG5baaaKqbakaacYcacaWG5baeaaaaaaaaa8qa caGGGcWdaiabg2da98qacaGGGcWdaiaaicdacaGGSaGaaGymaiaacY cacaaIYaGaaiilaiaac6cacaGGUaGaaiOlaiaacUdapeGaaiiOaiaa cckacaaIWaGaeyipaWJaamiCaiabgYda8iaaigdacaGGSaGaaiiOai aacckacaaIWaGaeyipaWJaamyCaiabg2da9iaaigdacqGHsislcaWG WbGaeyipaWJaaGymaaaa@61EE@ .

The moment generating function can be given by,

M Y ( t )= p 1q e t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaad2eada WgaaqcfasaaiaadMfaaKqbagqaamaabmaabaGaamiDaaGaayjkaiaa wMcaaiabg2da9maalaaabaGaamiCaaqaaiaaigdacqGHsislcaWGXb GaamyzamaaCaaabeqcfasaaiaadshaaaaaaaaa@4264@

the mean and the variance of this distribution are

E( Y )= μ Y = 1p p = q p  and V ar( Y )= 1p p 2 = q p 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadweada qadaqaaiaadMfaaiaawIcacaGLPaaacqGH9aqpcqaH8oqBdaWgaaqc fasaaiaadMfaaKqbagqaaiabg2da9maalaaabaGaaGymaiabgkHiTi aadchaaeaacaWGWbaaaiabg2da9maalaaabaGaamyCaaqaaiaadcha aaaeaaaaaaaaa8qacaGGGcGaamyyaiaad6gacaWGKbGaaiiOaiaadA facaGGGcGaamyyaiaadkhadaqadaqaaiaadMfaaiaawIcacaGLPaaa cqGH9aqpdaWcaaqaa8aacaaIXaGaeyOeI0IaamiCaaWdbeaacaWGWb WaaWbaaeqajuaibaGaaGOmaaaaaaqcfaOaeyypa0ZaaSaaaeaacaWG XbaabaGaamiCamaaCaaabeqcfasaaiaaikdaaaaaaaaa@5BA5@

An extension to the univariate geometric distribution is the bivariate geo­metric distribution which is discussed in the next section.

Bivariate Geometric Distribution

The joint probability mass function of a bivariate geometric distribution can be obtained by the product of a marginal and a conditional distribution, introduced by Phatak & Sreehari [1]. They considered a process from which the units could be classified as good, marginal and bad with probabilities q 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyCam aaBaaajuaibaGaaGymaaqcfayabaaaaa@3912@ , q 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyCam aaBaaajuaibaGaaGOmaaqcfayabaaaaa@3913@  and q 3 =( 1 q 1 q 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyCam aaBaaajuaibaGaaG4maaqcfayabaGaeyypa0ZaaeWaaeaacaaIXaGa eyOeI0IaamyCamaaBaaajuaibaGaaGymaaqcfayabaGaeyOeI0Iaam yCamaaBaaajuaibaGaaGOmaaqcfayabaaacaGLOaGaayzkaaaaaa@4355@  respectively. They proposed that the probability mass function of observing the first bad unit after several good and marginal units are passed as follows:

P( Y 1 = y 1 , Y 2 = y 2 )=( y 1 + y 2 y 1 ) q 1 y 1 q 2 y 2 ( 1 q 1 q 2 ), y 1 , y 2 =0,1,2,...; MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamiuamaabmaabaGaamywamaaBaaajuaibaGaaGymaaqcfaya baGaeyypa0JaamyEamaaBaaajuaibaGaaGymaaqcfayabaGaaiilai aadMfadaWgaaqcfasaaiaaikdaaKqbagqaaiabg2da9iaadMhadaWg aaqcfasaaiaaikdaaKqbagqaaaGaayjkaiaawMcaaiabg2da9maabm aaeaqabeaacaWG5bWaaSbaaKqbGeaacaaIXaaajuaGbeaacqGHRaWk caWG5bWaaSbaaKqbGeaacaaIYaaajuaGbeaaaeaacaWG5bWaaSbaaK qbGeaacaaIXaaabeaaaaqcfaOaayjkaiaawMcaaiaadghadaWgaaqc fasaaiaaigdaaKqbagqaamaaCaaabeqcfasaaiaadMhajuaGdaWgaa qcfasaaiaaigdaaeqaaaaajuaGcaWGXbWaaSbaaKqbGeaacaaIYaaa juaGbeaadaahaaqabeaajuaicaWG5bqcfa4aaSbaaKqbGeaacaaIYa aajuaGbeaaaaWaaeWaaeaacaaIXaGaeyOeI0IaamyCamaaBaaajuai baGaaGymaaqcfayabaGaeyOeI0IaamyCamaaBaaajuaibaGaaGOmaa qcfayabaaacaGLOaGaayzkaaGaaiilaiaadMhadaWgaaqcfasaaiaa igdaaKqbagqaaiaacYcacaWG5bWaaSbaaKqbGeaacaaIYaaajuaGbe aacqGH9aqpcaaIWaGaaiilaiaaigdacaGGSaGaaGOmaiaacYcacaGG UaGaaiOlaiaac6cacaGG7aaaaa@74F2@  (1)

0 <  q 1 + q 2 < 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaaGimaiaacckacqGH8aapcaGGGcGaamyCamaaBaaajuaibaGa aGymaaqcfayabaGaey4kaSIaamyCamaaBaaajuaibaGaaGOmaaqcfa yabaGaeyipaWJaaiiOaiaaigdaaaa@4381@ .

Here Y 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamywamaaBaaajuaibaGaaGymaaqcfayabaaaaa@390F@  and Y 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamywamaaBaaajuaibaGaaGOmaaqcfayabaaaaa@3910@  denote the number of good and marginal units respectively before the first bad unit is observed.

The marginal distribution of Y 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamywamaaBaaajuaibaGaaGymaaqcfayabaaaaa@390F@ is a geometric distributions with probability of success ( 1 q 1 q 2 1 q 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeWaaeWaaeaadaWcaaqaaiaaigdacqGHsislcaWGXbWaaSbaaKqb GeaacaaIXaaajuaGbeaacqGHsislcaWGXbWaaSbaaKqbGeaacaaIYa aajuaGbeaaaeaacaaIXaGaeyOeI0IaamyCamaaBaaajuaibaGaaGOm aaqcfayabaaaaaGaayjkaiaawMcaaaaa@441B@ , and can be written as follows,

P( Y 1 = y 1 )=( 1 q 1 q 2 1 q 2 ) ( q 1 1 q 2 ) y 1 ,  y 1 =0,1,2,... MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadcfada qadaqaaiaadMfadaWgaaqcfasaaiaaigdaaKqbagqaaiabg2da9iaa dMhadaWgaaqcfasaaiaaigdaaKqbagqaaaGaayjkaiaawMcaaiabg2 da9maabmaabaWaaSaaaeaacaaIXaGaeyOeI0IaamyCamaaBaaajuai baGaaGymaaqcfayabaGaeyOeI0IaamyCamaaBaaajuaibaGaaGOmaa qcfayabaaabaGaaGymaiabgkHiTiaadghadaWgaaqcfasaaiaaikda aKqbagqaaaaaaiaawIcacaGLPaaadaqadaqaamaalaaabaGaamyCam aaBaaajuaibaGaaGymaaqcfayabaaabaGaaGymaiabgkHiTiaadgha daWgaaqcfasaaiaaikdaaKqbagqaaaaaaiaawIcacaGLPaaadaahaa qabeaacaWG5bWaaSbaaKqbGeaacaaIXaaajuaGbeaaaaGaaiilaaba aaaaaaaapeGaaiiOaiaadMhadaWgaaqcfasaaiaaigdaaKqbagqaai abg2da9iaaicdacaGGSaGaaGymaiaacYcacaaIYaGaaiilaiaac6ca caGGUaGaaiOlaaaa@646E@

The conditional distribution of Y 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamywamaaBaaajuaibaGaaGOmaaqcfayabaaaaa@3910@ given Y 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamywamaaBaaajuaibaGaaGymaaqcfayabaaaaa@390F@  is

