Combining link-tracing sampling and cluster sampling to estimate the size of a hidden population in presence of heterogeneous link-probabilities 3. Maximum likelihood estimators of τ 1 , τ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbeqabeWaceGabiqabeqabmqabeabbaGcbaGaeqiXdq3aaS baaSqaaiaaigdaaeqaaOGaaiilaiabes8a0naaBaaaleaacaaIYaaa beaaaaa@3C52@  and τ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbeqabeWaceGabiqabeqabmqabeabbaGcbaGaeqiXdqhaaa@3804@

3.1 Probability models

To construct MLEs of the τ s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDie aacaWFzaIaae4Caaaa@3BE3@ we need to specify models for the observed variables. Thus, as in Félix-Medina and Thompson (2004), we will suppose that the numbers M 1 , , M N MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGnbWaaS baaSqaaiaaigdaaeqaaOGaaGilaiablAciljaaiYcacaWGnbWaaSba aSqaaiaad6eaaeqaaaaa@3E87@ of people who belong to the sites A 1 , , A N MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGbbWaaS baaSqaaiaaigdaaeqaaOGaaGilaiablAciljaaiYcacaWGbbWaaSba aSqaaiaad6eaaeqaaaaa@3E6F@ are independent Poisson random variables with mean λ 1 . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaH7oaBda WgaaWcbaGaaGymaaqabaGccaGGUaaaaa@3BBC@ Therefore, the joint conditional distribution of ( M 1 , , M n , τ 1 M ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaadaqadaqaai aad2eadaWgaaWcbaGaaGymaaqabaGccaaISaGaeSOjGSKaaGilaiaa d2eadaWgaaWcbaGaamOBaaqabaGccaaISaGaeqiXdq3aaSbaaSqaai aaigdaaeqaaOGaeyOeI0IaamytaaGaayjkaiaawMcaaaaa@4565@ given that 1 N M i = τ 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaadaaeWaqabS qaaiaaigdaaeaacaWGobaaniabggHiLdGccaWGnbWaaSbaaSqaaiaa dMgaaeqaaOGaeyypa0JaeqiXdq3aaSbaaSqaaiaaigdaaeqaaaaa@41A8@ is multinomial with probability mass function (pmf):

f( m 1 ,, m n , τ 1 m )= τ 1 ! 1 n m i !( τ 1 m )! ( 1 N ) m ( 1 n N ) τ 1 m .(3.1) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGMbWaae WaaeaacaWGTbWaaSbaaSqaaiaaigdaaeqaaOGaaGilaiablAciljaa iYcacaWGTbWaaSbaaSqaaiaad6gaaeqaaOGaaGilaiabes8a0naaBa aaleaacaaIXaaabeaakiabgkHiTiaad2gaaiaawIcacaGLPaaacqGH 9aqpdaWcaaqaaiabes8a0naaBaaaleaacaaIXaaabeaakiaacgcaae aadaqeWbqabSqaaiaaigdaaeaacaWGUbaaniabg+GivdGccaWGTbWa aSbaaSqaaiaadMgaaeqaaOGaaiyiamaabmaabaGaeqiXdq3aaSbaaS qaaiaaigdaaeqaaOGaeyOeI0IaamyBaaGaayjkaiaawMcaaiaacgca aaWaaeWaaeaadaWcaaqaaiaaigdaaeaacaWGobaaaaGaayjkaiaawM caamaaCaaaleqabaGaamyBaaaakmaabmaabaGaaGymaiabgkHiTmaa laaabaGaamOBaaqaaiaad6eaaaaacaGLOaGaayzkaaWaaWbaaSqabe aacqaHepaDdaWgaaadbaGaaGymaaqabaWccqGHsislcaWGTbaaaOGa aGOlaiaaywW7caaMf8UaaGzbVlaaywW7caaMf8Uaaiikaiaaiodaca GGUaGaaGymaiaacMcaaaa@72AE@

To model the links between the members of the population and the sampled sites we will define the following random variables: X ij ( k ) =1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGybWaa0 baaSqaaiaadMgacaWGQbaabaWaaeWaaeaacaWGRbaacaGLOaGaayzk aaaaaOGaeyypa0JaaGymaaaa@3F90@ if person j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGQbaaaa@3954@ in U k A i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGvbWaaS baaSqaaiaadUgaaeqaaOGaeyOeI0IaamyqamaaBaaaleaacaWGPbaa beaaaaa@3D32@ is linked to site A i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGbbWaaS baaSqaaiaadMgaaeqaaaaa@3A45@ and X ij ( k ) =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGybWaa0 baaSqaaiaadMgacaWGQbaabaWaaeWaaeaacaWGRbaacaGLOaGaayzk aaaaaOGaeyypa0JaaGimaaaa@3F8F@ if j A i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGQbGaey icI4SaamyqamaaBaaaleaacaWGPbaabeaaaaa@3CB8@ or that person is not linked to A i ,j=1,, τ k ,i=1,,n. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGbbWaaS baaSqaaiaadMgaaeqaaOGaaiilaiaadQgacqGH9aqpcaaIXaGaaiil aiablAciljaaiYcacqaHepaDdaWgaaWcbaGaam4AaaqabaGccaGGSa GaamyAaiabg2da9iaaigdacaGGSaGaeSOjGSKaaGilaiaad6gacaGG Uaaaaa@4AAE@ We will suppose that given the sample S A MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGtbWaaS baaSqaaiaadgeaaeqaaaaa@3A2F@ of sites the X i j ( k ) s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGybWaa0 baaSqaaiaadMgacaWGQbaabaWaaeWaaeaacaWGRbaacaGLOaGaayzk aaaaaGqaaOGaa8xgGiaabohaaaa@3F88@ are independent Bernoulli random variables with means p i j ( k ) s , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGWbWaa0 baaSqaaiaadMgacaWGQbaabaWaaeWaaeaacaWGRbaacaGLOaGaayzk aaaaaGqaaOGaa8xgGiaabohacaGGSaaaaa@4050@ where the link-probability p i j ( k ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGWbWaa0 baaSqaaiaadMgacaWGQbaabaWaaeWaaeaacaWGRbaacaGLOaGaayzk aaaaaaaa@3DDD@ satisfies the following Rasch model:

p ij ( k ) =Pr( X ij ( k ) =1| β j ( k ) , S A )= exp( α i ( k ) + β j ( k ) ) 1+exp( α i ( k ) + β j ( k ) ) , j U k A i ; i=1,,n.(3.2) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGWbWaa0 baaSqaaiaadMgacaWGQbaabaWaaeWaaeaacaWGRbaacaGLOaGaayzk aaaaaOGaeyypa0JaciiuaiaackhadaqadaqaaiaadIfadaqhaaWcba GaamyAaiaadQgaaeaadaqadaqaaiaadUgaaiaawIcacaGLPaaaaaGc cqGH9aqpcaaIXaWaaqqaaeaacaaMc8UaeqOSdi2aa0baaSqaaiaadQ gaaeaadaqadaqaaiaadUgaaiaawIcacaGLPaaaaaGccaaISaGaam4u amaaBaaaleaacaWGbbaabeaaaOGaay5bSdaacaGLOaGaayzkaaGaey ypa0ZaaSaaaeaaciGGLbGaaiiEaiaacchadaqadaqaaiabeg7aHnaa DaaaleaacaWGPbaabaWaaeWaaeaacaWGRbaacaGLOaGaayzkaaaaaO Gaey4kaSIaeqOSdi2aa0baaSqaaiaadQgaaeaadaqadaqaaiaadUga aiaawIcacaGLPaaaaaaakiaawIcacaGLPaaaaeaacaaIXaGaey4kaS IaciyzaiaacIhacaGGWbWaaeWaaeaacqaHXoqydaqhaaWcbaGaamyA aaqaamaabmaabaGaam4AaaGaayjkaiaawMcaaaaakiabgUcaRiabek 7aInaaDaaaleaacaWGQbaabaWaaeWaaeaacaWGRbaacaGLOaGaayzk aaaaaaGccaGLOaGaayzkaaaaaiaaiYcacaqGGaGaamOAaiabgIGiol aadwfadaWgaaWcbaGaam4AaaqabaGccqGHsislcaWGbbWaaSbaaSqa aiaadMgaaeqaaOGaai4oaiaayIW7caqGGaGaamyAaiabg2da9iaaig dacaGGSaGaeSOjGSKaaGilaiaad6gacaaIUaGaaGzbVlaaywW7caaM f8UaaGzbVlaacIcacaaIZaGaaiOlaiaaikdacaGGPaaaaa@92C2@

It is worth noting that this model was considered by Coull and Agresti (1999) in the context of capture-recapture sampling. In this model α i ( k ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHXoqyda qhaaWcbaGaamyAaaqaamaabmaabaGaam4AaaGaayjkaiaawMcaaaaa aaa@3D98@ is a fixed (not random) effect that represents the potential that the cluster A i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGbbWaaS baaSqaaiaadMgaaeqaaaaa@3A45@ has of forming links with the people in U k A i , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGvbWaaS baaSqaaiaadUgaaeqaaOGaeyOeI0IaamyqamaaBaaaleaacaWGPbaa beaakiaacYcaaaa@3DEC@ and β j ( k ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHYoGyda qhaaWcbaGaamOAaaqaamaabmaabaGaam4AaaGaayjkaiaawMcaaaaa aaa@3D9B@ is a random effect that represents the propensity of the person j U k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGQbGaey icI4SaamyvamaaBaaaleaacaWGRbaabeaaaaa@3CCE@ to be linked to a cluster. We will suppose that β j ( k ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHYoGyda qhaaWcbaGaamOAaaqaamaabmaabaGaam4AaaGaayjkaiaawMcaaaaa aaa@3D9B@ is normally distributed with mean 0 and unknown variance σ k 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHdpWCda qhaaWcbaGaam4Aaaqaaiaaikdaaaaaaa@3C01@ and that these variables are independent. The parameter σ k 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHdpWCda qhaaWcbaGaam4Aaaqaaiaaikdaaaaaaa@3C01@ determines the degree of heterogeneity of the p i j ( k ) s : MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGWbWaa0 baaSqaaiaadMgacaWGQbaabaWaaeWaaeaacaWGRbaacaGLOaGaayzk aaaaaGqaaOGaa8xgGiaabohacaGG6aaaaa@405E@ great values of σ k 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHdpWCda qhaaWcbaGaam4Aaaqaaiaaikdaaaaaaa@3C01@ imply high degree of heterogeneity.

Before we end this subsection, we will make some comments about the assumed models. First, the multinomial distribution of the observed M i s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGnbWaaS baaSqaaiaadMgaaeqaaGqaaOGaa8xgGiaabohaaaa@3C14@ (which is the one used in the likelihood function) implies that people are distributed independently and with equal probability on the sites of the sampling frame. This assumption is difficult to satisfy in actual situations; however, as will be shown later, the likelihood function depends on the observed M i s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGnbWaaS baaSqaaiaadMgaaeqaaGqaaOGaa8xgGiaabohaaaa@3C14@ basically through their sum M MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGnbaaaa@3937@ and since N M / n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaadaWcgaqaai aad6eacaWGnbaabaGaamOBaaaaaaa@3B13@ is a design-based estimator of τ 1 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDda WgaaWcbaGaaGymaaqabaGccaGGSaaaaa@3BCB@ that is, it is a distribution free estimator, it follows that the MLE of τ 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDda WgaaWcbaGaaGymaaqabaaaaa@3B11@ will be also robust to deviations from the multinomial distribution of the M i s . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGnbWaaS baaSqaaiaadMgaaeqaaGqaaOGaa8xgGiaabohacaqGUaaaaa@3CC5@ Nevertheless, deviations from this model will affect the performance of variance estimators and confidence intervals derived under this assumption. Second, the Rasch model given by (3.2) implies the following: ( i ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9pC0xbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca qGPbaacaGLOaGaayzkaaaaaa@3ACA@ the link-probability p i j ( k ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGWbWaa0 baaSqaaiaadMgacaWGQbaabaWaaeWaaeaacaWGRbaacaGLOaGaayzk aaaaaaaa@3DDD@ depends only on two effects: the sociability of the people in cluster A i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGbbWaaS baaSqaaiaadMgaaeqaaaaa@3A45@ and that of person j U k A i ; MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGQbGaey icI4SaamyvamaaBaaaleaacaWGRbaabeaakiabgkHiTiaadgeadaWg aaWcbaGaamyAaaqabaGccaGG7aaaaa@406E@ ( ii ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9pC0xbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca qGPbGaaeyAaaGaayjkaiaawMcaaaaa@3BB6@ the two effects are additive, and ( iii ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9pC0xbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca qGPbGaaeyAaiaabMgaaiaawIcacaGLPaaaaaa@3CA2@ for any site A i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGbbWaaS baaSqaaiaadMgaaeqaaaaa@3A45@ in the frame and any person jU A i , p ij ( k ) >0. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGQbGaey icI4SaamyvaiabgkHiTiaadgeadaWgaaWcbaGaamyAaaqabaGccaGG SaGaaGjbVlaadchadaqhaaWcbaGaamyAaiaadQgaaeaadaqadaqaai aadUgaaiaawIcacaGLPaaaaaGccaaMe8UaaeOpaiaaysW7caqGWaGa aeOlaaaa@4B87@ Model (3.2) is a particular case of a generalized linear mixed model. (See Agresti 2002, Section 2.1, for a brief review of this type of model.) Therefore, we could incorporate the network structures of the people in cluster A i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGbbWaaS baaSqaaiaadMgaaeqaaaaa@3A45@ and person j U k A i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGQbGaey icI4SaamyvamaaBaaaleaacaWGRbaabeaakiabgkHiTiaadgeadaWg aaWcbaGaamyAaaqabaaaaa@3FA5@ to model the link-probability p i j ( k ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGWbWaa0 baaSqaaiaadMgacaWGQbaabaWaaeWaaeaacaWGRbaacaGLOaGaayzk aaaaaaaa@3DDD@ by extending model (3.2) to one that includes covariates associated with person j , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGQbGaai ilaaaa@3A04@ with cluster A i , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGbbWaaS baaSqaaiaadMgaaeqaaOGaaiilaaaa@3AFF@ and their interaction terms. However, if we used a more general model than (3.2), we would make the problem of inference much more difficult than that we face in this work. Thus, in spite of the relative simplicity of model (3.2), we expect that it still captures the heterogeneity of the link-probabilities and allow us to make inferences about the τ s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDie aacaWFzaIaae4Caaaa@3BE3@ at least at the correct order of magnitude.

