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@
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
i j
(
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
i j
(
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
i j
(
k
)
= Pr (
X
i j
(
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
j ∈ U −
A
i
,
p
i j
(
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
) !
(
1 − n / 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
1 j
(
k
)
, … ,
X
n j
(
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
i j
(
k
)
]
x
i
[
1 −
p
i j
(
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
) d z ,
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
=
∑
x ∈ Ω − { 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
i j
( 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 )
}
x ∈ Ω − { 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
) !
∏
x ∈ Ω − { 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
) !
∏
x ∈ Ω − { 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
i − 1
,
x
i + 1
, … ,
x
n
) :
x
j
= 0 , 1 , j ≠ i , 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
i − 1
(
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
) =
∫
∏
j ≠ i
n
exp [
x
j
(
α
j
(
1
)
+
σ
1
z
) ]
1 + exp (
α
j
(
1
)
+
σ
1
z
)
ϕ (
z
) d z .
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
∏
j ≠ i
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 − (
1 − n / 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
)
}
x ∈ Ω − { 0 }
,
τ
1
− m −
r
1
| {
m
i
} ,
τ
1
,
α
1
,
σ
1
)
=
f (
{
r
x
(
1
)
}
x ∈ Ω − { 0 }
| {
m
i
} ,
τ
1
,
r
1
,
α
1
,
σ
1
) f (
r
1
|
{
m
i
} ,
τ
1
,
α
1
,
σ
1
)
∝
∏
x ∈ Ω − { 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
)
}
x ∈ Ω − { 0 }
,
τ
2
−
r
2
| {
m
i
} ,
τ
2
,
α
2
,
σ
2
)
=
f (
{
r
x
(
2
)
}
x ∈ Ω − { 0 }
| {
m
i
} ,
τ
2
,
r
2
,
α
2
,
σ
2
) f (
r
2
|
{
m
i
} ,
τ
2
,
α
2
,
σ
2
)
∝
∏
x ∈ Ω − { 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@
Editorial policy
Survey Methodology publishes articles dealing with various aspects of statistical development relevant to a statistical agency, such as design issues in the context of practical constraints, use of different data sources and collection techniques, total survey error, survey evaluation, research in survey methodology, time series analysis, seasonal adjustment, demographic studies, data integration, estimation and data analysis methods, and general survey systems development. The emphasis is placed on the development and evaluation of specific methodologies as applied to data collection or the data themselves. All papers will be refereed. However, the authors retain full responsibility for the contents of their papers and opinions expressed are not necessarily those of the Editorial Board or of Statistics Canada.
Submission of Manuscripts
Survey Methodology is published twice a year in electronic format. Authors are invited to submit their articles in English or French in electronic form, preferably in Word to the Editor, (statcan.smj-rte.statcan@canada.ca , Statistics Canada, 150 Tunney’s Pasture Driveway, Ottawa, Ontario, Canada, K1A 0T6). For formatting instructions, please see the guidelines provided in the journal and on the web site (www.statcan.gc.ca/SurveyMethodology).
Note of appreciation
Canada owes the success of its statistical system to a long-standing partnership between Statistics Canada, the citizens of Canada, its businesses, governments and other institutions. Accurate and timely statistical information could not be produced without their continued co-operation and goodwill.
Standards of service to the public
Statistics Canada is committed to serving its clients in a prompt, reliable and courteous manner. To this end, the Agency has developed standards of service which its employees observe in serving its clients.
Copyright
Published by authority of the Minister responsible for Statistics Canada.
© Minister of Industry, 2015
Catalogue no. 12-001-X
Frequency: semi-annual
Ottawa
Report a problem on this page
Is something not working? Is there information outdated? Can't find what you're looking for?
Please contact us and let us know how we can help you.
Privacy notice
Date modified:
2017-09-20