180
Appendix A
From Equation A.8, the log likelihood of given N, w(•), and data
x n is
=
=
+
∑
1
log
log (
) log (
)
n
j
n
j
L
f x
S x
(A.11)
where
l
l
∞
−
= ∫ 0
(
)
( )
G ( )
N n
n
f
n
S
d
x
r
(A.12)
Note the following points:
If w(
1.
y ) ≡ 1, the expectation in Equation A.4 is equal to
(N−n)! / N! and the joint density then reduces to the usual likelihood of n observations.
If the weight function is of the form w(
2.
y) 5 y
b for an unknown
parameter , the parameters and  are not always identifi -
able in the infi nite population selection-biased model shown
in Equation A.10. For instance, if f ( y | ) is lognormal with
parameters μ and s
2 , the sampling distribution is also lognormal with parameters μ 1 s
2 and s
2 . In the fi nite population
case, it is shown by Equation A.8 that if N 5 n (i.e., all units
have been selected), the likelihood separates into two parts.
The fi rst part contains only information about , which is the
usual likelihood of given the data; the second part contains
only information about , which is the probability of observing X 1 , . . . , X N , in that order. Note also that the second part
is just the marginal likelihood of  (Kalbfl eisch and Prentice,
1973) under the Cox model for survival data (Cox, 1972) when
there are no ties or censoring. When neither N 5 n nor  5 0,
information about and  is diffi cult to separate. There does
not appear to be a partial likelihood decomposition (Cox,
1975) for  or . Principally, this is the result of the fact that
unit values not included in the sample are unobservable. In the
extreme case, with n fi xed and N − n → ∞, information about
and  is so mixed up that they cannot be separated.
The joint density of the observations derived by Barouch and
3.
Kaufman (1976) when w( y) 5 y is the same as that shown in
Equation A.8, except that we derived it in terms of the general
gamma distribution. This form of Equation A.8 gives us the
interpretation that the likelihood function for and w consists
of the usual individual likelihoods and an adjustment term that
contains further but inseparable information about and w.
Appendix A
From Equation A.8, the log likelihood of given N, w(•), and data
x n is
=
=
+
∑
1
log
log (
) log (
)
n
j
n
j
L
f x
S x
(A.11)
where
l
l
∞
−
= ∫ 0
(
)
( )
G ( )
N n
n
f
n
S
d
x
r
(A.12)
Note the following points:
If w(
1.
y ) ≡ 1, the expectation in Equation A.4 is equal to
(N−n)! / N! and the joint density then reduces to the usual likelihood of n observations.
If the weight function is of the form w(
2.
y) 5 y
b for an unknown
parameter , the parameters and  are not always identifi -
able in the infi nite population selection-biased model shown
in Equation A.10. For instance, if f ( y | ) is lognormal with
parameters μ and s
2 , the sampling distribution is also lognormal with parameters μ 1 s
2 and s
2 . In the fi nite population
case, it is shown by Equation A.8 that if N 5 n (i.e., all units
have been selected), the likelihood separates into two parts.
The fi rst part contains only information about , which is the
usual likelihood of given the data; the second part contains
only information about , which is the probability of observing X 1 , . . . , X N , in that order. Note also that the second part
is just the marginal likelihood of  (Kalbfl eisch and Prentice,
1973) under the Cox model for survival data (Cox, 1972) when
there are no ties or censoring. When neither N 5 n nor  5 0,
information about and  is diffi cult to separate. There does
not appear to be a partial likelihood decomposition (Cox,
1975) for  or . Principally, this is the result of the fact that
unit values not included in the sample are unobservable. In the
extreme case, with n fi xed and N − n → ∞, information about
and  is so mixed up that they cannot be separated.
The joint density of the observations derived by Barouch and
3.
Kaufman (1976) when w( y) 5 y is the same as that shown in
Equation A.8, except that we derived it in terms of the general
gamma distribution. This form of Equation A.8 gives us the
interpretation that the likelihood function for and w consists
of the usual individual likelihoods and an adjustment term that
contains further but inseparable information about and w.