3.2 Likelihood function

The easiest way of constructing the likelihood function is to factorize it into different components. One of them is associated with the probability of selecting the initial sample S 0 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGtbWaaS baaSqaaiaaicdaaeqaaOGaaiilaaaa@3ADD@ which is given by the multinomial distribution (3.1), that is,

L MULT ( τ 1 ) τ 1 ! ( τ 1 m )! ( 1n/N ) τ 1 m . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGmbWaaS baaSqaaiaab2eacaqGvbGaaeitaiaabsfaaeqaaOWaaeWaaeaacqaH epaDdaWgaaWcbaGaaGymaaqabaaakiaawIcacaGLPaaacqGHDisTda Wcaaqaaiabes8a0naaBaaaleaacaaIXaaabeaakiaacgcaaeaadaqa daqaaiabes8a0naaBaaaleaacaaIXaaabeaakiabgkHiTiaad2gaai aawIcacaGLPaaacaGGHaaaamaabmaabaWaaSGbaeaacaaIXaGaeyOe I0IaamOBaaqaaiaad6eaaaaacaGLOaGaayzkaaWaaWbaaSqabeaacq aHepaDdaWgaaadbaGaaGymaaqabaWccqGHsislcaWGTbaaaOGaaGOl aaaa@5739@

Two other components are associated with the conditional probabilities of the configurations of links between the people in U k S 0 ,k=1,2, MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGvbWaaS baaSqaaiaadUgaaeqaaOGaeyOeI0Iaam4uamaaBaaaleaacaaIWaaa beaakiaacYcacaWGRbGaeyypa0JaaGymaiaacYcacaaIYaGaaiilaa aa@4297@ and the clusters A i S A , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGbbWaaS baaSqaaiaadMgaaeqaaOGaeyicI4Saam4uamaaBaaaleaacaWGbbaa beaakiaacYcaaaa@3E57@ given S A . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGtbWaaS baaSqaaiaadgeaaeqaaOGaaiOlaaaa@3AEB@ To derive these factors we need to compute the probabilities of some events. Let X j ( k ) =( X 1j ( k ) ,, X nj ( k ) ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWHybWaa0 baaSqaaiaadQgaaeaadaqadaqaaiaadUgaaiaawIcacaGLPaaaaaGc cqGH9aqpdaqadaqaaiaadIfadaqhaaWcbaGaaGymaiaadQgaaeaada qadaqaaiaadUgaaiaawIcacaGLPaaaaaGccaaISaGaeSOjGSKaaGil aiaadIfadaqhaaWcbaGaamOBaiaadQgaaeaadaqadaqaaiaadUgaai aawIcacaGLPaaaaaaakiaawIcacaGLPaaaaaa@4CA8@ be the n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGUbGaey OeI0caaa@3A45@ dimensional vector of link-indicator variables X i j ( k ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGybWaa0 baaSqaaiaadMgacaWGQbaabaWaaeWaaeaacaWGRbaacaGLOaGaayzk aaaaaaaa@3DC5@ associated with the j th MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGQbWaaW baaSqabeaacaqG0bGaaeiAaaaaaaa@3B63@ person in U k S 0 . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGvbWaaS baaSqaaiaadUgaaeqaaOGaeyOeI0Iaam4uamaaBaaaleaacaaIWaaa beaakiaac6caaaa@3DCC@ Notice that X j ( k ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWHybWaa0 baaSqaaiaadQgaaeaadaqadaqaaiaadUgaaiaawIcacaGLPaaaaaaa aa@3CDB@ indicates which clusters A i S A MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGbbWaaS baaSqaaiaadMgaaeqaaOGaeyicI4Saam4uamaaBaaaleaacaWGbbaa beaaaaa@3D9D@ are linked to that person. Let x=( x 1 ,, x n ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWH4bGaey ypa0ZaaeWaaeaacaWG4bWaaSbaaSqaaiaaigdaaeqaaOGaaGilaiab lAciljaaiYcacaWG4bWaaSbaaSqaaiaad6gaaeqaaaGccaGLOaGaay zkaaaaaa@4297@ be a vector whose i th MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGPbWaaW baaSqabeaacaqG0bGaaeiAaaaaaaa@3B62@ element is 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaaIWaaaaa@391F@ or 1,i=1,,n. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaaIXaGaai ilaiaadMgacqGH9aqpcaaIXaGaaiilaiablAciljaaiYcacaWGUbGa aiOlaaaa@40AC@ Because of the assumptions we made about the distributions of the variables X i j ( k ) s, MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGybWaa0 baaSqaaiaadMgacaWGQbaabaWaaeWaaeaacaWGRbaacaGLOaGaayzk aaaaaGqaaOGaa8xgGiaabohacaqGSaaaaa@4037@ we have that the conditional probability, given β j ( k ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHYoGyda qhaaWcbaGaamOAaaqaamaabmaabaGaam4AaaGaayjkaiaawMcaaaaa kiaacYcaaaa@3E55@ that X j ( k ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWHybWaa0 baaSqaaiaadQgaaeaadaqadaqaaiaadUgaaiaawIcacaGLPaaaaaaa aa@3CDB@ equals x , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWH4bGaai ilaaaa@3A16@ that is, the probability that the j th MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGQbWaaW baaSqabeaacaqG0bGaaeiAaaaaaaa@3B63@ person is linked to only those clusters A i S A MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGbbWaaS baaSqaaiaadMgaaeqaaOGaeyicI4Saam4uamaaBaaaleaacaWGbbaa beaaaaa@3D9D@ such that the i th MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGPbWaaW baaSqabeaacaqG0bGaaeiAaaaaaaa@3B62@ element x i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWG4bWaaS baaSqaaiaadMgaaeqaaaaa@3A7C@ of x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWH4baaaa@3966@ equals 1, is

Pr( X j ( k ) =x| β j ( k ) , S A )= i=1 n [ p ij ( k ) ] x i [ 1 p ij ( k ) ] 1 x i = i=1 n exp[ x i ( α i ( k ) + β j ( k ) ) ] 1+exp( α i ( k ) + β j ( k ) ) . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaaciGGqbGaai OCamaabmaabaGaaCiwamaaDaaaleaacaWGQbaabaWaaeWaaeaacaWG RbaacaGLOaGaayzkaaaaaOGaeyypa0JaaCiEamaaeeaabaGaaGPaVl abek7aInaaDaaaleaacaWGQbaabaWaaeWaaeaacaWGRbaacaGLOaGa ayzkaaaaaOGaaGilaiaadofadaWgaaWcbaGaamyqaaqabaaakiaawE a7aaGaayjkaiaawMcaaiabg2da9maarahabeWcbaGaamyAaiabg2da 9iaaigdaaeaacaWGUbaaniabg+GivdGcdaWadaqaaiaadchadaqhaa WcbaGaamyAaiaadQgaaeaadaqadaqaaiaadUgaaiaawIcacaGLPaaa aaaakiaawUfacaGLDbaadaahaaWcbeqaaiaadIhadaWgaaadbaGaam yAaaqabaaaaOWaamWaaeaacaaIXaGaeyOeI0IaamiCamaaDaaaleaa caWGPbGaamOAaaqaamaabmaabaGaam4AaaGaayjkaiaawMcaaaaaaO Gaay5waiaaw2faamaaCaaaleqabaGaaGymaiabgkHiTiaadIhadaWg aaadbaGaamyAaaqabaaaaOGaeyypa0ZaaebCaeqaleaacaWGPbGaey ypa0JaaGymaaqaaiaad6gaa0Gaey4dIunakmaalaaabaGaciyzaiaa cIhacaGGWbWaamWaaeaacaWG4bWaaSbaaSqaaiaadMgaaeqaaOWaae WaaeaacqaHXoqydaqhaaWcbaGaamyAaaqaamaabmaabaGaam4AaaGa ayjkaiaawMcaaaaakiabgUcaRiabek7aInaaDaaaleaacaWGQbaaba WaaeWaaeaacaWGRbaacaGLOaGaayzkaaaaaaGccaGLOaGaayzkaaaa caGLBbGaayzxaaaabaGaaGymaiabgUcaRiGacwgacaGG4bGaaiiCam aabmaabaGaeqySde2aa0baaSqaaiaadMgaaeaadaqadaqaaiaadUga aiaawIcacaGLPaaaaaGccqGHRaWkcqaHYoGydaqhaaWcbaGaamOAaa qaamaabmaabaGaam4AaaGaayjkaiaawMcaaaaaaOGaayjkaiaawMca aaaacaaIUaaaaa@97C5@

Therefore, the probability that the vector of link-indicator variables associated with a randomly selected person in U k S 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGvbWaaS baaSqaaiaadUgaaeqaaOGaeyOeI0Iaam4uamaaBaaaleaacaaIWaaa beaaaaa@3D10@ equals x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWH4baaaa@3966@ is

π x ( k ) ( α k , σ k )= i=1 n exp[ x i ( α i ( k ) + σ k z ) ] 1+exp( α i ( k ) + σ k z ) ϕ( z )dz, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHapaCda qhaaWcbaGaaCiEaaqaamaabmaabaGaam4AaaGaayjkaiaawMcaaaaa kmaabmaabaGaaCySdmaaBaaaleaacaWGRbaabeaakiaaiYcacqaHdp WCdaWgaaWcbaGaam4AaaqabaaakiaawIcacaGLPaaacqGH9aqpdaWd baqabSqabeqaniabgUIiYdGcdaqeWbqabSqaaiaadMgacqGH9aqpca aIXaaabaGaamOBaaqdcqGHpis1aOWaaSaaaeaaciGGLbGaaiiEaiaa cchadaWadaqaaiaadIhadaWgaaWcbaGaamyAaaqabaGcdaqadaqaai abeg7aHnaaDaaaleaacaWGPbaabaWaaeWaaeaacaWGRbaacaGLOaGa ayzkaaaaaOGaey4kaSIaeq4Wdm3aaSbaaSqaaiaadUgaaeqaaOGaam OEaaGaayjkaiaawMcaaaGaay5waiaaw2faaaqaaiaaigdacqGHRaWk ciGGLbGaaiiEaiaacchadaqadaqaaiabeg7aHnaaDaaaleaacaWGPb aabaWaaeWaaeaacaWGRbaacaGLOaGaayzkaaaaaOGaey4kaSIaeq4W dm3aaSbaaSqaaiaadUgaaeqaaOGaamOEaaGaayjkaiaawMcaaaaacq aHvpGzdaqadaqaaiaadQhaaiaawIcacaGLPaaacaWGKbGaamOEaiaa iYcaaaa@77CB@

where α k =( α 1 ( k ) ,, α n ( k ) ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWHXoWaaS baaSqaaiaadUgaaeqaaOGaeyypa0ZaaeWaaeaacqaHXoqydaqhaaWc baGaaGymaaqaamaabmaabaGaam4AaaGaayjkaiaawMcaaaaakiaaiY cacqWIMaYscaaISaGaeqySde2aa0baaSqaaiaad6gaaeaadaqadaqa aiaadUgaaiaawIcacaGLPaaaaaaakiaawIcacaGLPaaaaaa@4A31@ and ϕ() MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHvpGzca GGOaGaeyyXICTaaiykaaaa@3DD0@ denotes the probability density function of the standard normal distribution [ N ( 0 , 1 ) ] . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaadaWadaqaai aab6eadaqadaqaaiaaicdacaGGSaGaaGymaaGaayjkaiaawMcaaaGa ay5waiaaw2faaiaac6caaaa@3F88@