P( Y 2 = y 2 / Y 1 = y 2 )= ( y 1 + y 2 y 2 ) q 2 y 2 ( 1 q 2 ) y 1 +1 ,   y 1 , y 2 = 0,1,2,... MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadcfada qadaqaaiaadMfadaWgaaqcfasaaiaaikdaaKqbagqaaiabg2da9iaa dMhadaWgaaqcfasaaiaaikdaaKqbagqaaiaac+cacaWGzbWaaSbaaK qbGeaacaaIXaaajuaGbeaacqGH9aqpcaWG5bWaaSbaaKqbGeaacaaI YaaajuaGbeaaaiaawIcacaGLPaaacqGH9aqpqaaaaaaaaaWdbiaacc kapaWaaeWaaqaabeqaaiaadMhadaWgaaqcfasaaiaaigdaaKqbagqa aiabgUcaRiaadMhadaWgaaqcfasaaiaaikdaaKqbagqaaaqaaiaadM hadaWgaaqcfasaaiaaikdaaeqaaaaajuaGcaGLOaGaayzkaaGaamyC amaaBaaabaWaaSbaaKqbGeaacaaIYaaajuaGbeaaaeqaamaaCaaabe qaaKqbGiaadMhajuaGdaWgaaqcfasaaiaaikdaaKqbagqaaaaadaqa daqaaiaaigdacqGHsislcaWGXbWaaSbaaKqbGeaacaaIYaaajuaGbe aaaiaawIcacaGLPaaadaahaaqabeaacaWG5bWaaSbaaKqbGeaacaaI XaaajuaGbeaajuaicqGHRaWkcaaIXaaaaKqbakaacYcapeGaaiiOai aacckacaWG5bWaaSbaaKqbGeaacaaIXaaajuaGbeaacaGGSaGaamyE amaaBaaajuaibaGaaGOmaaqcfayabaGaeyypa0JaaiiOaiaaicdaca GGSaGaaGymaiaacYcacaaIYaGaaiilaiaac6cacaGGUaGaaiOlaaaa @749C@  (3)

The product of the marginal distribution of Y 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamywamaaBaaajuaibaGaaGymaaqcfayabaaaaa@390F@  in equation (2) and the con­ditional distribution of Y 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamywamaaBaaajuaibaGaaGOmaaqcfayabaaaaa@3910@  given Y 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamywamaaBaaajuaibaGaaGymaaqcfayabaaaaa@390F@  in equation (3) gives the mass function of bivariate geometric distribution in equation (1).

Maximum likelihood estimation

Estimation of parameters in the absence of regressors

In order to find the maximum likelihood estimators (mle)s from a saturated model (parameters are different for each pair of observations), it suffices to con­sider the likelihood functions based on the marginal and conditional mass func­tions. Let Y 1 ,...., Y n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamywam aaBaaajuaibaGaaGymaaqcfayabaGaaiilaiaac6cacaGGUaGaaiOl aiaac6cacaGGSaGaamywamaaBaaajuaibaGaamOBaaqcfayabaaaaa@3FD0@  be independent random vectors each having bivariate geo­metric distribution with different pairs of parameters ( q 1i , q 2i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaaba GaamyCamaaBaaajuaibaGaaGymaiaadMgaaKqbagqaaiaacYcacaWG XbWaaSbaaKqbGeaacaaIYaGaamyAaaqcfayabaaacaGLOaGaayzkaa aaaa@3FAB@ for i=1,2,....,n. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadMgacq GH9aqpcaaIXaGaaiilaiaaikdacaGGSaGaaiOlaiaac6cacaGGUaGa aiOlaiaacYcacaWGUbGaaiOlaaaa@4061@ .

The log likelihood function based on the conditional distribution of Y 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamywamaaBaaajuaibaGaaGOmaaqcfayabaaaaa@3910@  given Y 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamywamaaBaaajuaibaGaaGymaaqcfayabaaaaa@390F@  can be written as follows using (3):

l=   i=1 n [ ln( 1 q 1i q 2i )ln( 1 q 2i )+ y 1i ln q 1i y 1i ln( 1 q 2i ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadYgacq GH9aqpqaaaaaaaaaWdbiaacckacaGGGcWaaabCaeaadaWadaqaaiGa cYgacaGGUbWaaeWaaeaacaaIXaGaeyOeI0IaamyCamaaBaaajuaiba GaaGymaiaadMgaaKqbagqaaiabgkHiTiaadghadaWgaaqcfasaaiaa ikdacaWGPbaajuaGbeaaaiaawIcacaGLPaaacqGHsislciGGSbGaai OBamaabmaabaGaaGymaiabgkHiTiaadghadaWgaaqcfasaaiaaikda caWGPbaajuaGbeaaaiaawIcacaGLPaaacqGHRaWkcaWG5bWaaSbaaK qbGeaacaaIXaGaamyAaaqcfayabaGaciiBaiaac6gacaWGXbWaaSba aKqbGeaacaaIXaGaamyAaaqcfayabaGaeyOeI0IaamyEamaaBaaaju aibaGaaGymaiaadMgaaKqbagqaaiGacYgacaGGUbWaaeWaaeaacaaI XaGaeyOeI0IaamyCamaaBaaajuaibaGaaGOmaiaadMgaaKqbagqaaa GaayjkaiaawMcaaaGaay5waiaaw2faaaqcfasaaiaadMgacqGH9aqp caaIXaaabaGaamOBaaqcfaOaeyyeIuoaaaa@7095@  (4)

Differentiating (4) with respect to q 2i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadghada WgaaqcfasaaiaaikdacaWGPbaabeaaaaa@3968@ and setting it equal to zero, we get the mle of q 2i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadghada WgaaqcfasaaiaaikdacaWGPbaabeaaaaa@3968@  as,

q ^ 2i = y 2i y 1i + y 2i +1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadghaga qcamaaBaaajuaibaGaaGOmaiaadMgaaeqaaKqbakabg2da9maalaaa baGaamyEamaaBaaajuaibaGaaGOmaiaadMgaaKqbagqaaaqaaiaadM hadaWgaaqcfasaaiaaigdacaWGPbaajuaGbeaacqGHRaWkcaWG5bWa aSbaaKqbGeaacaaIYaGaamyAaaqcfayabaGaey4kaSIaaGymaaaaaa a@4829@  (5)

The log likelihood function based on the marginal distribution of Y 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamywamaaBaaajuaibaGaaGymaaqcfayabaaaaa@390F@  from (2) is,

l=   i=1 n [ ln( 1 q 1i q 2i )ln( 1 q 2i )+ y 1i ln q 1i y 1i ln( 1 q 2i ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadYgacq GH9aqpqaaaaaaaaaWdbiaacckacaGGGcWaaabCaeaadaWadaqaaiGa cYgacaGGUbWaaeWaaeaacaaIXaGaeyOeI0IaamyCamaaBaaajuaiba GaaGymaiaadMgaaKqbagqaaiabgkHiTiaadghadaWgaaqcfasaaiaa ikdacaWGPbaajuaGbeaaaiaawIcacaGLPaaacqGHsislciGGSbGaai OBamaabmaabaGaaGymaiabgkHiTiaadghadaWgaaqcfasaaiaaikda caWGPbaajuaGbeaaaiaawIcacaGLPaaacqGHRaWkcaWG5bWaaSbaaK qbGeaacaaIXaGaamyAaaqcfayabaGaciiBaiaac6gacaWGXbWaaSba aKqbGeaacaaIXaGaamyAaaqcfayabaGaeyOeI0IaamyEamaaBaaaju aibaGaaGymaiaadMgaaKqbagqaaiGacYgacaGGUbWaaeWaaeaacaaI XaGaeyOeI0IaamyCamaaBaaajuaibaGaaGOmaiaadMgaaKqbagqaaa GaayjkaiaawMcaaaGaay5waiaaw2faaaqcfasaaiaadMgacqGH9aqp caaIXaaabaGaamOBaaqcfaOaeyyeIuoaaaa@7095@

Differentiating (6) with respect to q 1i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamyCamaaBaaajuaibaGaaGymaiaadMgaaKqbagqaaaaa@3A15@ and setting it equal to zero, the mle of q 1i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamyCamaaBaaajuaibaGaaGymaiaadMgaaKqbagqaaaaa@3A15@ can be derived as,

q ^ 1i = y 1i y 1i + y 2i +1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGabmyCayaajaWaaSbaaKqbGeaacaaIXaGaamyAaaqcfayabaGa eyypa0ZaaSaaaeaacaWG5bWaaSbaaKqbGeaacaaIXaGaamyAaaqcfa yabaaabaGaamyEamaaBaaajuaibaGaaGymaiaadMgaaKqbagqaaiab gUcaRiaadMhadaWgaaqcfasaaiaaikdacaWGPbaajuaGbeaacqGHRa WkcaaIXaaaaaaa@4847@  (7)

Here, q ^ 1i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGabmyCayaajaWaaSbaaKqbGeaacaaIXaGaamyAaaqcfayabaaa aa@3A25@ and q 1i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamyCamaaBaaajuaibaGaaGymaiaadMgaaKqbagqaaaaa@3A15@ are the maximum likelihood estimators of q 1i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamyCamaaBaaajuaibaGaaGymaiaadMgaaKqbagqaaaaa@3A15@ and q 2i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadghada WgaaqcfasaaiaaikdacaWGPbaabeaaaaa@3968@ , i=1,...,n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyAai abg2da9iaaigdacaGGSaGaaiOlaiaac6cacaGGUaGaaiilaiaad6ga aaa@3D9C@ respectively under the saturated model.

