Estimation of Superpopulation Parameters
197
because an explorationist quite frequently has other pertinent geological information about the number of fi elds/pools that could exist in the
play. This information is usually summarized as a subjective distribution for N.
In a situation in which it is known that N has a fi nite support
N 1 , ..., N k , but unknown probability masses g 1 , ..., g k with S g i = 1, the
maximum-likelihood estimation procedure for (␪, r) is equivalent to the
likelihood profi le method on the support N 1 , ..., N k ; hence, no advantage is gained. This equivalence can be seen as follows: For fi xed ␪, the
likelihood is maximized by setting all g 1 ’s to zero except for the one g j
with an associated N j that maximizes the conditional likelihoods of ␪
given N = N i for i = 1, 2, ..., k.
Inference for the Weight Function
In this section we shall consider the weight function w( y, ␤) where
␤ is a vector of parameters b 1 , … b k , and look at the joint maximumlikelihood estimation of ␪ and ␤. To this end, let r f (l|␪, ␤) be the Laplace
transform of w( y, ␤) with respect to f ( y| ␪), and let d G n (l|␤) be the
general gamma density with parameters b j (␤), where j = 1, 2, … , n. Let
S(␪, ␤|x n ) be the integral given by Equation A.12 in terms of r f (l| ␪, ␤)
and dG n (l | ␤). Therefore, the log likelihood of (␪, ␤) is
( )
( )
( )
(
)
=
=


=
+
−
+


∑
∑
1
1
log
w ,
log b
log
,
n
n
j
j
j
n
j
j
L
f x
x
S
x
␪
␤
␤
␪␤
(A.77)
We shall assume that w( y, ␤) is suffi ciently smooth so that its partial
derivatives with respect to b l , l = 1, 2, ... , k, all exist and are continuous.
Defi ne
w ( )
w( , )
lj
j
l
x
∂
= ∂b
␤
␤
and
w ( )
( )
b ( )
n
lj
lj
i j
j
A
=
= ∑
␤
␤
␤
l 5 1, 2, … , k; j 5 1, 2, …, n