As in Coull and Agresti (1999), instead of using π x ( k ) ( α k , σ k ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHapaCda qhaaWcbaGaaCiEaaqaamaabmaabaGaam4AaaGaayjkaiaawMcaaaaa kmaabmaabaGaaCySdmaaBaaaleaacaWGRbaabeaakiaaiYcacqaHdp WCdaWgaaWcbaGaam4AaaqabaaakiaawIcacaGLPaaaaaa@455E@ in the likelihood function we will use its Gaussian quadrature approximation π ˜ x ( k ) ( α k , σ k ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHapaCga acamaaDaaaleaacaWH4baabaWaaeWaaeaacaWGRbaacaGLOaGaayzk aaaaaOWaaeWaaeaacaWHXoWaaSbaaSqaaiaadUgaaeqaaOGaaGilai abeo8aZnaaBaaaleaacaWGRbaabeaaaOGaayjkaiaawMcaaaaa@456D@ given by

π ˜ x ( k ) ( α k , σ k )= t=1 q i=1 n exp[ x i ( α i ( k ) + σ k z t ) ] 1+exp( α i ( k ) + σ k z t ) ν t ,(3.3) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHapaCga acamaaDaaaleaacaWH4baabaWaaeWaaeaacaWGRbaacaGLOaGaayzk aaaaaOWaaeWaaeaacaWHXoWaaSbaaSqaaiaadUgaaeqaaOGaaGilai abeo8aZnaaBaaaleaacaWGRbaabeaaaOGaayjkaiaawMcaaiabg2da 9maaqahabeWcbaGaamiDaiabg2da9iaaigdaaeaacaWGXbaaniabgg HiLdGcdaqeWbqabSqaaiaadMgacqGH9aqpcaaIXaaabaGaamOBaaqd cqGHpis1aOWaaSaaaeaaciGGLbGaaiiEaiaacchadaWadaqaaiaadI hadaWgaaWcbaGaamyAaaqabaGcdaqadaqaaiabeg7aHnaaDaaaleaa caWGPbaabaWaaeWaaeaacaWGRbaacaGLOaGaayzkaaaaaOGaey4kaS Iaeq4Wdm3aaSbaaSqaaiaadUgaaeqaaOGaamOEamaaBaaaleaacaWG 0baabeaaaOGaayjkaiaawMcaaaGaay5waiaaw2faaaqaaiaaigdacq GHRaWkciGGLbGaaiiEaiaacchadaqadaqaaiabeg7aHnaaDaaaleaa caWGPbaabaWaaeWaaeaacaWGRbaacaGLOaGaayzkaaaaaOGaey4kaS Iaeq4Wdm3aaSbaaSqaaiaadUgaaeqaaOGaamOEamaaBaaaleaacaWG 0baabeaaaOGaayjkaiaawMcaaaaacqaH9oGBdaWgaaWcbaGaamiDaa qabaGccaaISaGaaGzbVlaaywW7caaMf8UaaGzbVlaaywW7caGGOaGa aG4maiaac6cacaaIZaGaaiykaaaa@8629@

where q MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGXbaaaa@395B@ is a fixed constant and { z t } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaadaGadaqaai aadQhadaWgaaWcbaGaamiDaaqabaaakiaawUhacaGL9baaaaa@3CC4@ and { ν t } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaadaGadaqaai abe27aUnaaBaaaleaacaWG0baabeaaaOGaay5Eaiaaw2haaaaa@3D7D@ are obtained from tables.

We are now in conditions of computing the two above mentioned factors of the likelihood function. Let Ω={ ( x 1 ,, x n ): x i =0,1;i=1,,n }, MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqqHPoWvcq GH9aqpdaGadaqaamaabmaabaGaamiEamaaBaaaleaacaaIXaaabeaa kiaaiYcacqWIMaYscaaISaGaamiEamaaBaaaleaacaWGUbaabeaaaO GaayjkaiaawMcaaiaacQdacaWG4bWaaSbaaSqaaiaadMgaaeqaaOGa eyypa0JaaGimaiaacYcacaaIXaGaai4oaiaadMgacqGH9aqpcaaIXa GaaiilaiablAciljaaiYcacaWGUbaacaGL7bGaayzFaaGaaiilaaaa @52F8@ the set of all n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGUbGaey OeI0caaa@3A45@ dimensional vectors such that each one of their elements is 0 or 1. For x=( x 1 ,, x n )Ω, MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWH4bGaey ypa0ZaaeWaaeaacaWG4bWaaSbaaSqaaiaaigdaaeqaaOGaaGilaiab lAciljaaiYcacaWG4bWaaSbaaSqaaiaad6gaaeqaaaGccaGLOaGaay zkaaGaeyicI4SaeuyQdCLaaiilaaaa@4659@ let R x ( k ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGsbWaa0 baaSqaaiaahIhaaeaadaqadaqaaiaadUgaaiaawIcacaGLPaaaaaaa aa@3CE3@ be the random variable that indicates the number of distinct people in U k S 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGvbWaaS baaSqaaiaadUgaaeqaaOGaeyOeI0Iaam4uamaaBaaaleaacaaIWaaa beaaaaa@3D10@ whose vectors of link-indicator variables are equal to x . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWH4bGaai Olaaaa@3A18@ Finally, let R k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGsbWaaS baaSqaaiaadUgaaeqaaaaa@3A58@ be the random variable that indicates the number of distinct people in U k S 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGvbWaaS baaSqaaiaadUgaaeqaaOGaeyOeI0Iaam4uamaaBaaaleaacaaIWaaa beaaaaa@3D10@ that are linked to at least one cluster A i S A . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGbbWaaS baaSqaaiaadMgaaeqaaOGaeyicI4Saam4uamaaBaaaleaacaWGbbaa beaakiaac6caaaa@3E59@ Notice that R k = xΩ{0} R x (k) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGsbWaaS baaSqaaiaadUgaaeqaaOGaeyypa0ZaaabeaeaacaWGsbWaa0baaSqa aiaahIhaaeaacaGGOaacbiGaa83AaiaacMcaaaaabaGaaCiEaiabgI GiolabfM6axjabgkHiTiaabUhacaWHWaGaaeyFaaqab0GaeyyeIuoa kiaacYcaaaa@4A06@ where 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWHWaaaaa@391E@ denotes the n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGUbGaey OeI0caaa@3A45@ dimensional vector of zeros.

Because of the assumptions we made about the distributions of the variables X ij (k) s, MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGybWaa0 baaSqaaiaadMgacaWGQbaabaGaaiikaiaadUgacaGGPaaaaGqaaOGa a8xgGiaabohacaGGSaaaaa@4008@ we have that the conditional joint probability distribution of the variables { R x (1) } xΩ{ 0 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaadaGadaqaai aadkfadaqhaaWcbaGaaCiEaaqaaiaacIcacaaIXaGaaiykaaaaaOGa ay5Eaiaaw2haamaaBaaaleaacaWH4bGaeyicI4SaeuyQdCLaeyOeI0 YaaiWaaeaacaWHWaaacaGL7bGaayzFaaaabeaaaaa@46CF@ and τ 1 m R 1 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDda WgaaWcbaGaaGymaaqabaGccqGHsislcaWGTbGaeyOeI0IaamOuamaa BaaaleaacaaIXaaabeaakiaacYcaaaa@405F@ given that { M i = m i } i=1 n , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaadaGadaqaai aad2eadaWgaaWcbaGaamyAaaqabaGccqGH9aqpcaWGTbWaaSbaaSqa aiaadMgaaeqaaaGccaGL7bGaayzFaaWaa0baaSqaaiaadMgacqGH9a qpcaaIXaaabaGaamOBaaaakiaacYcaaaa@4431@ is a multinomial distribution with parameter of size τ 1 m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDda WgaaWcbaGaaGymaaqabaGccqGHsislcaWGTbaaaa@3CFA@ and probabilities { π x ( 1 ) ( α 1 , σ 1 ) } x Ω { 0 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaadaGadaqaai abec8aWnaaDaaaleaacaWH4baabaWaaeWaaeaacaaIXaaacaGLOaGa ayzkaaaaaOWaaeWaaeaacaWHXoWaaSbaaSqaaiaaigdaaeqaaOGaaG ilaiabeo8aZnaaBaaaleaacaaIXaaabeaaaOGaayjkaiaawMcaaaGa ay5Eaiaaw2haamaaBaaaleaacaWH4bGaeyicI4SaeuyQdCLaeyOeI0 YaaiWaaeaacaWHWaaacaGL7bGaayzFaaaabeaaaaa@4F06@ and π 0 ( 1 ) ( α 1 , σ 1 ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHapaCda qhaaWcbaGaaCimaaqaamaabmaabaGaaGymaaGaayjkaiaawMcaaaaa kmaabmaabaGaaCySdmaaBaaaleaacaaIXaaabeaakiaaiYcacqaHdp WCdaWgaaWcbaGaaGymaaqabaaakiaawIcacaGLPaaacaGGSaaaaa@4527@ and that of the variables { R x ( 2 ) } x Ω { 0 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaadaGadaqaai aadkfadaqhaaWcbaGaaCiEaaqaamaabmaabaGaaGOmaaGaayjkaiaa wMcaaaaaaOGaay5Eaiaaw2haamaaBaaaleaacaWH4bGaeyicI4Saeu yQdCLaeyOeI0YaaiWaaeaacaWHWaaacaGL7bGaayzFaaaabeaaaaa@4700@ and τ 2 R 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDda WgaaWcbaGaaGOmaaqabaGccqGHsislcaWGsbWaaSbaaSqaaiaaikda aeqaaaaa@3DC8@ is a multinomial distribution with parameter of size τ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDda WgaaWcbaGaaGOmaaqabaaaaa@3B12@ and probabilities { π x ( 2 ) ( α 2 , σ 2 ) } x Ω { 0 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaadaGadaqaai abec8aWnaaDaaaleaacaWH4baabaWaaeWaaeaacaaIYaaacaGLOaGa ayzkaaaaaOWaaeWaaeaacaWHXoWaaSbaaSqaaiaaikdaaeqaaOGaaG ilaiabeo8aZnaaBaaaleaacaaIYaaabeaaaOGaayjkaiaawMcaaaGa ay5Eaiaaw2haamaaBaaaleaacaWH4bGaeyicI4SaeuyQdCLaeyOeI0 YaaiWaaeaacaWHWaaacaGL7bGaayzFaaaabeaaaaa@4F09@ and π 0 ( 2 ) ( α 2 , σ 2 ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHapaCda qhaaWcbaGaaCimaaqaamaabmaabaGaaGOmaaGaayjkaiaawMcaaaaa kmaabmaabaGaaCySdmaaBaaaleaacaaIYaaabeaakiaaiYcacqaHdp WCdaWgaaWcbaGaaGOmaaqabaaakiaawIcacaGLPaaacaGGUaaaaa@452C@

Therefore, the factors associated with the probabilities of the configurations of links between the people in U k S 0 ,k=1,2, MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGvbWaaS baaSqaaiaadUgaaeqaaOGaeyOeI0Iaam4uamaaBaaaleaacaaIWaaa beaakiaacYcacaWGRbGaeyypa0JaaGymaiaacYcacaaIYaGaaiilaa aa@4297@ and the clusters A i S A MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGbbWaaS baaSqaaiaadMgaaeqaaOGaeyicI4Saam4uamaaBaaaleaacaWGbbaa beaaaaa@3D9D@ are