Similarly the maximum likelihood estimators (mle)s from a reduced model (parameters are the same for each pair of observations) can be obtained as:

q ^ 2 = y 2 y 1 + y 2 +1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGabmyCayaajaWaaSbaaKqbGeaacaaIYaaajuaGbeaacqGH9aqp daWcaaqaaiqadMhagaWdamaaBaaajuaibaGaaGOmaaqcfayabaaaba GabmyEayaaoaWaaSbaaKqbGeaacaaIXaaajuaGbeaacqGHRaWkceWG 5bGba8aadaWgaaqcfasaaiaaikdaaKqbagqaaiabgUcaRiaaigdaaa aaaa@44D2@  (8)

q ^ 1 = y 1 y 1 + y 2 +1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGabmyCayaajaWaaSbaaKqbGeaacaaIXaaajuaGbeaacqGH9aqp daWcaaqaaiqadMhagaWdamaaBaaajuaibaGaaGymaaqcfayabaaaba GabmyEayaaoaWaaSbaaKqbGeaacaaIXaaajuaGbeaacqGHRaWkceWG 5bGba8aadaWgaaqcfasaaiaaikdaaKqbagqaaiabgUcaRiaaigdaaa aaaa@44D0@  (9)

Where q ^ 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGabmyCayaajaWaaSbaaKqbGeaacaaIXaaajuaGbeaaaaa@3937@ and q ^ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGabmyCayaajaWaaSbaaKqbGeaacaaIYaaajuaGbeaaaaa@3938@  are the maximum likelihood estimators of q 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyCam aaBaaajuaibaGaaGymaaqcfayabaaaaa@3912@  and q 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyCam aaBaaajuaibaGaaGOmaaqcfayabaaaaa@3913@ respectively under the reduced model.

Estimation of parameters in the presence of regressors: In the presence of regressors, one can employ a generalized linear model and hence estimate the parameters in terms of the estimated model parameters. The conditional distribution of Y 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamywamaaBaaajuaibaGaaGOmaaqcfayabaaaaa@3910@  given Y 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamywamaaBaaajuaibaGaaGymaaqcfayabaaaaa@390F@  in (3) can be set as exponential family representation as follows,

P( Y 2 = y 2 / Y 1 = y 1 ) =  exp[ y 2 ln  q 2 { ( y 1 +1 )ln( 1 q 2 ) }+ln ( y 1 + y 2 )! y 1 ! y 2 ! ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamiuamaabmaabaGaamywamaaBaaajuaibaGaaGOmaaqcfaya baGaeyypa0JaamyEamaaBaaajuaibaGaaGOmaaqcfayabaGaai4lai aadMfadaWgaaqcfasaaiaaigdaaKqbagqaaiabg2da9iaadMhadaWg aaqcfasaaiaaigdaaKqbagqaaaGaayjkaiaawMcaaiaacckacqGH9a qpcaGGGcGaaiiOaiGacwgacaGG4bGaaiiCamaadmaabaGaamyEamaa BaaajuaibaGaaGOmaaqcfayabaGaciiBaiaac6gacaGGGcGaamyCam aaBaaajuaibaGaaGOmaaqcfayabaGaeyOeI0YaaiWaaeaacqGHsisl daqadaqaaiaadMhadaWgaaqcfasaaiaaigdaaKqbagqaaiabgUcaRi aaigdaaiaawIcacaGLPaaaciGGSbGaaiOBamaabmaabaGaaGymaiab gkHiTiaadghadaWgaaqcfasaaiaaikdaaKqbagqaaaGaayjkaiaawM caaaGaay5Eaiaaw2haaiabgUcaRiGacYgacaGGUbWaaSaaaeaadaqa daqaaiaadMhadaWgaaqcfasaaiaaigdaaKqbagqaaiabgUcaRiaadM hadaWgaaqcfasaaiaaikdaaKqbagqaaaGaayjkaiaawMcaaiaacgca aeaacaWG5bWaaSbaaKqbGeaacaaIXaaajuaGbeaacaGGHaGaamyEam aaBaaajuaibaGaaGOmaaqcfayabaGaaiyiaaaaaiaawUfacaGLDbaa aaa@7A32@

Here the natural parameter and the function of the natural parameter respec­tively are,

θ = ln q 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaeqiUdeNaaiiOaiabg2da9iaacckaciGGSbGaaiOBaiaadgha daWgaaqcfasaaiaaikdaaKqbagqaaaaa@4010@

b( θ ) =   ( y 1 +1 )ln( 1 q 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOyamaabmaabaGaeqiUdehacaGLOaGaayzkaaGaaiiOaiab g2da9iaacckacaGGGcGaeyOeI0IaaiiOamaabmaabaGaamyEamaaBa aajuaibaGaaGymaaqcfayabaGaey4kaSIaaGymaaGaayjkaiaawMca aiGacYgacaGGUbWaaeWaaeaacaaIXaGaeyOeI0IaamyCamaaBaaaju aibaGaaGOmaaqcfayabaaacaGLOaGaayzkaaaaaa@4EA2@

Thus the mean of the conditional distribution of Y 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamywamaaBaaajuaibaGaaGOmaaqcfayabaaaaa@3910@  given Y 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamywamaaBaaajuaibaGaaGymaaqcfayabaaaaa@390F@ is

μ i = E[ Y 2 / Y 1 = y 1 ]/= b'( θ )= y 1 +1 1 q 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaeqiVd02aaSbaaKqbGeaacaWGPbaajuaGbeaacqGH9aqpcaGG GcGaamyramaadmaabaGaamywamaaBaaajuaibaGaaGOmaaqcfayaba Gaai4laiaadMfadaWgaaqcfasaaiaaigdaaKqbagqaaiabg2da9iaa dMhadaWgaaqcfasaaiaaigdaaKqbagqaaaGaay5waiaaw2faaiaac+ cacqGH9aqpcaGGGcGaamOyaiaacEcadaqadaqaaiabeI7aXbGaayjk aiaawMcaaiabg2da9maalaaabaGaamyEamaaBaaajuaibaGaaGymaa qcfayabaGaey4kaSIaaGymaaqaaiaaigdacqGHsislcaWGXbWaaSba aKqbGeaacaaIYaaajuaGbeaaaaaaaa@596A@

A generalized linear model based on the conditional distribution of Y 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamywamaaBaaajuaibaGaaGOmaaqcfayabaaaaa@3910@ given Y 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamywamaaBaaajuaibaGaaGymaaqcfayabaaaaa@390F@ can be written as,

g( μ i ) =  ln μ i =/ n = j=1 p x 2ij β 2i ; i =   1,2,....,n;   n>p MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaam4zamaabmaabaGaeqiVd02aaSbaaKqbGeaacaWGPbaajuaG beaaaiaawIcacaGLPaaacaGGGcGaeyypa0JaaiiOaiaacckacaGGSb GaaiOBaiabeY7aTnaaBaaajuaibaGaamyAaaqcfayabaGaeyypa0Ja ai4laiaacckacaWGUbGaaiiOaiabg2da9maaqahabaGaamiEamaaBa aajuaibaGaaGOmaiaadMgacaWGQbaajuaGbeaacqaHYoGydaWgaaqc fasaaiaaikdacaWGPbaajuaGbeaacaGG7aGaaiiOaiaadMgacaGGGc Gaeyypa0JaaiiOaaqcfasaaiaadQgacqGH9aqpcaaIXaaabaGaamiC aaqcfaOaeyyeIuoacaGGGcGaaGymaiaacYcacaaIYaGaaiilaiaac6 cacaGGUaGaaiOlaiaac6cacaGGSaGaamOBaiaacUdacaGGGcGaaiiO aiaacckacaWGUbGaeyOpa4JaamiCaaaa@7124@

Since, Y 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamywamaaBaaajuaibaGaaGOmaaqcfayabaaaaa@3910@ represents the number of trials before a certain event can occur it is considered as count response, the linear predictor can be written as the log­arithm of the mean μ i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaeqiVd02aaSbaaKqbGeaacaWGPbaajuaGbeaaaaa@3A1A@ . Thus the conditional link function can be expressed as,

g( μ i )=ln y 1i +1 1 q 2i = j=1 p x 2ij β 2i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadEgada qadaqaaiabeY7aTnaaBaaajuaibaGaamyAaaqcfayabaaacaGLOaGa ayzkaaGaeyypa0JaciiBaiaac6gadaWcaaqaaiaadMhadaWgaaqcfa saaiaaigdacaWGPbaajuaGbeaacqGHRaWkcaaIXaaabaGaaGymaiab gkHiTiaadghadaWgaaqcfasaaiaaikdacaWGPbaajuaGbeaaaaGaey ypa0ZaaabCaeaacaWG4bWaaSbaaKqbGeaacaaIYaGaamyAaiaadQga aKqbagqaaiabek7aInaaBaaajuaibaGaaGOmaiaadMgaaKqbagqaaa qcfasaaiaadQgacqGH9aqpcaaIXaaabaGaamiCaaqcfaOaeyyeIuoa aaa@59DB@

MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyO0H4 naaa@38E1@ q ^ 2i = 1  y 1i +1 exp{ j=1 p x 2ij β 2j } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadghaga qcamaaBaaajuaibaGaaGOmaiaadMgaaKqbagqaaiabg2da9abaaaaa aaaapeGaaiiOa8aacaaIXaGaeyOeI0YdbiaacckapaWaaSaaaeaaca WG5bWaaSbaaKqbGeaacaaIXaGaamyAaaqcfayabaGaey4kaSIaaGym aaqaaiGacwgacaGG4bGaaiiCamaacmaabaWaaabmaeaacaWG4bWaaS baaKqbGeaacaaIYaGaamyAaiaadQgaaKqbagqaaiabek7aInaaBaaa juaibaGaaGOmaiaadQgaaKqbagqaaaqcfasaaiaadQgacqGH9aqpca aIXaaabaGaamiCaaqcfaOaeyyeIuoaaiaawUhacaGL9baaaaaaaa@586E@  (10)

Here, β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabek7aIb aa@381A@  is an element of the matrix β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabek7aIb aa@381A@ corresponding to the covariate x 2i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIhada WgaaqcfasaaiaaikdacaWGPbaajuaGbeaaaaa@39FD@ which represents the effect of covariate to the mean responses through the link function g( μ i )= n i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadEgada qadaqaaiabeY7aTnaaBaaajuaibaGaamyAaaqcfayabaaacaGLOaGa ayzkaaGaeyypa0JaamOBamaaBaaajuaibaGaamyAaaqcfayabaaaaa@4033@ .

Differentiating (6) again with respect to q 1i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamyCamaaBaaajuaibaGaaGymaiaadMgaaKqbagqaaaaa@3A15@ , setting it to zero and using (10) we get,

q ^ 1i = y 1i exp{ j=1 p x 2ij β 2j } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadghaga qcamaaBaaajuaibaGaaGymaiaadMgaaKqbagqaaiabg2da9maalaaa baGaamyEamaaBaaajuaibaGaaGymaiaadMgaaKqbagqaaaqaaiGacw gacaGG4bGaaiiCamaacmaabaWaaabCaeaacaWG4bWaaSbaaKqbGeaa caaIYaGaamyAaiaadQgaaKqbagqaaiabek7aInaaBaaajuaibaGaaG OmaiaadQgaaKqbagqaaaqcfasaaiaadQgacqGH9aqpcaaIXaaabaGa amiCaaqcfaOaeyyeIuoaaiaawUhacaGL9baaaaaaaa@52D2@ (11)

Hypothesis Testing

In order to test the identical parameter assumption across each pair of observed data, we derived deviance as a goodness of fit statistics. Additional deviance statistics are derived for generalized linear model (glm) to compare two nested glms.

Deviance for reduced model with identical parameter assumption

The log likelihood function for the saturated model can be written using (1) and the maximum likelihood estimates of the parameters q 1i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamyCamaaBaaajuaibaGaaGymaiaadMgaaKqbagqaaaaa@3A15@ and q 2i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadghada WgaaqcfasaaiaaikdacaWGPbaabeaaaaa@3968@ from equations (5) and (7) respectively as follows,

l( b max ;y )= i=1 n [ y 1i ln y 1i y 1i ln( y 1i + y 2i +1 )+ y 2i ln y 2i y 2i ln( y 1i + y 2i +1 )ln( y 1i + y 2i +1 )+ln( y 1i + y 2i )!ln y 1i !ln y 2i ! ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadYgada qadaqaaiaadkgadaWgaaqcfasaaiGac2gacaGGHbGaaiiEaaqcfaya baGaai4oaiaadMhaaiaawIcacaGLPaaacqGH9aqpdaaeWbqaamaadm aabaGaamyEamaaBaaajuaibaGaaGymaiaadMgaaKqbagqaaiGacYga caGGUbGaamyEamaaBaaajuaibaGaaGymaiaadMgaaKqbagqaaiabgk HiTiaadMhadaWgaaqcfasaaiaaigdacaWGPbaajuaGbeaaciGGSbGa aiOBamaabmaabaGaamyEamaaBaaajuaibaGaaGymaiaadMgaaKqbag qaaiabgUcaRiaadMhadaWgaaqcfasaaiaaikdacaWGPbaajuaGbeaa cqGHRaWkcaaIXaaacaGLOaGaayzkaaGaey4kaSIaamyEamaaBaaaju aibaGaaGOmaiaadMgaaKqbagqaaiGacYgacaGGUbGaamyEamaaBaaa juaibaGaaGOmaiaadMgaaKqbagqaaiabgkHiTiaadMhadaWgaaqcfa saaiaaikdacaWGPbaajuaGbeaaciGGSbGaaiOBamaabmaabaGaamyE amaaBaaajuaibaGaaGymaiaadMgaaKqbagqaaiabgUcaRiaadMhada WgaaqcfasaaiaaikdacaWGPbaajuaGbeaacqGHRaWkcaaIXaaacaGL OaGaayzkaaGaeyOeI0IaciiBaiaac6gadaqadaqaaiaadMhadaWgaa qcfasaaiaaigdacaWGPbaajuaGbeaacqGHRaWkcaWG5bWaaSbaaKqb GeaacaaIYaGaamyAaaqcfayabaGaey4kaSIaaGymaaGaayjkaiaawM caaiabgUcaRiGacYgacaGGUbWaaeWaaeaacaWG5bWaaSbaaKqbGeaa caaIXaGaamyAaaqcfayabaGaey4kaSIaamyEamaaBaaajuaibaGaaG OmaiaadMgaaKqbagqaaaGaayjkaiaawMcaaiaacgcacqGHsislciGG SbGaaiOBaiaadMhadaWgaaqcfasaaiaaigdacaWGPbaajuaGbeaaca GGHaGaeyOeI0IaciiBaiaac6gacaWG5bWaaSbaaKqbGeaacaaIYaGa amyAaaqcfayabaGaaiyiaaGaay5waiaaw2faaaqcfasaaiaadMgacq GH9aqpcaaIXaaabaGaamOBaaqcfaOaeyyeIuoaaaa@A707@ (12)

Similarly, the log likelihood function of the reduced model can be written using (1) and the maximum likelihood estimates of q 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyCam aaBaaajuaibaGaaGymaaqcfayabaaaaa@3912@ and q 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyCam aaBaaajuaibaGaaGOmaaqcfayabaaaaa@3913@ from equations (8) and (9) respectively as follows,

l( b;y )= i=1 i=n [ y 1i ln y ^ y 1i ln( y ^ 1 + y ^ 2 +1 )+ y 2i ln y ^ 2 y 2i ln( y ^ 1 + y ^ 2 +1 )ln( y ^ 1 + y ^ 2 +1 )+ln( y 1i + y 2i )!ln y 1i !ln y 2i ! ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadYgada qadaqaaiaadkgacaGG7aGaamyEaaGaayjkaiaawMcaaiabg2da9maa qahabaWaamWaaeaacaWG5bWaaSbaaKqbGeaacaaIXaGaamyAaaqcfa yabaGaciiBaiaac6gaceWG5bGbaKaacqGHsislcaWG5bWaaSbaaKqb GeaacaaIXaGaamyAaaqcfayabaGaciiBaiaac6gadaqadaqaaiqadM hagaqcamaaBaaajuaibaGaaGymaaqcfayabaGaey4kaSIabmyEayaa jaWaaSbaaKqbGeaacaaIYaaajuaGbeaacqGHRaWkcaaIXaaacaGLOa GaayzkaaGaey4kaSIaamyEamaaBaaajuaibaGaaGOmaiaadMgaaKqb agqaaiGacYgacaGGUbGabmyEayaajaWaaSbaaKqbGeaacaaIYaaaju aGbeaacqGHsislcaWG5bWaaSbaaKqbGeaacaaIYaGaamyAaaqcfaya baGaciiBaiaac6gadaqadaqaaiqadMhagaqcamaaBaaajqwba9FaaK qbaoaaBaaajuaibaGaaGymaaqabaaajuaGbeaacqGHRaWkceWG5bGb aKaadaWgaaqcfasaaiaaikdaaKqbagqaaiabgUcaRiaaigdaaiaawI cacaGLPaaacqGHsislciGGSbGaaiOBamaabmaabaGabmyEayaajaWa aSbaaKqbGeaajuaGdaWgaaqcfasaaiaaigdaaKqbagqaaaqabaGaey 4kaSIabmyEayaajaWaaSbaaKqbGeaacaaIYaaajuaGbeaacqGHRaWk caaIXaaacaGLOaGaayzkaaGaey4kaSIaciiBaiaac6gadaqadaqaai aadMhadaWgaaqcfasaaiaaigdacaWGPbaajuaGbeaacqGHRaWkcaWG 5bWaaSbaaKqbGeaacaaIYaGaamyAaaqcfayabaaacaGLOaGaayzkaa GaaiyiaiabgkHiTiGacYgacaGGUbGaamyEamaaBaaajuaibaGaaGym aiaadMgaaKqbagqaaiaacgcacqGHsislciGGSbGaaiOBaiaadMhada WgaaqcfasaaiaaikdacaWGPbaajuaGbeaacaGGHaaacaGLBbGaayzx aaaajuaibaGaamyAaiabg2da9iaaigdaaeaacaWGPbGaeyypa0Jaam OBaaqcfaOaeyyeIuoaaaa@9FFF@  (13)

Thus the deviance statistic for testing the identical parameter for each observed pair of data can be expressed as follows,