L 1 ( τ 1 , α 1 , σ 1 ) ( τ 1 m )! ( τ 1 m r 1 )! xΩ{ 0 } [ π ˜ x ( 1 ) ( α 1 , σ 1 ) ] r x ( 1 ) [ π ˜ 0 ( 1 ) ( α 1 , σ 1 ) ] τ 1 m r 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGmbWaaS baaSqaaiaaigdaaeqaaOWaaeWaaeaacqaHepaDdaWgaaWcbaGaaGym aaqabaGccaaISaGaaCySdmaaBaaaleaacaaIXaaabeaakiaaiYcacq aHdpWCdaWgaaWcbaGaaGymaaqabaaakiaawIcacaGLPaaacqGHDisT daWcaaqaamaabmaabaGaeqiXdq3aaSbaaSqaaiaaigdaaeqaaOGaey OeI0IaamyBaaGaayjkaiaawMcaaiaacgcaaeaadaqadaqaaiabes8a 0naaBaaaleaacaaIXaaabeaakiabgkHiTiaad2gacqGHsislcaWGYb WaaSbaaSqaaiaaigdaaeqaaaGccaGLOaGaayzkaaGaaiyiaaaadaqe qbqabSqaaiaahIhacqGHiiIZcqqHPoWvcqGHsisldaGadaqaaiaahc daaiaawUhacaGL9baaaeqaniabg+GivdGcdaWadaqaaiqbec8aWzaa iaWaa0baaSqaaiaahIhaaeaadaqadaqaaiaaigdaaiaawIcacaGLPa aaaaGcdaqadaqaaiaahg7adaWgaaWcbaGaaGymaaqabaGccaaISaGa eq4Wdm3aaSbaaSqaaiaaigdaaeqaaaGccaGLOaGaayzkaaaacaGLBb GaayzxaaWaaWbaaSqabeaacaWGYbWaa0baaWqaaiaahIhaaeaadaqa daqaaiaaigdaaiaawIcacaGLPaaaaaaaaOWaamWaaeaacuaHapaCga acamaaDaaaleaacaWHWaaabaWaaeWaaeaacaaIXaaacaGLOaGaayzk aaaaaOWaaeWaaeaacaWHXoWaaSbaaSqaaiaaigdaaeqaaOGaaGilai abeo8aZnaaBaaaleaacaaIXaaabeaaaOGaayjkaiaawMcaaaGaay5w aiaaw2faamaaCaaaleqabaGaeqiXdq3aaSbaaWqaaiaaigdaaeqaaS GaeyOeI0IaamyBaiabgkHiTiaadkhadaWgaaadbaGaaGymaaqabaaa aaaa@8944@

and

L 2 ( τ 2 , α 2 , σ 2 ) τ 2 ! ( τ 2 r 2 )! xΩ{ 0 } [ π ˜ x ( 2 ) ( α 2 , σ 2 ) ] r x ( 2 ) [ π ˜ 0 ( 2 ) ( α 2 , σ 2 ) ] τ 2 r 2 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGmbWaaS baaSqaaiaaikdaaeqaaOWaaeWaaeaacqaHepaDdaWgaaWcbaGaaGOm aaqabaGccaaISaGaaCySdmaaBaaaleaacaaIYaaabeaakiaaiYcacq aHdpWCdaWgaaWcbaGaaGOmaaqabaaakiaawIcacaGLPaaacqGHDisT daWcaaqaaiabes8a0naaBaaaleaacaaIYaaabeaakiaacgcaaeaada qadaqaaiabes8a0naaBaaaleaacaaIYaaabeaakiabgkHiTiaadkha daWgaaWcbaGaaGOmaaqabaaakiaawIcacaGLPaaacaGGHaaaamaara fabeWcbaGaaCiEaiabgIGiolabfM6axjabgkHiTmaacmaabaGaaCim aaGaay5Eaiaaw2haaaqab0Gaey4dIunakmaadmaabaGafqiWdaNbaG aadaqhaaWcbaGaaCiEaaqaamaabmaabaGaaGOmaaGaayjkaiaawMca aaaakmaabmaabaGaaCySdmaaBaaaleaacaaIYaaabeaakiaaiYcacq aHdpWCdaWgaaWcbaGaaGOmaaqabaaakiaawIcacaGLPaaaaiaawUfa caGLDbaadaahaaWcbeqaaiaadkhadaqhaaadbaGaaCiEaaqaamaabm aabaGaaGOmaaGaayjkaiaawMcaaaaaaaGcdaWadaqaaiqbec8aWzaa iaWaa0baaSqaaiaahcdaaeaadaqadaqaaiaaikdaaiaawIcacaGLPa aaaaGcdaqadaqaaiaahg7adaWgaaWcbaGaaGOmaaqabaGccaaISaGa eq4Wdm3aaSbaaSqaaiaaikdaaeqaaaGccaGLOaGaayzkaaaacaGLBb GaayzxaaWaaWbaaSqabeaacqaHepaDdaWgaaadbaGaaGOmaaqabaWc cqGHsislcaWGYbWaaSbaaWqaaiaaikdaaeqaaaaakiaai6caaaa@82F0@

The last component of the likelihood function is associated with the conditional probability, given S A , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGtbWaaS baaSqaaiaadgeaaeqaaOGaaiilaaaa@3AE9@ of the configuration of links between the people in S 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGtbWaaS baaSqaaiaaicdaaeqaaaaa@3A23@ and the clusters A i S A . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGbbWaaS baaSqaaiaadMgaaeqaaOGaeyicI4Saam4uamaaBaaaleaacaWGbbaa beaakiaac6caaaa@3E59@ To derive this factor firstly observe that by the definition of the indicator variables X i j ( k ) s , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGybWaa0 baaSqaaiaadMgacaWGQbaabaWaaeWaaeaacaWGRbaacaGLOaGaayzk aaaaaGqaaOGaa8xgGiaabohacaGGSaaaaa@4038@ the i th MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGPbWaaW baaSqabeaacaqG0bGaaeiAaaaaaaa@3B62@ element of the vector of link-indicator variables associated with a person in A i S A MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGbbWaaS baaSqaaiaadMgaaeqaaOGaeyicI4Saam4uamaaBaaaleaacaWGbbaa beaaaaa@3D9D@ is equal to zero. Thus, let Ω i ={ ( x 1 ,, x i1 , x i+1 ,, x n ): x j =0,1, ji, j=1,,n }, MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqqHPoWvda WgaaWcbaGaeyOeI0IaamyAaaqabaGccqGH9aqpdaGadaqaamaabmaa baGaamiEamaaBaaaleaacaaIXaaabeaakiaaiYcacqWIMaYscaaISa GaamiEamaaBaaaleaacaWGPbGaeyOeI0IaaGymaaqabaGccaaISaGa amiEamaaBaaaleaacaWGPbGaey4kaSIaaGymaaqabaGccaaISaGaeS OjGSKaaGilaiaadIhadaWgaaWcbaGaamOBaaqabaaakiaawIcacaGL PaaacaGG6aGaamiEamaaBaaaleaacaWGQbaabeaakiabg2da9iaaic dacaGGSaGaaGymaiaacYcacaqGGaGaamOAaiabgcMi5kaadMgacaaI SaGaaeiiaiaadQgacqGH9aqpcaaIXaGaaiilaiablAciljaaiYcaca WGUbaacaGL7bGaayzFaaGaaiilaaaa@6567@ that is, the set of all ( n 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaadaqadaqaai aad6gacqGHsislcaaIXaaacaGLOaGaayzkaaGaeyOeI0caaa@3D76@ dimensional vectors obtained from the vectors in Ω MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqqHPoWvaa a@39F3@ by omitting their i th MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGPbWaaW baaSqabeaacaqG0bGaaeiAaaaaaaa@3B62@ coordinate. For x=( x 1 ,, x i=1 , x i+1 ,, x n ) Ω i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWH4bGaey ypa0ZaaeWaaeaacaWG4bWaaSbaaSqaaiaaigdaaeqaaOGaaGilaiab lAciljaaiYcacaWG4bWaaSbaaSqaaiaadMgacqGH9aqpcaaIXaaabe aakiaaiYcacaWG4bWaaSbaaSqaaiaadMgacqGHRaWkcaaIXaaabeaa kiaaiYcacqWIMaYscaaISaGaamiEamaaBaaaleaacaWGUbaabeaaaO GaayjkaiaawMcaaiabgIGiolabfM6axnaaBaaaleaacqGHsislcaWG Pbaabeaaaaa@5294@ let R x ( A i ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGsbWaa0 baaSqaaiaahIhaaeaadaqadaqaaiaadgeadaWgaaadbaGaamyAaaqa baaaliaawIcacaGLPaaaaaaaaa@3DDF@ be the random variable that indicates the number of distinct people in A i S A MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGbbWaaS baaSqaaiaadMgaaeqaaOGaeyicI4Saam4uamaaBaaaleaacaWGbbaa beaaaaa@3D9D@ such that their vectors of link-indicator variables, when the i th MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGPbWaaW baaSqabeaacaqG0bGaaeiAaaaaaaa@3B62@ coordinate is omitted, equal x . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWH4bGaai Olaaaa@3A18@ Finally, let R ( A i ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGsbWaaW baaSqabeaadaqadaqaaiaadgeadaWgaaadbaGaamyAaaqabaaaliaa wIcacaGLPaaaaaaaaa@3CDE@ be the random variable that indicates the number of distinct people in A i S A MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGbbWaaS baaSqaaiaadMgaaeqaaOGaeyicI4Saam4uamaaBaaaleaacaWGbbaa beaaaaa@3D9D@ that are linked to at least one site A j S A , j i . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGbbWaaS baaSqaaiaadQgaaeqaaOGaeyicI4Saam4uamaaBaaaleaacaWGbbaa beaakiaacYcacaWGQbGaeyiyIKRaamyAaiaac6caaaa@42AE@ Notice that R ( A i ) = x Ω i { 0 } R x ( A i ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGsbWaaW baaSqabeaadaqadaqaaiaadgeadaWgaaadbaGaamyAaaqabaaaliaa wIcacaGLPaaaaaGccqGH9aqpdaaeqaqaaiaadkfadaqhaaWcbaGaaC iEaaqaamaabmaabaGaamyqamaaBaaameaacaWGPbaabeaaaSGaayjk aiaawMcaaaaaaeaacaWH4bGaeyicI4SaeuyQdC1aaSbaaWqaaiabgk HiTiaadMgaaeqaaSGaeyOeI0YaaiWaaeaacaWHWaaacaGL7bGaayzF aaaabeqdcqGHris5aOGaaiilaaaa@4FF7@ where 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWHWaaaaa@391E@ denotes the ( n 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaadaqadaqaai aad6gacqGHsislcaaIXaaacaGLOaGaayzkaaGaeyOeI0caaa@3D76@ dimensional vector of zeros. Then, as in the previous cases, the conditional joint probability distribution of the variables { R x ( A i ) } x Ω i { 0 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaadaGadaqaai aadkfadaqhaaWcbaGaaCiEaaqaamaabmaabaGaamyqamaaBaaameaa caWGPbaabeaaaSGaayjkaiaawMcaaaaaaOGaay5Eaiaaw2haamaaBa aaleaacaWH4bGaeyicI4SaeuyQdC1aaSbaaWqaaiabgkHiTiaadMga aeqaaSGaeyOeI0YaaiWaaeaacaWHWaaacaGL7bGaayzFaaaabeaaaa a@4A43@ and m i R ( A i ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGTbWaaS baaSqaaiaadMgaaeqaaOGaeyOeI0IaamOuamaaCaaaleqabaWaaeWa aeaacaWGbbWaaSbaaWqaaiaadMgaaeqaaaWccaGLOaGaayzkaaaaaO Gaaiilaaaa@409B@ given that { M i = m i } i=1 n , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaadaGadaqaai aad2eadaWgaaWcbaGaamyAaaqabaGccqGH9aqpcaWGTbWaaSbaaSqa aiaadMgaaeqaaaGccaGL7bGaayzFaaWaa0baaSqaaiaadMgacqGH9a qpcaaIXaaabaGaamOBaaaakiaacYcaaaa@4431@ is a multinomial distribution with parameter of size m i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGTbWaaS baaSqaaiaadMgaaeqaaaaa@3A71@ and probabilities { π x ( A i ) ( α 1 ( i ) , σ 1 ) } x Ω i { 0 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaadaGadaqaai abec8aWnaaDaaaleaacaWH4baabaWaaeWaaeaacaWGbbWaaSbaaWqa aiaadMgaaeqaaaWccaGLOaGaayzkaaaaaOWaaeWaaeaacaWHXoWaa0 baaSqaaiaaigdaaeaadaqadaqaaiabgkHiTiaadMgaaiaawIcacaGL PaaaaaGccaaISaGaeq4Wdm3aaSbaaSqaaiaaigdaaeqaaaGccaGLOa GaayzkaaaacaGL7bGaayzFaaWaaSbaaSqaaiaahIhacqGHiiIZcqqH PoWvdaWgaaadbaGaeyOeI0IaamyAaaqabaWccqGHsisldaGadaqaai aahcdaaiaawUhacaGL9baaaeqaaaaa@55AF@ and π 0 ( A i ) ( α 1 ( i ) , σ 1 ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHapaCda qhaaWcbaGaaCimaaqaamaabmaabaGaamyqamaaBaaameaacaWGPbaa beaaaSGaayjkaiaawMcaaaaakmaabmaabaGaaCySdmaaDaaaleaaca aIXaaabaWaaeWaaeaacqGHsislcaWGPbaacaGLOaGaayzkaaaaaOGa aGilaiabeo8aZnaaBaaaleaacaaIXaaabeaaaOGaayjkaiaawMcaai aacYcaaaa@49BD@ where α 1 ( i ) =( α 1 ( 1 ) ,, α i1 ( 1 ) , α i+1 ( 1 ) ,, α n ( 1 ) ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWHXoWaa0 baaSqaaiaaigdaaeaadaqadaqaaiabgkHiTiaadMgaaiaawIcacaGL PaaaaaGccqGH9aqpdaqadaqaaiabeg7aHnaaDaaaleaacaaIXaaaba WaaeWaaeaacaaIXaaacaGLOaGaayzkaaaaaOGaaGilaiablAciljaa iYcacqaHXoqydaqhaaWcbaGaamyAaiabgkHiTiaaigdaaeaadaqada qaaiaaigdaaiaawIcacaGLPaaaaaGccaaISaGaeqySde2aa0baaSqa aiaadMgacqGHRaWkcaaIXaaabaWaaeWaaeaacaaIXaaacaGLOaGaay zkaaaaaOGaaGilaiablAciljaaiYcacqaHXoqydaqhaaWcbaGaamOB aaqaamaabmaabaGaaGymaaGaayjkaiaawMcaaaaaaOGaayjkaiaawM caaaaa@5D90@ and