D I =2[ l( b max ;y )l( b;y ) ]=2 i=1 i=n [ y 1i ln y 1i y ^ 1 y 1i ln y 1i + y 2i +1 y ^ 1 + y ^ 2 +1 + y 2i ln y 2i y ^ 2 y 2i ln y 1i + y 2i +1 y ^ 1 + y ^ 2 +1 ln y 1i + y 2i +1 y ^ 1 + y ^ 2 +1 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadseada WgaaqcfasaaiaadMeaaKqbagqaaiabg2da9iaaikdadaWadaqaaiaa dYgadaqadaqaaiaadkgadaWgaaqcfasaaiGac2gacaGGHbGaaiiEaa qcfayabaGaai4oaiaadMhaaiaawIcacaGLPaaacqGHsislcaWGSbWa aeWaaeaacaWGIbGaai4oaiaadMhaaiaawIcacaGLPaaaaiaawUfaca GLDbaacqGH9aqpcaaIYaWaaabCaeaadaWadaqaaiaadMhadaWgaaqc fasaaiaaigdacaWGPbaajuaGbeaaciGGSbGaaiOBamaalaaabaGaam yEamaaBaaajuaibaGaaGymaiaadMgaaKqbagqaaaqaaiqadMhagaqc amaaBaaabaWaaSbaaKqbGeaacaaIXaaajuaGbeaaaeqaaaaacqGHsi slcaWG5bWaaSbaaKqbGeaacaaIXaGaamyAaaqcfayabaGaciiBaiaa c6gadaWcaaqaaiaadMhadaWgaaqcfasaaiaaigdacaWGPbaajuaGbe aacqGHRaWkcaWG5bWaaSbaaKqbGeaacaaIYaGaamyAaaqcfayabaGa ey4kaSIaaGymaaqaaiqadMhagaqcamaaBaaajuaibaGaaGymaaqcfa yabaGaey4kaSIabmyEayaajaWaaSbaaKqbGeaacaaIYaaajuaGbeaa cqGHRaWkcaaIXaaaaiabgUcaRiaadMhadaWgaaqcfasaaiaaikdaca WGPbaajuaGbeaaciGGSbGaaiOBamaalaaabaGaamyEamaaBaaajuai baGaaGOmaiaadMgaaKqbagqaaaqaaiqadMhagaqcamaaBaaajuaiba GaaGOmaaqcfayabaaaaiabgkHiTiaadMhadaWgaaqcfasaaiaaikda caWGPbaajuaGbeaaciGGSbGaaiOBamaalaaabaGaamyEamaaBaaaju aibaGaaGymaiaadMgaaKqbagqaaiabgUcaRiaadMhadaWgaaqcfasa aiaaikdacaWGPbaajuaGbeaacqGHRaWkcaaIXaaabaGabmyEayaaja WaaSbaaKqbGeaacaaIXaaajuaGbeaacqGHRaWkceWG5bGbaKaadaWg aaqcfasaaiaaikdaaKqbagqaaiabgUcaRiaaigdaaaGaeyOeI0Iaci iBaiaac6gadaWcaaqaaiaadMhadaWgaaqcfasaaiaaigdacaWGPbaa juaGbeaacqGHRaWkcaWG5bWaaSbaaKqbGeaacaaIYaGaamyAaaqcfa yabaGaey4kaSIaaGymaaqaaiqadMhagaqcamaaBaaajuaibaGaaGym aaqcfayabaGaey4kaSIabmyEayaajaWaaSbaaKqbGeaacaaIYaaaju aGbeaacqGHRaWkcaaIXaaaaaGaay5waiaaw2faaaqcfasaaiaadMga cqGH9aqpcaaIXaaabaGaamyAaiabg2da9iaad6gaaKqbakabggHiLd aaaa@B39D@  (14)

According to Dobson (2001), D I MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadseada WgaaqcfasaaiaadMeaaKqbagqaaaaa@38ED@ follows a X 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIfada ahaaqabKqbGeaacaaIYaaaaaaa@3862@ distribution with ( 2n2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaaba GaaGOmaiaad6gacqGHsislcaaIYaaacaGLOaGaayzkaaaaaa@3B5A@ degrees of freedom.

Deviance for a GLM

The deviance statistic for the glm of interest can be written using (1) and the maximum likelihood estimates of q 1i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamyCamaaBaaajuaibaGaaGymaiaadMgaaKqbagqaaaaa@3A15@ and q 2i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadghada WgaaqcfasaaiaaikdacaWGPbaabeaaaaa@3968@ based on the glm from equations (10) and (11) respectively as follows,

l( b;y )= i=1 i=n [ y 1i ln y 1i exp{ j=1 j=p x 2ij β 2j } + y 2i ln( 1 y 1i +1 exp{ j=1 j=p x 2ij β 2j } )+ln 1 exp{ j=1 j=p x 2ij β 2j } +ln( y 1i + y 2i )!ln y 1i !ln y 2i ! ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadYgada qadaqaaiaadkgacaGG7aGaamyEaaGaayjkaiaawMcaaiabg2da9maa qahabaWaamWaaeaacaWG5bWaaSbaaKqbGeaacaaIXaGaamyAaaqcfa yabaGaciiBaiaac6gadaWcaaqaaiaadMhadaWgaaqcfasaaiaaigda caWGPbaajuaGbeaaaeaaciGGLbGaaiiEaiaacchadaGadaqaamaaqa dabaGaamiEamaaBaaajuaibaGaaGOmaiaadMgacaWGQbaajuaGbeaa aKqbGeaacaWGQbGaeyypa0JaaGymaaqaaiaadQgacqGH9aqpcaWGWb aajuaGcqGHris5aiabek7aInaaBaaajuaibaGaaGOmaiaadQgaaKqb agqaaaGaay5Eaiaaw2haaaaacqGHRaWkcaWG5bWaaSbaaKqbGeaaca aIYaGaamyAaaqcfayabaGaciiBaiaac6gadaqadaqaaiaaigdacqGH sisldaWcaaqaaiaadMhadaWgaaqaamaaBaaajuaibaGaaGymaiaadM gaaKqbagqaaiabgUcaRiaaigdaaeqaaaqaaiaacwgacaGG4bGaaiiC amaacmaabaWaaabmaeaacaWG4bWaaSbaaKqbGeaacaaIYaGaamyAai aadQgaaKqbagqaaaqcfasaaiaadQgacqGH9aqpcaaIXaaabaGaamOA aiabg2da9iaadchaaKqbakabggHiLdGaeqOSdi2aaSbaaKqbGeaaca aIYaGaamOAaaqcfayabaaacaGL7bGaayzFaaaaaaGaayjkaiaawMca aiabgUcaRiGacYgacaGGUbWaaSaaaeaacaaIXaaabaGaaiyzaiaacI hacaGGWbWaaiWaaeaadaaeWaqaaiaadIhadaWgaaqcfasaaiaaikda caWGPbGaamOAaaqcfayabaaajuaibaGaamOAaiabg2da9iaaigdaae aacaWGQbGaeyypa0JaamiCaaqcfaOaeyyeIuoacqaHYoGydaWgaaqc fasaaiaaikdacaWGQbaajuaGbeaaaiaawUhacaGL9baaaaGaey4kaS IaciiBaiaac6gadaqadaqaaiaadMhadaWgaaqcfasaaiaaigdacaWG PbaajuaGbeaacqGHRaWkcaWG5bWaaSbaaKqbGeaacaaIYaGaamyAaa qcfayabaaacaGLOaGaayzkaaGaaiyiaiabgkHiTiGacYgacaGGUbGa amyEamaaBaaajuaibaGaaGymaiaadMgaaKqbagqaaiaacgcacqGHsi slciGGSbGaaiOBaiaadMhadaWgaaqcfasaaiaaikdacaWGPbaajuaG beaacaGGHaaacaGLBbGaayzxaaaajuaibaGaamyAaiabg2da9iaaig daaeaacaWGPbGaeyypa0JaamOBaaqcfaOaeyyeIuoaaaa@BEEE@ (15)