π x ( A i ) ( α 1 ( i ) , σ 1 )= ji n exp[ x j ( α j ( 1 ) + σ 1 z ) ] 1+exp( α j ( 1 ) + σ 1 z ) ϕ( z )dz. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHapaCda qhaaWcbaGaaCiEaaqaamaabmaabaGaamyqamaaBaaameaacaWGPbaa beaaaSGaayjkaiaawMcaaaaakmaabmaabaGaaCySdmaaDaaaleaaca aIXaaabaWaaeWaaeaacqGHsislcaWGPbaacaGLOaGaayzkaaaaaOGa aGilaiabeo8aZnaaBaaaleaacaaIXaaabeaaaOGaayjkaiaawMcaai abg2da9maapeaabeWcbeqab0Gaey4kIipakmaarahabeWcbaGaamOA aiabgcMi5kaadMgaaeaacaWGUbaaniabg+GivdGcdaWcaaqaaiGacw gacaGG4bGaaiiCamaadmaabaGaamiEamaaBaaaleaacaWGQbaabeaa kmaabmaabaGaeqySde2aa0baaSqaaiaadQgaaeaadaqadaqaaiaaig daaiaawIcacaGLPaaaaaGccqGHRaWkcqaHdpWCdaWgaaWcbaGaaGym aaqabaGccaWG6baacaGLOaGaayzkaaaacaGLBbGaayzxaaaabaGaaG ymaiabgUcaRiGacwgacaGG4bGaaiiCamaabmaabaGaeqySde2aa0ba aSqaaiaadQgaaeaadaqadaqaaiaaigdaaiaawIcacaGLPaaaaaGccq GHRaWkcqaHdpWCdaWgaaWcbaGaaGymaaqabaGccaWG6baacaGLOaGa ayzkaaaaaiabew9aMnaabmaabaGaamOEaaGaayjkaiaawMcaaiaads gacaWG6bGaaGOlaaaa@7BE8@

Therefore, the probability of the configuration of links between the people in S 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGtbWaaS baaSqaaiaaicdaaeqaaaaa@3A23@ and the clusters A j S A MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGbbWaaS baaSqaaiaadQgaaeqaaOGaeyicI4Saam4uamaaBaaaleaacaWGbbaa beaaaaa@3D9E@ is given by the product of the previous multinomial probabilities (one for each A i S A ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGbbWaaS baaSqaaiaadMgaaeqaaOGaeyicI4Saam4uamaaBaaaleaacaWGbbaa beaakiaacMcacaGGSaaaaa@3F04@ and consequently the factor of the likelihood associated with that probability is

L 0 ( α 1 , σ 1 ) i=1 n x Ω i { 0 } [ π ˜ x ( A i ) ( α 1 ( i ) , σ 1 ) ] r x ( A i ) [ π ˜ 0 ( A i ) ( α 1 ( i ) , σ 1 ) ] m i r ( A i ) , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGmbWaaS baaSqaaiaaicdaaeqaaOWaaeWaaeaacaWHXoWaaSbaaSqaaiaaigda aeqaaOGaaGilaiabeo8aZnaaBaaaleaacaaIXaaabeaaaOGaayjkai aawMcaaiabg2Hi1oaarahabeWcbaGaamyAaiabg2da9iaaigdaaeaa caWGUbaaniabg+GivdGcdaqeqbqabSqaaiaahIhacqGHiiIZcqqHPo WvdaWgaaadbaGaeyOeI0IaamyAaaqabaWccqGHsisldaGadaqaaiaa hcdaaiaawUhacaGL9baaaeqaniabg+GivdGcdaWadaqaaiqbec8aWz aaiaWaa0baaSqaaiaahIhaaeaadaqadaqaaiaadgeadaWgaaadbaGa amyAaaqabaaaliaawIcacaGLPaaaaaGcdaqadaqaaiaahg7adaqhaa WcbaGaaGymaaqaamaabmaabaGaeyOeI0IaamyAaaGaayjkaiaawMca aaaakiaaiYcacqaHdpWCdaWgaaWcbaGaaGymaaqabaaakiaawIcaca GLPaaaaiaawUfacaGLDbaadaahaaWcbeqaaiaadkhadaqhaaadbaGa aCiEaaqaamaabmaabaGaamyqamaaBaaabaGaamyAaaqabaaacaGLOa GaayzkaaaaaaaakmaadmaabaGafqiWdaNbaGaadaqhaaWcbaGaaCim aaqaamaabmaabaGaamyqamaaBaaameaacaWGPbaabeaaaSGaayjkai aawMcaaaaakmaabmaabaGaaCySdmaaDaaaleaacaaIXaaabaWaaeWa aeaacqGHsislcaWGPbaacaGLOaGaayzkaaaaaOGaaGilaiabeo8aZn aaBaaaleaacaaIXaaabeaaaOGaayjkaiaawMcaaaGaay5waiaaw2fa amaaCaaaleqabaGaamyBamaaBaaameaacaWGPbaabeaaliabgkHiTi aadkhadaahaaadbeqaamaabmaabaGaamyqamaaBaaabaGaamyAaaqa baaacaGLOaGaayzkaaaaaaaakiaacYcaaaa@8882@

where

π ˜ x ( A i ) ( α 1 ( i ) , σ 1 )= t=1 q ji n exp[ x j ( α j ( 1 ) + σ 1 z t ) ] 1+exp( α j ( 1 ) + σ 1 z t ) ν t ,(3.4) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHapaCga acamaaDaaaleaacaWH4baabaWaaeWaaeaacaWGbbWaaSbaaWqaaiaa dMgaaeqaaaWccaGLOaGaayzkaaaaaOWaaeWaaeaacaWHXoWaa0baaS qaaiaaigdaaeaadaqadaqaaiabgkHiTiaadMgaaiaawIcacaGLPaaa aaGccaaISaGaeq4Wdm3aaSbaaSqaaiaaigdaaeqaaaGccaGLOaGaay zkaaGaeyypa0ZaaabCaeqaleaacaWG0bGaeyypa0JaaGymaaqaaiaa dghaa0GaeyyeIuoakmaarahabeWcbaGaamOAaiabgcMi5kaadMgaae aacaWGUbaaniabg+GivdGcdaWcaaqaaiGacwgacaGG4bGaaiiCamaa dmaabaGaamiEamaaBaaaleaacaWGQbaabeaakmaabmaabaGaeqySde 2aa0baaSqaaiaadQgaaeaadaqadaqaaiaaigdaaiaawIcacaGLPaaa aaGccqGHRaWkcqaHdpWCdaWgaaWcbaGaaGymaaqabaGccaWG6bWaaS baaSqaaiaadshaaeqaaaGccaGLOaGaayzkaaaacaGLBbGaayzxaaaa baGaaGymaiabgUcaRiGacwgacaGG4bGaaiiCamaabmaabaGaeqySde 2aa0baaSqaaiaadQgaaeaadaqadaqaaiaaigdaaiaawIcacaGLPaaa aaGccqGHRaWkcqaHdpWCdaWgaaWcbaGaaGymaaqabaGccaWG6bWaaS baaSqaaiaadshaaeqaaaGccaGLOaGaayzkaaaaaiabe27aUnaaBaaa leaacaWG0baabeaakiaaiYcacaaMf8UaaGzbVlaaywW7caaMf8UaaG zbVlaacIcacaaIZaGaaiOlaiaaisdacaGGPaaaaa@8A45@

is the Gaussian quadrature approximation to the probability π x ( A i ) ( α 1 ( i ) , σ 1 ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHapaCda qhaaWcbaGaaCiEaaqaamaabmaabaGaamyqamaaBaaameaacaWGPbaa beaaaSGaayjkaiaawMcaaaaakmaabmaabaGaaCySdmaaDaaaleaaca aIXaaabaWaaeWaaeaacqGHsislcaWGPbaacaGLOaGaayzkaaaaaOGa aGilaiabeo8aZnaaBaaaleaacaaIXaaabeaaaOGaayjkaiaawMcaai aac6caaaa@4A07@

From the previous results we have that the likelihood function is given by

L( τ 1 , τ 2 , α 1 , α 2 , σ 1 , σ 2 )= L ( 1 ) ( τ 1 , α 1 , σ 1 ) L ( 2 ) ( τ 2 , α 2 , σ 2 ), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGmbWaae WaaeaacqaHepaDdaWgaaWcbaGaaGymaaqabaGccaaISaGaeqiXdq3a aSbaaSqaaiaaikdaaeqaaOGaaGilaiabeg7aHnaaBaaaleaacaaIXa aabeaakiaaiYcacqaHXoqydaWgaaWcbaGaaGOmaaqabaGccaaISaGa eq4Wdm3aaSbaaSqaaiaaigdaaeqaaOGaaGilaiabeo8aZnaaBaaale aacaaIYaaabeaaaOGaayjkaiaawMcaaiabg2da9iaadYeadaWgaaWc baWaaeWaaeaacaaIXaaacaGLOaGaayzkaaaabeaakmaabmaabaGaeq iXdq3aaSbaaSqaaiaaigdaaeqaaOGaaGilaiaahg7adaWgaaWcbaGa aGymaaqabaGccaaISaGaeq4Wdm3aaSbaaSqaaiaaigdaaeqaaaGcca GLOaGaayzkaaGaamitamaaBaaaleaadaqadaqaaiaaikdaaiaawIca caGLPaaaaeqaaOWaaeWaaeaacqaHepaDdaWgaaWcbaGaaGOmaaqaba GccaaISaGaaCySdmaaBaaaleaacaaIYaaabeaakiaaiYcacqaHdpWC daWgaaWcbaGaaGOmaaqabaaakiaawIcacaGLPaaacaaISaaaaa@6BB3@

where

L ( 1 ) ( τ 1 , α 1 , σ 1 )= L MULT ( τ 1 ) L 1 ( τ 1 , α 1 , σ 1 ) L 0 ( α 1 , σ 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGmbWaaS baaSqaamaabmaabaGaaGymaaGaayjkaiaawMcaaaqabaGcdaqadaqa aiabes8a0naaBaaaleaacaaIXaaabeaakiaaiYcacaWHXoWaaSbaaS qaaiaaigdaaeqaaOGaaGilaiabeo8aZnaaBaaaleaacaaIXaaabeaa aOGaayjkaiaawMcaaiabg2da9iaadYeadaWgaaWcbaGaaeytaiaabw facaqGmbGaaeivaaqabaGcdaqadaqaaiabes8a0naaBaaaleaacaaI XaaabeaaaOGaayjkaiaawMcaaiaadYeadaWgaaWcbaGaaGymaaqaba Gcdaqadaqaaiabes8a0naaBaaaleaacaaIXaaabeaakiaaiYcacaWH XoWaaSbaaSqaaiaaigdaaeqaaOGaaGilaiabeo8aZnaaBaaaleaaca aIXaaabeaaaOGaayjkaiaawMcaaiaadYeadaWgaaWcbaGaaGimaaqa baGcdaqadaqaaiaahg7adaWgaaWcbaGaaGymaaqabaGccaaISaGaeq 4Wdm3aaSbaaSqaaiaaigdaaeqaaaGccaGLOaGaayzkaaaaaa@6507@

and

L ( 2 ) ( τ 2 , α 2 , σ 2 )= L 2 ( τ 2 , α 2 , σ 2 ). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGmbWaaS baaSqaamaabmaabaGaaGOmaaGaayjkaiaawMcaaaqabaGcdaqadaqa aiabes8a0naaBaaaleaacaaIYaaabeaakiaaiYcacaWHXoWaaSbaaS qaaiaaikdaaeqaaOGaaGilaiabeo8aZnaaBaaaleaacaaIYaaabeaa aOGaayjkaiaawMcaaiabg2da9iaadYeadaWgaaWcbaGaaGOmaaqaba Gcdaqadaqaaiabes8a0naaBaaaleaacaaIYaaabeaakiaaiYcacaWH XoWaaSbaaSqaaiaaikdaaeqaaOGaaGilaiabeo8aZnaaBaaaleaaca aIYaaabeaaaOGaayjkaiaawMcaaiaai6caaaa@5451@