Thus the deviance can be expressed as follows,

D II =2[ l( b max ;y )l( b;y ) ]=2 i=1 i=n [ y 2i ln y 2i ( y 1i + y 2i +1 )ln( y 1i + y 2i +1 )+{ j=1 j=p x 2ij β 2j }( y 1i + y 2i +1 ) y 2i ln( exp{ j=1 j=p x 2ij β 2j } y 1i 1 ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadseada WgaaqcfasaaiaadMeacaWGjbaajuaGbeaacqGH9aqpcaaIYaWaamWa aeaacaWGSbWaaeWaaeaacaWGIbWaaSbaaKqbGeaaciGGTbGaaiyyai aacIhaaKqbagqaaiaacUdacaWG5baacaGLOaGaayzkaaGaeyOeI0Ia amiBamaabmaabaGaamOyaiaacUdacaWG5baacaGLOaGaayzkaaaaca GLBbGaayzxaaGaeyypa0JaaGOmamaaqahabaWaamWaaeaacaWG5bWa aSbaaKqbGeaacaaIYaGaamyAaaqcfayabaGaciiBaiaac6gacaWG5b WaaSbaaKqbGeaacaaIYaGaamyAaaqcfayabaGaeyOeI0YaaeWaaeaa caWG5bWaaSbaaKqbGeaacaaIXaGaamyAaaqcfayabaGaey4kaSIaam yEamaaBaaajuaibaGaaGOmaiaadMgaaKqbagqaaiabgUcaRiaaigda aiaawIcacaGLPaaaciGGSbGaaiOBamaabmaabaGaamyEamaaBaaaju aibaGaaGymaiaadMgaaKqbagqaaiabgUcaRiaadMhadaWgaaqcfasa aiaaikdacaWGPbaajuaGbeaacqGHRaWkcaaIXaaacaGLOaGaayzkaa Gaey4kaSYaaiWaaeaadaaeWbqaaiaadIhadaWgaaqcfasaaiaaikda caWGPbGaamOAaaqcfayabaGaeqOSdi2aaSbaaKqbGeaacaaIYaGaam OAaaqcfayabaaajuaibaGaamOAaiabg2da9iaaigdaaeaacaWGQbGa eyypa0JaamiCaaqcfaOaeyyeIuoaaiaawUhacaGL9baadaqadaqaai aadMhadaWgaaqcfasaaiaaigdacaWGPbaajuaGbeaacqGHRaWkcaWG 5bWaaSbaaKqbGeaacaaIYaGaamyAaaqcfayabaGaey4kaSIaaGymaa GaayjkaiaawMcaaiabgkHiTiaadMhadaWgaaqcfasaaiaaikdacaWG PbaajuaGbeaaciGGSbGaaiOBamaabmaabaGaciyzaiaacIhacaGGWb WaaiWaaeaadaaeWbqaaiaadIhadaWgaaqcfasaaiaaikdacaWGPbGa amOAaaqcfayabaGaeqOSdi2aaSbaaKqbGeaacaaIYaGaamOAaaqcfa yabaaajuaibaGaamOAaiabg2da9iaaigdaaeaacaWGQbGaeyypa0Ja amiCaaqcfaOaeyyeIuoaaiaawUhacaGL9baacqGHsislcaWG5bWaaS baaKqbGeaacaaIXaGaamyAaaqcfayabaGaeyOeI0IaaGymaaGaayjk aiaawMcaaaGaay5waiaaw2faaaqcfasaaiaadMgacqGH9aqpcaaIXa aabaGaamyAaiabg2da9iaad6gaaKqbakabggHiLdaaaa@BDBA@ (16)

According to Dobson (2001), D II MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiram aaBaaajuaibaGaamysaiaadMeaaKqbagqaaaaa@39C6@ follows χ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4Xdm 2aaWbaaeqajuaibaGaaGOmaaaaaaa@3947@ distribution with ( 2np ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacaaIYaGaamOBaiabgkHiTiaadchaaiaawIcacaGLPaaaaaa@3B9E@ degrees of freedom.

Comparison between two GLMs

In order to compare two nested generalized linear models, we consider the fol­lowing hypotheses. The null hypothesis corresponding to a smaller model ( M 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacaWGnbWaaSbaaKqbGeaacaaIWaaajuaGbeaaaiaawIcacaGLPaaa aaa@3A76@ in terms of number of regression parameters is

H 0 :β= β 0 =[ β 1 β 2 . . . β q ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamisam aaBaaajuaibaGaaGimaaqcfayabaGaaiOoaiabek7aIjabg2da9iab ek7aInaaBaaajuaibaGaaGimaaqcfayabaGaeyypa0ZaamWaaqaabe qaaiabek7aInaaBaaajuaibaGaaGymaaqcfayabaaabaGaeqOSdi2a aSbaaKqbGeaacaaIYaaajuaGbeaaaeaacaGGUaaabaGaaiOlaaqaai aac6caaeaacqaHYoGydaWgaaqcfasaaiaadghaaKqbagqaaaaacaGL BbGaayzxaaaaaa@4E85@

The alternative hypothesis corresponding to a bigger model (M1 with q < p < n) within which the smaller model is nested can be written as,

H 1 :β= β 1 =[ β 1 β 2 . . . β p ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamisam aaBaaajuaibaGaaGymaaqabaqcfaOaaiOoaiabek7aIjabg2da9iab ek7aInaaBaaajuaibaGaaGymaaqcfayabaGaeyypa0ZaamWaaqaabe qaaiabek7aInaaBaaajuaibaGaaGymaaqcfayabaaabaGaeqOSdi2a aSbaaKqbGeaacaaIYaaajuaGbeaaaeaacaGGUaaabaGaaiOlaaqaai aac6caaeaacqaHYoGydaWgaaqcfasaaiaadchaaKqbagqaaaaacaGL BbGaayzxaaaaaa@4E86@

We can test H 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamisam aaBaaajuaibaGaaGimaaqcfayabaaaaa@38E8@ against H 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamisam aaBaaajuaibaGaaGymaaqcfayabaaaaa@38E9@ using the difference of the deviance statistics. Here, l( b 0 ;y ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiBam aabmaabaGaamOyamaaBaaajuaibaGaaGimaaqcfayabaGaai4oaiaa dMhaaiaawIcacaGLPaaaaaa@3D39@ is used to denote the likelihood function corresponding to the model M 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamytam aaBaaajuaibaGaaGimaaqcfayabaaaaa@38ED@ and l( b 1 ;y ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiBam aabmaabaGaamOyamaaBaaajuaibaGaaGymaaqcfayabaGaai4oaiaa dMhaaiaawIcacaGLPaaaaaa@3D3A@  to denote the likelihood function corresponding to the model M 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamytam aaBaaajuaibaGaaGymaaqcfayabaaaaa@38EE@ . Hence the deviance difference can be written as,