In the comments at the end of Subsection 3.1 was indicated that the likelihood function depends on the M i s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGnbWaaS baaSqaaiaadMgaaeqaaGqaaOGaa8xgGiaabohaaaa@3C14@ basically through their sum M . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGnbGaai Olaaaa@39E9@ This can be seen by noting that only the factor L 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGmbWaaS baaSqaaiaaicdaaeqaaaaa@3A1C@ depends directly through the M i s . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGnbWaaS baaSqaaiaadMgaaeqaaGqaaOGaa8xgGiaabohacaqGUaaaaa@3CC5@ The factors L MULT MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGmbWaaS baaSqaaiaab2eacaqGvbGaaeitaiaabsfaaeqaaaaa@3CB0@ and L 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGmbWaaS baaSqaaiaaigdaaeqaaaaa@3A1D@ depend on the M i s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGnbWaaS baaSqaaiaadMgaaeqaaGqaaOGaa8xgGiaabohaaaa@3C14@ through M , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGnbGaai ilaaaa@39E7@ whereas the factor L ( 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGmbWaaS baaSqaamaabmaabaGaaGOmaaGaayjkaiaawMcaaaqabaaaaa@3BA7@ does not depend on the M i s . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGnbWaaS baaSqaaiaadMgaaeqaaGqaaOGaa8xgGiaabohacaGGUaaaaa@3CC6@

3.3 Unconditional maximum likelihood estimators

Numerical maximization of the likelihood function L ( τ 1 , τ 2 , α 1 , α 2 , σ 1 , σ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGmbWaae WaaeaacqaHepaDdaWgaaWcbaGaaGymaaqabaGccaaISaGaeqiXdq3a aSbaaSqaaiaaikdaaeqaaOGaaGilaiabeg7aHnaaBaaaleaacaaIXa aabeaakiaaiYcacqaHXoqydaWgaaWcbaGaaGOmaaqabaGccaaISaGa eq4Wdm3aaSbaaSqaaiaaigdaaeqaaOGaaGilaiabeo8aZnaaBaaale aacaaIYaaabeaaaOGaayjkaiaawMcaaaaa@4E44@ with respect to the parameters yields the ordinary or unconditional maximum likelihood estimators (UMLEs) τ ^ k ( U ) , α ^ k ( U ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga qcamaaDaaaleaacaWGRbaabaWaaeWaaeaacaWGvbaacaGLOaGaayzk aaaaaOGaaiilaiqbeg7aHzaajaWaa0baaSqaaiaadUgaaeaadaqada qaaiaadwfaaiaawIcacaGLPaaaaaaaaa@43A3@ and σ ^ k ( U ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHdpWCga qcamaaDaaaleaacaWGRbaabaWaaeWaaeaacaWGvbaacaGLOaGaayzk aaaaaaaa@3DB8@ of τ k , α k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDda WgaaWcbaGaam4AaaqabaGccaGGSaGaeqySde2aaSbaaSqaaiaadUga aeqaaaaa@3EBB@ and σ k ,k=1,2. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHdpWCda WgaaWcbaGaam4AaaqabaGccaGGSaGaam4Aaiabg2da9iaaigdacaGG SaGaaGOmaiaac6caaaa@40CD@ Consequently the UMLE of τ= τ 1 + τ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDcq GH9aqpcqaHepaDdaWgaaWcbaGaaGymaaqabaGccqGHRaWkcqaHepaD daWgaaWcbaGaaGOmaaqabaaaaa@4175@ is τ ^ ( U ) = τ ^ 1 ( U ) + τ ^ 2 ( U ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga qcamaaCaaaleqabaWaaeWaaeaacaWGvbaacaGLOaGaayzkaaaaaOGa eyypa0JafqiXdqNbaKaadaqhaaWcbaGaaGymaaqaamaabmaabaGaam yvaaGaayjkaiaawMcaaaaakiabgUcaRiqbes8a0zaajaWaa0baaSqa aiaaikdaaeaadaqadaqaaiaadwfaaiaawIcacaGLPaaaaaGccaGGUa aaaa@49C3@ Closed forms for the UMLEs do not exist; however, using the asymptotic approximation ln( τ k ! )/ τ k ln( τ k ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaadaWcgaqaai abgkGi2kGacYgacaGGUbWaaeWaaeaacqaHepaDdaWgaaWcbaGaam4A aaqabaGccaGGHaaacaGLOaGaayzkaaaabaGaeyOaIyRaeqiXdq3aaS baaSqaaiaadUgaaeqaaOGaeyisISRaciiBaiaac6gadaqadaqaaiab es8a0naaBaaaleaacaWGRbaabeaaaOGaayjkaiaawMcaaaaacaGGSa aaaa@4DE8@ we get the following approximations to τ ^ 1 ( U ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga qcamaaDaaaleaacaaIXaaabaWaaeWaaeaacaWGvbaacaGLOaGaayzk aaaaaaaa@3D85@ and τ ^ 2 ( U ) : MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga qcamaaDaaaleaacaaIYaaabaWaaeWaaeaacaWGvbaacaGLOaGaayzk aaaaaOGaaiOoaaaa@3E4E@

τ ^ 1 ( U ) = M+ R 1 1( 1n/N ) π ˜ 0 ( 1 ) ( α ^ 1 ( U ) , σ ^ 1 ( U ) )    and    τ ^ 2 ( U ) = R 2 1 π ˜ 0 ( 2 ) ( α ^ 2 ( U ) , σ ^ 2 ( U ) ) .(3.5) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga qcamaaDaaaleaacaaIXaaabaWaaeWaaeaacaWGvbaacaGLOaGaayzk aaaaaOGaeyypa0ZaaSaaaeaacaWGnbGaey4kaSIaamOuamaaBaaale aacaaIXaaabeaaaOqaaiaaigdacqGHsisldaqadaqaamaalyaabaGa aGymaiabgkHiTiaad6gaaeaacaWGobaaaaGaayjkaiaawMcaaiqbec 8aWzaaiaWaa0baaSqaaiaahcdaaeaadaqadaqaaiaaigdaaiaawIca caGLPaaaaaGcdaqadaqaaiqahg7agaqcamaaDaaaleaacaaIXaaaba WaaeWaaeaacaWGvbaacaGLOaGaayzkaaaaaOGaaGilaiqbeo8aZzaa jaWaa0baaSqaaiaaigdaaeaadaqadaqaaiaadwfaaiaawIcacaGLPa aaaaaakiaawIcacaGLPaaaaaGaaeiiaiaabccacaqGGaGaaeyyaiaa b6gacaqGKbGaaeiiaiaabccacaqGGaGafqiXdqNbaKaadaqhaaWcba GaaGOmaaqaamaabmaabaGaamyvaaGaayjkaiaawMcaaaaakiabg2da 9maalaaabaGaamOuamaaBaaaleaacaaIYaaabeaaaOqaaiaaigdacq GHsislcuaHapaCgaacamaaDaaaleaacaWHWaaabaWaaeWaaeaacaaI YaaacaGLOaGaayzkaaaaaOWaaeWaaeaaceWHXoGbaKaadaqhaaWcba GaaGOmaaqaamaabmaabaGaamyvaaGaayjkaiaawMcaaaaakiaaiYca cuaHdpWCgaqcamaaDaaaleaacaaIYaaabaWaaeWaaeaacaWGvbaaca GLOaGaayzkaaaaaaGccaGLOaGaayzkaaaaaiaai6cacaaMf8UaaGzb VlaaywW7caaMf8UaaiikaiaaiodacaGGUaGaaGynaiaacMcaaaa@85A1@

Notice that these expressions are not closed forms since α ^ k ( U ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHXoqyga qcamaaDaaaleaacaWGRbaabaWaaeWaaeaacaWGvbaacaGLOaGaayzk aaaaaaaa@3D94@ and σ ^ k ( U ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHdpWCga qcamaaDaaaleaacaWGRbaabaWaaeWaaeaacaWGvbaacaGLOaGaayzk aaaaaaaa@3DB8@ depend on τ ^ k ( U ) , k=1,2. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga qcamaaDaaaleaacaWGRbaabaWaaeWaaeaacaWGvbaacaGLOaGaayzk aaaaaOGaaiilaiaabccacaWGRbGaeyypa0JaaGymaiaacYcacaaIYa GaaiOlaaaa@43E6@ Nevertheless, these expressions are useful to get formulae for the asymptotic variances of τ ^ 1 ( U ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga qcamaaDaaaleaacaaIXaaabaWaaeWaaeaacaWGvbaacaGLOaGaayzk aaaaaaaa@3D85@ and τ ^ 2 ( U ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga qcamaaDaaaleaacaaIYaaabaWaaeWaaeaacaWGvbaacaGLOaGaayzk aaaaaOGaaiOlaaaa@3E42@

3.4 Conditional maximum likelihood estimators

Another way to get MLEs of τ k , α k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDda WgaaWcbaGaam4AaaqabaGccaGGSaGaeqySde2aaSbaaSqaaiaadUga aeqaaaaa@3EBB@ and σ k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHdpWCda WgaaWcbaGaam4Aaaqabaaaaa@3B44@ is by using Sanathanan’s (1972) approach, which yields conditional maximum likelihood estimators (CMLEs). These estimators are numerically simpler to compute than UMLEs. In addition, if covariates were used in the model for the link-probability p i j ( k ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGWbWaa0 baaSqaaiaadMgacaWGQbaabaWaaeWaaeaacaWGRbaacaGLOaGaayzk aaaaaOGaaiilaaaa@3E97@ this approach could still be used to get estimators of τ k , α k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDda WgaaWcbaGaam4AaaqabaGccaGGSaGaeqySde2aaSbaaSqaaiaadUga aeqaaaaa@3EBB@ and σ k , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHdpWCda WgaaWcbaGaam4AaaqabaGccaGGSaaaaa@3BFE@ whereas the unconditional likelihood approach could not since the values of the covariates associated with the non sampled elements would be unknown.

The idea in Sanathanan’s approach is to factorize the pmf of the multinomial distributions of the frequencies R x ( k ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGsbWaa0 baaSqaaiaahIhaaeaadaqadaqaaiaadUgaaiaawIcacaGLPaaaaaaa aa@3CE3@ of the different configurations of links as follows:

L 1 ( τ 1 , α 1 , σ 1 ) f( { r x ( 1 ) } xΩ{ 0 } , τ 1 m r 1 |{ m i }, τ 1 , α 1 , σ 1 ) = f( { r x ( 1 ) } xΩ{ 0 } |{ m i }, τ 1 , r 1 , α 1 , σ 1 )f( r 1 | { m i }, τ 1 , α 1 , σ 1 ) xΩ{ 0 } [ π ˜ x ( 1 ) ( α 1 , σ 1 ) 1 π ˜ 0 ( 1 ) ( α 1 , σ 1 ) ] r x ( 1 ) × ( τ 1 m )! ( τ 1 m r 1 )! [ 1 π ˜ 0 ( 1 ) ( α 1 , σ 1 ) ] r 1 [ π ˜ 0 ( 1 ) ( α 1 , σ 1 ) ] τ 1 m r 1 = L 11 ( α 1 , σ 1 ) L 12 ( τ 1 , α 1 , σ 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=eFD0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqbaeaabsWaaa aabaGaamitamaaBaaaleaacaaIXaaabeaakmaabmaabaGaeqiXdq3a aSbaaSqaaiaaigdaaeqaaOGaaGilaiaahg7adaWgaaWcbaGaaGymaa qabaGccaaISaGaeq4Wdm3aaSbaaSqaaiaaigdaaeqaaaGccaGLOaGa ayzkaaaabaGaeyyhIulabaGaamOzamaabmaabaWaaqGaaeaadaGada qaaiaadkhadaqhaaWcbaGaaCiEaaqaamaabmaabaGaaGymaaGaayjk aiaawMcaaaaaaOGaay5Eaiaaw2haamaaBaaaleaacaWH4bGaeyicI4 SaeuyQdCLaeyOeI0YaaiWaaeaacaWHWaaacaGL7bGaayzFaaaabeaa kiaaiYcacqaHepaDdaWgaaWcbaGaaGymaaqabaGccqGHsislcaWGTb GaeyOeI0IaamOCamaaBaaaleaacaaIXaaabeaaaOGaayjcSdWaaiWa aeaacaWGTbWaaSbaaSqaaiaadMgaaeqaaaGccaGL7bGaayzFaaGaaG ilaiabes8a0naaBaaaleaacaaIXaaabeaakiaaiYcacaWHXoWaaSba aSqaaiaaigdaaeqaaOGaaGilaiabeo8aZnaaBaaaleaacaaIXaaabe aaaOGaayjkaiaawMcaaaqaaaqaaiabg2da9aqaaiaadAgadaqadaqa amaaeiaabaWaaiWaaeaacaWGYbWaa0baaSqaaiaahIhaaeaadaqada qaaiaaigdaaiaawIcacaGLPaaaaaaakiaawUhacaGL9baadaWgaaWc baGaaCiEaiabgIGiolabfM6axjabgkHiTmaacmaabaGaaCimaaGaay 5Eaiaaw2haaaqabaaakiaawIa7amaacmaabaGaamyBamaaBaaaleaa caWGPbaabeaaaOGaay5Eaiaaw2haaiaaiYcacqaHepaDdaWgaaWcba GaaGymaaqabaGccaaISaGaamOCamaaBaaaleaacaaIXaaabeaakiaa iYcacaWHXoWaaSbaaSqaaiaaigdaaeqaaOGaaGilaiabeo8aZnaaBa aaleaacaaIXaaabeaaaOGaayjkaiaawMcaaiaadAgadaqadaqaaiaa dkhadaWgaaWcbaGaaGymaaqabaGcdaabbaqaamaacmaabaGaamyBam aaBaaaleaacaWGPbaabeaaaOGaay5Eaiaaw2haaiaaiYcacqaHepaD daWgaaWcbaGaaGymaaqabaGccaaISaGaaCySdmaaBaaaleaacaaIXa aabeaakiaaiYcacqaHdpWCdaWgaaWcbaGaaGymaaqabaaakiaawEa7 aaGaayjkaiaawMcaaaqaaaqaaiabg2Hi1cqaamaarafabeWcbaGaaC iEaiabgIGiolabfM6axjabgkHiTmaacmaabaGaaCimaaGaay5Eaiaa w2haaaqab0Gaey4dIunakmaadmaabaWaaSaaaeaacuaHapaCgaacam aaDaaaleaacaWH4baabaWaaeWaaeaacaaIXaaacaGLOaGaayzkaaaa aOWaaeWaaeaacaWHXoWaaSbaaSqaaiaaigdaaeqaaOGaaGilaiabeo 8aZnaaBaaaleaacaaIXaaabeaaaOGaayjkaiaawMcaaaqaaiaaigda cqGHsislcuaHapaCgaacamaaDaaaleaacaWHWaaabaWaaeWaaeaaca aIXaaacaGLOaGaayzkaaaaaOWaaeWaaeaacaWHXoWaaSbaaSqaaiaa igdaaeqaaOGaaGilaiabeo8aZnaaBaaaleaacaaIXaaabeaaaOGaay jkaiaawMcaaaaaaiaawUfacaGLDbaadaahaaWcbeqaaiaadkhadaqh aaadbaGaaCiEaaqaamaabmaabaGaaGymaaGaayjkaiaawMcaaaaaaa GccqGHxdaTdaWcaaqaamaabmaabaGaeqiXdq3aaSbaaSqaaiaaigda aeqaaOGaeyOeI0IaamyBaaGaayjkaiaawMcaaiaacgcaaeaadaqada qaaiabes8a0naaBaaaleaacaaIXaaabeaakiabgkHiTiaad2gacqGH sislcaWGYbWaaSbaaSqaaiaaigdaaeqaaaGccaGLOaGaayzkaaGaai yiaaaadaWadaqaaiaaigdacqGHsislcuaHapaCgaacamaaDaaaleaa caWHWaaabaWaaeWaaeaacaaIXaaacaGLOaGaayzkaaaaaOWaaeWaae aacaWHXoWaaSbaaSqaaiaaigdaaeqaaOGaaGilaiabeo8aZnaaBaaa leaacaaIXaaabeaaaOGaayjkaiaawMcaaaGaay5waiaaw2faamaaCa aaleqabaGaamOCamaaBaaameaacaaIXaaabeaaaaGcdaWadaqaaiqb ec8aWzaaiaWaa0baaSqaaiaahcdaaeaadaqadaqaaiaaigdaaiaawI cacaGLPaaaaaGcdaqadaqaaiaahg7adaWgaaWcbaGaaGymaaqabaGc caaISaGaeq4Wdm3aaSbaaSqaaiaaigdaaeqaaaGccaGLOaGaayzkaa aacaGLBbGaayzxaaWaaWbaaSqabeaacqaHepaDdaWgaaadbaGaaGym aaqabaWccqGHsislcaWGTbGaeyOeI0IaamOCamaaBaaameaacaaIXa aabeaaaaaakeaaaeaacqGH9aqpaeaacaWGmbWaaSbaaSqaaiaaigda caaIXaaabeaakmaabmaabaGaaCySdmaaBaaaleaacaaIXaaabeaaki aaiYcacqaHdpWCdaWgaaWcbaGaaGymaaqabaaakiaawIcacaGLPaaa caWGmbWaaSbaaSqaaiaaigdacaaIYaaabeaakmaabmaabaGaeqiXdq 3aaSbaaSqaaiaaigdaaeqaaOGaaGilaiaahg7adaWgaaWcbaGaaGym aaqabaGccaaISaGaeq4Wdm3aaSbaaSqaaiaaigdaaeqaaaGccaGLOa Gaayzkaaaaaaaa@23F4@

and

L 2 ( τ 2 , α 2 , σ 2 ) f( { r x ( 2 ) } xΩ{0} , τ 2 r 2 |{ m i }, τ 2 , α 2 , σ 2 ) = f( { r x ( 2 ) } xΩ{ 0 } |{ m i }, τ 2 , r 2 , α 2 , σ 2 )f( r 2 | { m i }, τ 2 , α 2 , σ 2 ) xΩ{ 0 } [ π ˜ x ( 2 ) ( α 2 , σ 2 ) 1 π ˜ 0 ( 2 ) ( α 2 , σ 2 ) ] r x ( 2 ) × τ 2 ! ( τ 2 r 2 )! [ 1 π ˜ 0 ( 2 ) ( α 2 , σ 2 ) ] r 2 [ π ˜ 0 ( 2 ) ( α 2 , σ 2 ) ] τ 2 r 2 = L 21 ( α 2 , σ 2 ) L 22 ( τ 2 , α 2 , σ 2 ). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=eFD0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqbaeaabsWaaa aabaGaamitamaaBaaaleaacaaIYaaabeaakmaabmaabaGaeqiXdq3a aSbaaSqaaiaaikdaaeqaaOGaaGilaiaahg7adaWgaaWcbaGaaGOmaa qabaGccaaISaGaeq4Wdm3aaSbaaSqaaiaaikdaaeqaaaGccaGLOaGa ayzkaaaabaGaeyyhIulabaGaamOzamaabmaabaWaaqGaaeaadaGada qaaiaadkhadaqhaaWcbaGaaCiEaaqaamaabmaabaGaaGOmaaGaayjk aiaawMcaaaaaaOGaay5Eaiaaw2haamaaBaaaleaacaWH4bGaeyicI4 SaeuyQdCLaeyOeI0Iaae4EaiaahcdacaqG9baabeaakiaaiYcacqaH epaDdaWgaaWcbaGaaGOmaaqabaGccqGHsislcaWGYbWaaSbaaSqaai aaikdaaeqaaaGccaGLiWoadaGadaqaaiaad2gadaWgaaWcbaGaamyA aaqabaaakiaawUhacaGL9baacaaISaGaeqiXdq3aaSbaaSqaaiaaik daaeqaaOGaaGilaiaahg7adaWgaaWcbaGaaGOmaaqabaGccaaISaGa eq4Wdm3aaSbaaSqaaiaaikdaaeqaaaGccaGLOaGaayzkaaaabaaaba Gaeyypa0dabaGaamOzamaabmaabaWaaqGaaeaadaGadaqaaiaadkha daqhaaWcbaGaaCiEaaqaamaabmaabaGaaGOmaaGaayjkaiaawMcaaa aaaOGaay5Eaiaaw2haamaaBaaaleaacaWH4bGaeyicI4SaeuyQdCLa eyOeI0YaaiWaaeaacaWHWaaacaGL7bGaayzFaaaabeaaaOGaayjcSd WaaiWaaeaacaWGTbWaaSbaaSqaaiaadMgaaeqaaaGccaGL7bGaayzF aaGaaGilaiabes8a0naaBaaaleaacaaIYaaabeaakiaaiYcacaWGYb WaaSbaaSqaaiaaikdaaeqaaOGaaGilaiaahg7adaWgaaWcbaGaaGOm aaqabaGccaaISaGaeq4Wdm3aaSbaaSqaaiaaikdaaeqaaaGccaGLOa GaayzkaaGaamOzamaabmaabaGaamOCamaaBaaaleaacaaIYaaabeaa kmaaeeaabaWaaiWaaeaacaWGTbWaaSbaaSqaaiaadMgaaeqaaaGcca GL7bGaayzFaaGaaGilaiabes8a0naaBaaaleaacaaIYaaabeaakiaa iYcacaWHXoWaaSbaaSqaaiaaikdaaeqaaOGaaGilaiabeo8aZnaaBa aaleaacaaIYaaabeaaaOGaay5bSdaacaGLOaGaayzkaaaabaaabaGa eyyhIulabaWaaebuaeqaleaacaWH4bGaeyicI4SaeuyQdCLaeyOeI0 YaaiWaaeaacaWHWaaacaGL7bGaayzFaaaabeqdcqGHpis1aOWaamWa aeaadaWcaaqaaiqbec8aWzaaiaWaa0baaSqaaiaahIhaaeaadaqada qaaiaaikdaaiaawIcacaGLPaaaaaGcdaqadaqaaiaahg7adaWgaaWc baGaaGOmaaqabaGccaaISaGaeq4Wdm3aaSbaaSqaaiaaikdaaeqaaa GccaGLOaGaayzkaaaabaGaaGymaiabgkHiTiqbec8aWzaaiaWaa0ba aSqaaiaahcdaaeaadaqadaqaaiaaikdaaiaawIcacaGLPaaaaaGcda qadaqaaiaahg7adaWgaaWcbaGaaGOmaaqabaGccaaISaGaeq4Wdm3a aSbaaSqaaiaaikdaaeqaaaGccaGLOaGaayzkaaaaaaGaay5waiaaw2 faamaaCaaaleqabaGaamOCamaaDaaameaacaWH4baabaWaaeWaaeaa caaIYaaacaGLOaGaayzkaaaaaaaakiabgEna0oaalaaabaGaeqiXdq 3aaSbaaSqaaiaaikdaaeqaaOGaaiyiaaqaamaabmaabaGaeqiXdq3a aSbaaSqaaiaaikdaaeqaaOGaeyOeI0IaamOCamaaBaaaleaacaaIYa aabeaaaOGaayjkaiaawMcaaiaacgcaaaWaamWaaeaacaaIXaGaeyOe I0IafqiWdaNbaGaadaqhaaWcbaGaaCimaaqaamaabmaabaGaaGOmaa GaayjkaiaawMcaaaaakmaabmaabaGaaCySdmaaBaaaleaacaaIYaaa beaakiaaiYcacqaHdpWCdaWgaaWcbaGaaGOmaaqabaaakiaawIcaca GLPaaaaiaawUfacaGLDbaadaahaaWcbeqaaiaadkhadaWgaaadbaGa aGOmaaqabaaaaOWaamWaaeaacuaHapaCgaacamaaDaaaleaacaWHWa aabaWaaeWaaeaacaaIYaaacaGLOaGaayzkaaaaaOWaaeWaaeaacaWH XoWaaSbaaSqaaiaaikdaaeqaaOGaaGilaiabeo8aZnaaBaaaleaaca aIYaaabeaaaOGaayjkaiaawMcaaaGaay5waiaaw2faamaaCaaaleqa baGaeqiXdq3aaSbaaWqaaiaaikdaaeqaaSGaeyOeI0IaamOCamaaBa aameaacaaIYaaabeaaaaaakeaaaeaacqGH9aqpaeaacaWGmbWaaSba aSqaaiaaikdacaaIXaaabeaakmaabmaabaGaaCySdmaaBaaaleaaca aIYaaabeaakiaaiYcacqaHdpWCdaWgaaWcbaGaaGOmaaqabaaakiaa wIcacaGLPaaacaWGmbWaaSbaaSqaaiaaikdacaaIYaaabeaakmaabm aabaGaeqiXdq3aaSbaaSqaaiaaikdaaeqaaOGaaGilaiaahg7adaWg aaWcbaGaaGOmaaqabaGccaaISaGaeq4Wdm3aaSbaaSqaaiaaikdaae qaaaGccaGLOaGaayzkaaGaaGOlaaaaaaa@1BA1@