D= D 0 D 1 =2[ l( b max ;y )l( b 0 ;y ) ]2[ l( b max ;y )l( b 1 ;y ) ]=2[ l( b 1 ;y )l( b 0 ;y ) ] =2 i=1 i=n [ { j=1 j=p x 2ij β 2j }( y 1i + y 2i +1 ) y 2i ln( exp{ j=1 j=p x 2ij β 2j } y 1i 1 ){ j=1 j=q x 2ij β 2j }( y 1i + y 2i +1 )+ y 2i ln( exp{ j=1 j=q x 2ij β 2j } y 1i 1 ){ j=1 j=q x 2ij β 2j }( y 1i + y 2i +1 )+ y 2i ln( exp{ j=1 j=q x 2ij β 2j } y 1i 1 ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaqcfaOaeS 4SLyLaamiraiabg2da9iaadseadaWgaaqcfasaaiaaicdaaKqbagqa aiabgkHiTiaadseadaWgaaqcfasaaiaaigdaaKqbagqaaiabg2da9i aaikdadaWadaqaaiaadYgadaqadaqaaiaadkgadaWgaaqcfasaaiGa c2gacaGGHbGaaiiEaaqcfayabaGaai4oaiaadMhaaiaawIcacaGLPa aacqGHsislcaWGSbWaaeWaaeaacaWGIbWaaSbaaKqbGeaacaaIWaaa juaGbeaacaGG7aGaamyEaaGaayjkaiaawMcaaaGaay5waiaaw2faai abgkHiTiaaikdadaWadaqaaiaadYgadaqadaqaaiaadkgadaWgaaqc fasaaiGac2gacaGGHbGaaiiEaaqcfayabaGaai4oaiaadMhaaiaawI cacaGLPaaacqGHsislcaWGSbWaaeWaaeaacaWGIbWaaSbaaKqbGeaa caaIXaaajuaGbeaacaGG7aGaamyEaaGaayjkaiaawMcaaaGaay5wai aaw2faaiabg2da9iaaikdadaWadaqaaiaadYgadaqadaqaaiaadkga daWgaaqcfasaaiaaigdaaKqbagqaaiaacUdacaWG5baacaGLOaGaay zkaaGaeyOeI0IaamiBamaabmaabaGaamOyamaaBaaajuaibaGaaGim aaqcfayabaGaai4oaiaadMhaaiaawIcacaGLPaaaaiaawUfacaGLDb aaaOqaaKqbakabg2da9iaaikdadaaeWbqaamaadmaabaWaaiWaaeaa daaeWbqaaiaadIhadaWgaaqcfasaaiaaikdacaWGPbGaamOAaaqcfa yabaGaeqOSdi2aaSbaaKqbGeaacaaIYaGaamOAaaqcfayabaaajuai baGaamOAaiabg2da9iaaigdaaeaacaWGQbGaeyypa0JaamiCaaqcfa OaeyyeIuoaaiaawUhacaGL9baadaqadaqaaiaadMhadaWgaaqcfasa aiaaigdacaWGPbaajuaGbeaacqGHRaWkcaWG5bWaaSbaaKqbGeaaca aIYaGaamyAaaqcfayabaGaey4kaSIaaGymaaGaayjkaiaawMcaaiab gkHiTiaadMhadaWgaaqcfasaaiaaikdacaWGPbaajuaGbeaaciGGSb GaaiOBamaabmaabaGaciyzaiaacIhacaGGWbWaaiWaaeaadaaeWbqa aiaadIhadaWgaaqcfasaaiaaikdacaWGPbGaamOAaaqcfayabaGaeq OSdi2aaSbaaKqbGeaacaaIYaGaamOAaaqcfayabaaajuaibaGaamOA aiabg2da9iaaigdaaeaacaWGQbGaeyypa0JaamiCaaqcfaOaeyyeIu oaaiaawUhacaGL9baacqGHsislcaWG5bWaaSbaaKqbGeaacaaIXaGa amyAaaqcfayabaGaeyOeI0IaaGymaaGaayjkaiaawMcaaiabgkHiTm aacmaabaWaaabCaeaacaWG4bWaaSbaaKqbGeaacaaIYaGaamyAaiaa dQgaaKqbagqaaiabek7aInaaBaaajuaibaGaaGOmaiaadQgaaKqbag qaaaqcfasaaiaadQgacqGH9aqpcaaIXaaabaGaamOAaiabg2da9iaa dghaaKqbakabggHiLdaacaGL7bGaayzFaaWaaeWaaeaacaWG5bWaaS baaKqbGeaacaaIXaGaamyAaaqcfayabaGaey4kaSIaamyEamaaBaaa juaibaGaaGOmaiaadMgaaKqbagqaaiabgUcaRiaaigdaaiaawIcaca GLPaaacqGHRaWkcaWG5bWaaSbaaKqbGeaacaaIYaGaamyAaaqcfaya baGaciiBaiaac6gadaqadaqaaiGacwgacaGG4bGaaiiCamaacmaaba WaaabCaeaacaWG4bWaaSbaaKqbGeaacaaIYaGaamyAaiaadQgaaKqb agqaaiabek7aInaaBaaajuaibaGaaGOmaiaadQgaaKqbagqaaaqcfa saaiaadQgacqGH9aqpcaaIXaaabaGaamOAaiabg2da9iaadghaaKqb akabggHiLdaacaGL7bGaayzFaaGaeyOeI0IaamyEamaaBaaajuaiba GaaGymaiaadMgaaKqbagqaaiabgkHiTiaaigdaaiaawIcacaGLPaaa cqGHsisldaGadaqaamaaqahabaGaamiEamaaBaaajuaibaGaaGOmai aadMgacaWGQbaajuaGbeaacqaHYoGydaWgaaqcfasaaiaaikdacaWG QbaajuaGbeaaaKqbGeaacaWGQbGaeyypa0JaaGymaaqaaiaadQgacq GH9aqpcaWGXbaajuaGcqGHris5aaGaay5Eaiaaw2haamaabmaabaGa amyEamaaBaaajuaibaGaaGymaiaadMgaaKqbagqaaiabgUcaRiaadM hadaWgaaqcfasaaiaaikdacaWGPbaajuaGbeaacqGHRaWkcaaIXaaa caGLOaGaayzkaaGaey4kaSIaamyEamaaBaaajuaibaGaaGOmaiaadM gaaKqbagqaaiGacYgacaGGUbWaaeWaaeaaciGGLbGaaiiEaiaaccha daGadaqaamaaqahabaGaamiEamaaBaaajuaibaGaaGOmaiaadMgaca WGQbaajuaGbeaacqaHYoGydaWgaaqcfasaaiaaikdacaWGQbaajuaG beaaaKqbGeaacaWGQbGaeyypa0JaaGymaaqaaiaadQgacqGH9aqpca WGXbaajuaGcqGHris5aaGaay5Eaiaaw2haaiabgkHiTiaadMhadaWg aaqcfasaaiaaigdacaWGPbaajuaGbeaacqGHsislcaaIXaaacaGLOa GaayzkaaaacaGLBbGaayzxaaaajuaibaGaamyAaiabg2da9iaaigda aeaacaWGPbGaeyypa0JaamOBaaqcfaOaeyyeIuoaaaaa@5004@

According to Dobson [12] this ΔD MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeuiLdq Kaamiraaaa@38B3@ follows χ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4Xdm 2aaWbaaeqajuaibaGaaGOmaaaaaaa@3947@ distribution with pq MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiCai abgkHiTiaadghaaaa@395C@ degrees of freedom.

If the value of ΔD MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeuiLdq Kaamiraaaa@38B3@ is consistent with the χ ( pq ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4Xdm 2aa0baaKqbGeaajuaGdaqadaqcfasaaiaadchacqGHsislcaWGXbaa caGLOaGaayzkaaaabaGaaGOmaaaaaaa@3E64@ distribution we would generally choose the M 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamytam aaBaaajuaibaGaaGimaaqcfayabaaaaa@38ED@ corresponding to H 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamisam aaBaaajuaibaGaaGimaaqcfayabaaaaa@38E8@ because it is simpler. On the other hand, if the value of ΔD MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeuiLdq Kaamiraaaa@38B3@ is in the critical region i.e., greater than the upper tail 100×α% MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaaGymai aaicdacaaIWaGaey41aqRaeqySdeMaaiyjaaaa@3D12@ point of the χ ( pq ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4Xdm 2aa0baaKqbGeaajuaGdaqadaqcfasaaiaadchacqGHsislcaWGXbaa caGLOaGaayzkaaaabaGaaGOmaaaaaaa@3E64@  distribution then would reject H 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamisam aaBaaajuaibaGaaGimaaqcfayabaaaaa@38E8@ in favor of H 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamisam aaBaaajuaibaGaaGymaaqcfayabaaaaa@38E9@ on the grounds that model M 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamytam aaBaaajuaibaGaaGymaaqcfayabaaaaa@38EE@ provides a significantly better description of the data.

Data Simulation and Analysis

To determine the efficiency of our derived deviances we need to have data with known parameters. However, we cannot generate data directly from bivariate geometric distribution using the available computer software packages. Kr­ishna and Pundir suggested an algorithm based on a theorem given by Hogg et al. [13] to generate random numbers from bivariate geometric distribution. According to this, paired values can be generated from a bivariate geometric distribution using the following steps,

  1. Step 1: Generate krandom numbers from univariate geometric distribu­tion with probability of success ( 1 q 1 q 2 1 q 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aadaWcaaqaaiaaigdacqGHsislcaWGXbWaaSbaaKqbGeaacaaIXaaa juaGbeaacqGHsislcaWGXbWaaSbaaKqbGeaacaaIYaaajuaGbeaaae aacaaIXaGaeyOeI0IaamyCamaaBaaajuaibaGaaGOmaaqcfayabaaa aaGaayjkaiaawMcaaaaa@4406@ .
  2. Step 2: Suppose that our generated random numbers from the geometric distribution are x 1 , x 2 ,..., x k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEam aaBaaajuaibaGaaGymaaqcfayabaGaaiilaiaadIhadaWgaaqcfasa aiaaikdaaKqbagqaaiaacYcacaGGUaGaaiOlaiaac6cacaGGSaGaam iEamaaBaaajuaibaGaam4Aaaqcfayabaaaaa@429F@ .
  3. Step 3: Generate k random numbers y ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyEam aaBaaajuaibaGaamyAaiaadQgaaKqbagqaaaaa@3A3C@ , k times each from a negative binomial distribution with parameters x i +1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIhada WgaaqcfasaaiaadMgaaKqbagqaaiabgUcaRiaaigdaaaa@3ADE@ and ( 1 q 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaaba GaaGymaiabgkHiTiaadghadaWgaaqcfasaaiaaikdaaKqbagqaaaGa ayjkaiaawMcaaaaa@3C39@ .
  4. Step 4: These generated pairs are from the bivariate geometric distribu­tion with parameters q 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyCam aaBaaajuaibaGaaGymaaqcfayabaaaaa@3912@ and q 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyCam aaBaaajuaibaGaaGOmaaqcfayabaaaaa@3913@ .
Deviance Checking for Reduced Model

In this subsection, we use the following steps to check our derived deviance for the reduced model with identical values of parameters ( q 1 , q 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacaWGXbWaaSbaaKqbGeaacaaIXaaajuaGbeaacaGGSaGaamyCamaa BaaajuaibaGaaGOmaaqcfayabaaacaGLOaGaayzkaaaaaa@3DDA@  for each observed pair of data.

  1. Step 1: Assume some fixed values of q 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyCam aaBaaajuaibaGaaGymaaqcfayabaaaaa@3912@ and q 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyCam aaBaaajuaibaGaaGOmaaqcfayabaaaaa@3913@ .
  2. Step 2: Generate k random numbers from univariate geometric distribu­tion with probability of success ( 1 q 1 q 2 1 q 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aadaWcaaqaaiaaigdacqGHsislcaWGXbWaaSbaaKqbGeaacaaIXaaa juaGbeaacqGHsislcaWGXbWaaSbaaKqbGeaacaaIYaaajuaGbeaaae aacaaIXaGaeyOeI0IaamyCamaaBaaajuaibaGaaGOmaaqcfayabaaa aaGaayjkaiaawMcaaaaa@4406@ using the assumed values of q 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyCam aaBaaajuaibaGaaGymaaqcfayabaaaaa@3912@ and q 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyCam aaBaaajuaibaGaaGOmaaqcfayabaaaaa@3913@ from Step 1.
  3. Step 3: Suppose that our generated random numbers from the geometric distribution are x 1 , x 2 ,..., x k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEam aaBaaajuaibaGaaGymaaqcfayabaGaaiilaiaadIhadaWgaaqcfasa aiaaikdaaKqbagqaaiaacYcacaGGUaGaaiOlaiaac6cacaGGSaGaam iEamaaBaaajuaibaGaam4Aaaqcfayabaaaaa@429F@ .
  4. Step 4: Generate k random numbers y ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadMhada WgaaqcfasaaiaadMgacaWGQbaajuaGbeaaaaa@3A31@ , k times each from the negative binomial distribution with parameters x i +1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIhada WgaaqcfasaaiaadMgaaKqbagqaaiabgUcaRiaaigdaaaa@3ADE@ and ( 1 q 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaaba GaaGymaiabgkHiTiaadghadaWgaaqcfasaaiaaikdaaKqbagqaaaGa ayjkaiaawMcaaaaa@3C39@ .
  5. Step 5: The generated pairs are from the bivariate geometric distribution with parameters q 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyCam aaBaaajuaibaGaaGymaaqcfayabaaaaa@3912@ and q 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyCam aaBaaajuaibaGaaGOmaaqcfayabaaaaa@3913@ .
  6. Step 6: Estimate deviance which is derived in (14).

We take the values of q 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyCam aaBaaajuaibaGaaGymaaqcfayabaaaaa@3912@ and q 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyCam aaBaaajuaibaGaaGOmaaqcfayabaaaaa@3913@ ranging from 0.10 to 0.90 and satisfying the constraint q 1 + q 2 <1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadghada WgaaqcfasaaiaaigdaaKqbagqaaiabgUcaRiaadghadaWgaaqcfasa aiaaikdaaKqbagqaaiabgYda8iaaigdaaaa@3E37@ . We considered several values for the pair ( q 1 , q 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacaWGXbWaaSbaaKqbGeaacaaIXaaajuaGbeaacaGGSaGaamyCamaa BaaajuaibaGaaGOmaaqcfayabaaacaGLOaGaayzkaaaaaa@3DDA@ and generate random pairs to observe the efficiency of our derived deviance under different parametric values. For each specified pairs of parameters ( q 1 , q 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacaWGXbWaaSbaaKqbGeaacaaIXaaajuaGbeaacaGGSaGaamyCamaa BaaajuaibaGaaGOmaaqcfayabaaacaGLOaGaayzkaaaaaa@3DDA@ , we ran this experiment twice to see whether there is a change in our decision due to randomness. The values of the pair of parameters and the corresponding deviance values are tabulated as follows.

The deviance we derived to test the parameters of the reduced model works well as we see that all, but four of the values of the deviances are smaller than ­ X 198 2 ( 0.95 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIfada qhaaqcfasaaiaaigdacaaI5aGaaGioaaqaaiaaikdaaaqcfa4aaeWa aeaacaaIWaGaaiOlaiaaiMdacaaI1aaacaGLOaGaayzkaaaaaa@3FA7@ . However, among these four values of the deviances three are greater than ­ X 198 2 ( 0.95 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIfada qhaaqcfasaaiaaigdacaaI5aGaaGioaaqaaiaaikdaaaqcfa4aaeWa aeaacaaIWaGaaiOlaiaaiMdacaaI1aaacaGLOaGaayzkaaaaaa@3FA7@ , but less than X 198 2 ( 0.99 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIfada qhaaqcfasaaiaaigdacaaI5aGaaGioaaqaaiaaikdaaaqcfa4aaeWa aeaacaaIWaGaaiOlaiaaiMdacaaI5aaacaGLOaGaayzkaaaaaa@3FAB@ . So, it can be concluded that our derived deviance works well. On the other hand, if most of the values of the deviances had a larger value than our desired ­ χ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4Xdm 2aaWbaaeqajuaibaGaaGOmaaaaaaa@3947@ value, then we had to conclude that our derived deviance does not work in testing hypothesis regarding the parameters of the reduced model.

Parameters

Deviance

χ 198 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeE8aJn aaBaaajuaibaGaaGymaiaaiMdacaaI4aaajuaGbeaaaaa@3B4D@ (0.95)

χ 198 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeE8aJn aaBaaajuaibaGaaGymaiaaiMdacaaI4aaajuaGbeaaaaa@3B4D@ (0.975)

χ 198 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeE8aJn aaBaaajuaibaGaaGymaiaaiMdacaaI4aaajuaGbeaaaaa@3B4D@ (0.99)

q1=0.30,q2=0.30

177.4164

231.8292

238.8612

247.2118

q1=0.30,q2=0.30

172.3071

231.8292

238.8612

247.2118

q1=0.30,q2=0.40

185.3107

231.8292

238.8612

247.2118

q1=0.30,q2=0.40

159.5293

231.8292

238.8612

247.2118

q1=0.30,q2=0.50

193.8942

231.8292

238.8612

247.2118

q1=0.30,q2=0.50

158.266

231.8292

238.8612

247.2118

q1=0.30,q2=0.60

223.1697

231.8292

238.8612

247.2118

q1=0.30,q2=0.60

193.667

231.8292

238.8612

247.2118

q1=0.40,q2=0.30

216.3456

231.8292

238.8612

247.2118

q1=0.40,q2=0.30

211.828

231.8292

238.8612

247.2118

q1=0.50,q2=0.30

148.1757

231.8292

238.8612

247.2118

q1=0.50,q2=0.30

254.3887

231.8292

238.8612

247.2118

q1=0.60,q2=0.30

239.3245

231.8292

238.8612

247.2118

q1=0.60,q2=0.30

215.4915

231.8292

238.8612

247.2118

q1=0.30,q2=0.50

232.1984

231.8292

238.8612

247.2118

q1=0.30,q2=0.50

191.7516

231.8292

238.8612

247.2118

q1=0.30,q2=0.60

184.1803

231.8292

238.8612

247.2118

q1=0.30,q2=0.60

236.0869

231.8292

238.8612

247.2118

q1=0.10,q2=0.10

97.9206

231.8292

238.8612

247.2118

q1=0.10,q2=0.10

85.10731

231.8292

238.8612

247.2118

Table 1: Estimation of deviance for different parameters under consideration.

Parameters

Deviance

χ 198 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeE8aJn aaBaaajuaibaGaaGymaiaaiMdacaaI4aaajuaGbeaaaaa@3B4D@  
(0.95)

χ 198 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeE8aJn aaBaaajuaibaGaaGymaiaaiMdacaaI4aaajuaGbeaaaaa@3B4D@  
(0.975)

χ 198 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeE8aJn aaBaaajuaibaGaaGymaiaaiMdacaaI4aaajuaGbeaaaaa@3B4D@
 (0.99)

q1=0.10,q2=0.20

100.8624

231.8292

238.8612

247.2118

q1=0.10,q2=0.20

155.157

231.8292

238.8612

247.2118

q1=0.10,q2=0.30

155.157

231.8292

238.8612

247.2118

q1=0.10,q2=0.30

123.3245

231.8292

238.8612

247.2118

q1=0.20,q2=0.20

113.3245

231.8292

238.8612

247.2118

q1=0.20,q2=0.20

147.3637

231.8292

238.8612

247.2118

q1=0.20,q2=0.30

166.6306

231.8292

238.8612

247.2118

q1=0.20,q2=0.30

157.8232

231.8292

238.8612

247.2118

q1=0.30,q2=0.10

133.2772

231.8292

238.8612

247.2118

q1=0.30,q2=0.10

131.2191

231.8292

238.8612

247.2118

q1=0.10,q2=0.80

183.8584

231.8292

238.8612

247.2118

q1=0.10,q2=0.80

218.8224

231.8292

238.8612

247.2118

q1=0.80,q2=0.10

203.6515

231.8292

238.8612

247.2118

q1=0.80,q2=0.10

177.6116

231.8292

238.8612

247.2118

q1=0.10,q2=0.40

144.1728

231.8292

238.8612

247.2118

q1=0.10,q2=0.40

168.524

231.8292

238.8612

247.2118

q1=0.70,q2=0.10

169.3248

231.8292

238.8612

247.2118

q1=0.70,q2=0.10

177.8397

231.8292

238.8612

247.2118

q1=0.60,q2=0.10

177.1335

231.8292

238.8612

247.2118

q1=0.70,q2=0.10

197.0526

231.8292

238.8612

247.2118

q1=0.50,q2=0.10

159.3473

231.8292

238.8612

247.2118

q1=0.50,q2=0.10

146.7018

231.8292

238.8612

247.2118

Table 2: Estimation of deviance for different parameters under consideration (contd).

Conclusion

In this paper, we addressed an important problem of inference regarding bivariate geometric distribution and developed testing procedure for the parameters of this distribution with and without covariate information. Our method depends on deriving the deviance statistics using maximum likelihood estimators (mle) of parameters. Our mles of the parameters of the bivariate geometric distribution are obtained using the conditional and the marginal distributions.

We conducted a numerical analysis based on simulated data for the testing the 11 identical parameter assumption for each pair of observed data. Our numerical example did not consider any covariate information. We found that without covariate information our derived deviance worked well in most cases.

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