Observe that in each case the first factor L k 1 ( α k , σ k ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGmbWaaS baaSqaaiaadUgacaaIXaaabeaakmaabmaabaGaaCySdmaaBaaaleaa caWGRbaabeaakiaaiYcacqaHdpWCdaWgaaWcbaGaam4Aaaqabaaaki aawIcacaGLPaaaaaa@42A2@ is proportional to the conditional joint pmf of the { R x ( k ) } x Ω 0 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaadaGadaqaai aadkfadaqhaaWcbaGaaCiEaaqaamaabmaabaGaam4AaaGaayjkaiaa wMcaaaaaaOGaay5Eaiaaw2haamaaBaaaleaacaWH4bGaeyicI4Saeu yQdCLaeyOeI0IaaCimaaqabaGccaGGSaaaaa@45BD@ given that { M i = m i } 1 n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaadaGadaqaai aad2eadaWgaaWcbaGaamyAaaqabaGccqGH9aqpcaWGTbWaaSbaaSqa aiaadMgaaeqaaaGccaGL7bGaayzFaaWaa0baaSqaaiaaigdaaeaaca WGUbaaaaaa@4183@ and R k = r k , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGsbWaaS baaSqaaiaadUgaaeqaaOGaeyypa0JaamOCamaaBaaaleaacaWGRbaa beaakiaacYcaaaa@3E35@ which is the multinomial distribution with parameter of size r k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGYbWaaS baaSqaaiaadUgaaeqaaaaa@3A78@ and probabilities { π ˜ x ( k ) / [ 1 π ˜ 0 ( k ) ] } x Ω 0 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaadaGadaqaam aalyaabaGafqiWdaNbaGaadaqhaaWcbaGaaCiEaaqaamaabmaabaGa am4AaaGaayjkaiaawMcaaaaaaOqaamaadmaabaGaaGymaiabgkHiTi qbec8aWzaaiaWaa0baaSqaaiaahcdaaeaadaqadaqaaiaadUgaaiaa wIcacaGLPaaaaaaakiaawUfacaGLDbaaaaaacaGL7bGaayzFaaWaaS baaSqaaiaahIhacqGHiiIZcqqHPoWvcqGHsislcaWHWaaabeaakiaa cYcaaaa@4F97@ and that this distribution does not depend on τ k . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDda WgaaWcbaGaam4AaaqabaGccaGGUaaaaa@3C02@ Notice also that the second factors L 12 ( τ 1 , α 1 , σ 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGmbWaaS baaSqaaiaaigdacaaIYaaabeaakmaabmaabaGaeqiXdq3aaSbaaSqa aiaaigdaaeqaaOGaaGilaiaahg7adaWgaaWcbaGaaGymaaqabaGcca aISaGaeq4Wdm3aaSbaaSqaaiaaigdaaeqaaaGccaGLOaGaayzkaaaa aa@4570@ and L 22 ( τ 2 , α 2 , σ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGmbWaaS baaSqaaiaaikdacaaIYaaabeaakmaabmaabaGaeqiXdq3aaSbaaSqa aiaaikdaaeqaaOGaaGilaiaahg7adaWgaaWcbaGaaGOmaaqabaGcca aISaGaeq4Wdm3aaSbaaSqaaiaaikdaaeqaaaGccaGLOaGaayzkaaaa aa@4574@ are proportional to the conditional pmfs of R 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGsbWaaS baaSqaaiaaigdaaeqaaaaa@3A23@ and R 2 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGsbWaaS baaSqaaiaaikdaaeqaaOGaaiilaaaa@3ADE@ given that { M i = m i } 1 n , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaadaGadaqaai aad2eadaWgaaWcbaGaamyAaaqabaGccqGH9aqpcaWGTbWaaSbaaSqa aiaadMgaaeqaaaGccaGL7bGaayzFaaWaa0baaSqaaiaaigdaaeaaca WGUbaaaOGaaiilaaaa@423D@ which are the distributions Bin ( τ 1 m ,1 π ˜ 0 ( 1 ) ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaqGcbGaae yAaiaab6gadaqadaqaaiabes8a0naaBaaaleaacaaIXaaabeaakiab gkHiTiaad2gacaaISaGaaGymaiabgkHiTiqbec8aWzaaiaWaa0baaS qaaiaahcdaaeaadaqadaqaaiaaigdaaiaawIcacaGLPaaaaaaakiaa wIcacaGLPaaaaaa@4883@ and Bin ( τ 2 ,1 π ˜ 0 ( 2 ) ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaqGcbGaae yAaiaab6gadaqadaqaaiabes8a0naaBaaaleaacaaIYaaabeaakiaa iYcacaaIXaGaeyOeI0IafqiWdaNbaGaadaqhaaWcbaGaaCimaaqaam aabmaabaGaaGOmaaGaayjkaiaawMcaaaaaaOGaayjkaiaawMcaaiaa cYcaaaa@4756@ respectively, where Bin ( τ , θ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaqGcbGaae yAaiaab6gadaqadaqaaiabes8a0jaaiYcacqaH4oqCaiaawIcacaGL Paaaaaa@40C1@ denotes the Binomial distribution with parameter of size τ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDaa a@3A2A@ and probability θ . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaH4oqCca GGUaaaaa@3ACD@

The CMLEs α ^ k ( C ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaaceWHXoGbaK aadaqhaaWcbaGaam4AaaqaamaabmaabaGaam4qaaGaayjkaiaawMca aaaaaaa@3D20@ and σ ^ k ( C ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHdpWCga qcamaaDaaaleaacaWGRbaabaWaaeWaaeaacaWGdbaacaGLOaGaayzk aaaaaaaa@3DA6@ of α k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWHXoWaaS baaSqaaiaadUgaaeqaaaaa@3ABE@ and σ k ,k=1,2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHdpWCda WgaaWcbaGaam4AaaqabaGccaGGSaGaam4Aaiabg2da9iaaigdacaGG SaGaaGOmaaaa@401B@ are obtained by maximizing numerically

L 11 ( α 1 , σ 1 ) L 0 ( α 1 , σ 1 )    and    L 21 ( α 2 , σ 2 ) ( 3.6 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGmbWaaS baaSqaaiaaigdacaaIXaaabeaakmaabmaabaGaaCySdmaaBaaaleaa caaIXaaabeaakiaaiYcacqaHdpWCdaWgaaWcbaGaaGymaaqabaaaki aawIcacaGLPaaacaWGmbWaaSbaaSqaaiaaicdaaeqaaOWaaeWaaeaa caWHXoWaaSbaaSqaaiaaigdaaeqaaOGaaGilaiabeo8aZnaaBaaale aacaaIXaaabeaaaOGaayjkaiaawMcaaiaabccacaqGGaGaaeiiaiaa bggacaqGUbGaaeizaiaabccacaqGGaGaaeiiaiaadYeadaWgaaWcba GaaGOmaiaaigdaaeqaaOWaaeWaaeaacaWHXoWaaSbaaSqaaiaaikda aeqaaOGaaGilaiabeo8aZnaaBaaaleaacaaIYaaabeaaaOGaayjkai aawMcaaiaaywW7caaMf8UaaGzbVlaaywW7caaMf8Uaaiikaiaaioda caGGUaGaaGOnaiaacMcaaaa@6661@

with respect to ( α 1 , σ 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaadaqadaqaai aahg7adaWgaaWcbaGaaGymaaqabaGccaaISaGaeq4Wdm3aaSbaaSqa aiaaigdaaeqaaaGccaGLOaGaayzkaaaaaa@3F86@ and ( α 2 , σ 2 ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaadaqadaqaai aahg7adaWgaaWcbaGaaGOmaaqabaGccaaISaGaeq4Wdm3aaSbaaSqa aiaaikdaaeqaaaGccaGLOaGaayzkaaGaaiilaaaa@4038@ respectively. Note that the factors in (3.6) do not depend on τ k ,k=1,2. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDda WgaaWcbaGaam4AaaqabaGccaGGSaGaam4Aaiabg2da9iaaigdacaGG SaGaaGOmaiaac6caaaa@40CF@

Finally, by plugging the estimates α ^ k ( C ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaaceWHXoGbaK aadaqhaaWcbaGaam4AaaqaamaabmaabaGaam4qaaGaayjkaiaawMca aaaaaaa@3D20@ and σ ^ k ( C ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHdpWCga qcamaaDaaaleaacaWGRbaabaWaaeWaaeaacaWGdbaacaGLOaGaayzk aaaaaaaa@3DA6@ into the factors of the likelihood function that depend on τ k ,k=1,2, MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDda WgaaWcbaGaam4AaaqabaGccaGGSaGaam4Aaiabg2da9iaaigdacaGG SaGaaGOmaiaacYcaaaa@40CD@ and maximizing these factors, that is, maximizing L 12 ( τ 1 , α ^ 1 ( C ) , σ ^ 1 ( C ) ) L MULT ( τ 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGmbWaaS baaSqaaiaaigdacaaIYaaabeaakmaabmaabaGaeqiXdq3aaSbaaSqa aiaaigdaaeqaaOGaaGilaiqahg7agaqcamaaDaaaleaacaaIXaaaba WaaeWaaeaacaWGdbaacaGLOaGaayzkaaaaaOGaaGilaiqbeo8aZzaa jaWaa0baaSqaaiaaigdaaeaadaqadaqaaiaadoeaaiaawIcacaGLPa aaaaaakiaawIcacaGLPaaacaWGmbWaaSbaaSqaaiaab2eacaqGvbGa aeitaiaabsfaaeqaaOWaaeWaaeaacqaHepaDdaWgaaWcbaGaaGymaa qabaaakiaawIcacaGLPaaaaaa@52C8@ and L 22 ( τ 2 , α ^ 2 ( C ) , σ ^ 2 ( C ) ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGmbWaaS baaSqaaiaaikdacaaIYaaabeaakmaabmaabaGaeqiXdq3aaSbaaSqa aiaaikdaaeqaaOGaaGilaiqahg7agaqcamaaDaaaleaacaaIYaaaba WaaeWaaeaacaWGdbaacaGLOaGaayzkaaaaaOGaaGilaiqbeo8aZzaa jaWaa0baaSqaaiaaikdaaeaadaqadaqaaiaadoeaaiaawIcacaGLPa aaaaaakiaawIcacaGLPaaacaGGSaaaaa@4AE8@ with respect to τ 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDda WgaaWcbaGaaGymaaqabaaaaa@3B11@ and τ 2 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDda WgaaWcbaGaaGOmaaqabaGccaGGSaaaaa@3BCC@ respectively, we get that the CMLEs τ ^ 1 ( C ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga qcamaaDaaaleaacaaIXaaabaWaaeWaaeaacaWGdbaacaGLOaGaayzk aaaaaaaa@3D73@ and τ ^ 2 ( C ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga qcamaaDaaaleaacaaIYaaabaWaaeWaaeaacaWGdbaacaGLOaGaayzk aaaaaaaa@3D74@ of τ 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDda WgaaWcbaGaaGymaaqabaaaaa@3B11@ and τ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDda WgaaWcbaGaaGOmaaqabaaaaa@3B12@ are given by (3.5) but replacing α ^ k ( U ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaaceWHXoGbaK aadaqhaaWcbaGaam4AaaqaamaabmaabaGaamyvaaGaayjkaiaawMca aaaaaaa@3D32@ and σ ^ k ( U ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHdpWCga qcamaaDaaaleaacaWGRbaabaWaaeWaaeaacaWGvbaacaGLOaGaayzk aaaaaaaa@3DB8@ by α ^ k ( C ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaaceWHXoGbaK aadaqhaaWcbaGaam4AaaqaamaabmaabaGaam4qaaGaayjkaiaawMca aaaaaaa@3D20@ and σ ^ k ( C ) ,k=1,2. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHdpWCga qcamaaDaaaleaacaWGRbaabaWaaeWaaeaacaWGdbaacaGLOaGaayzk aaaaaOGaaiilaiaadUgacqGH9aqpcaaIXaGaaiilaiaaikdacaGGUa aaaa@432F@ Observe that these expressions for τ ^ 1 ( C ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga qcamaaDaaaleaacaaIXaaabaWaaeWaaeaacaWGdbaacaGLOaGaayzk aaaaaaaa@3D73@ and τ ^ 2 ( C ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga qcamaaDaaaleaacaaIYaaabaWaaeWaaeaacaWGdbaacaGLOaGaayzk aaaaaaaa@3D74@ are closed forms. The CMLE of τ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDaa a@3A2A@ is τ ^ ( C ) = τ ^ 1 ( C ) + τ ^ 2 ( C ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga qcamaaCaaaleqabaWaaeWaaeaacaWGdbaacaGLOaGaayzkaaaaaOGa eyypa0JafqiXdqNbaKaadaqhaaWcbaGaaGymaaqaamaabmaabaGaam 4qaaGaayjkaiaawMcaaaaakiabgUcaRiqbes8a0zaajaWaa0baaSqa aiaaikdaaeaadaqadaqaaiaadoeaaiaawIcacaGLPaaaaaGccaGGUa aaaa@498D@

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