178
Appendix A
maximum-likelihood estimation of ␪ under the sampling model given
in Equation A.1. The two-parameter lognormal distribution is of special interest, because petroleum geologists commonly use it for resource
evaluation. Estimation of the coeffi cient of discoverability b and prediction of the population size N are also considered.
In the following section of Appendix A, we derive the likelihood
function and propose a computational method for its evaluation.
Maximum-likelihood estimations for ␪ are next considered in the
section “Maximum-Likelihood Estimation,” together with examples
of some specifi c forms of f ( y| ␪). The section “Inference for ␪ and N”
introduces a method of prediction for the population size N. In the fi nal
section, “Inference for the Weight Function,” we consider the weight
function w( y, ␤), where ␤ is a vector of parameters, and examine the
joint maximum-likelihood estimation of ␪ and ␤.
The Likelihood Function
Defi ne X N 5 (X 1 , . . . , X N ) as the vector of observations in order of occurrence so that X j is the value observed for the jth draw. Upon relabeling the elements of Y N 5 (X 1 , . . . , X N ) so that X j 5 Y j , j 5 1, 2, . . . , n, the
probability of observing x N 5 (x 1 , . . . , x N ), given Y N 5 y N , is
(
)
1
1
w ( )
P 1, 2, ,
b w(
)
w( )
n
j
N
j
j
n
N
x
n
=
+


…
=


+
+ +
∏
y
y
y
"
(A.3)
where b j 5 w(x j ) 1 · · · 1 w(x n ). Multiplying Equation A.3 by the joint
density of Y N and integrating over the unobserved values (Y n11 , . . . , Y N )
of Y N , the joint density of X 1 , . . . , X n is given as
(
)
( )
1
1
1
w ( )
!
E
!
b w (
)
w ( )
n
n
j
j
j
j
j
n
N
x
N
f x
N n
Y
Y
=
=
+




−
+
+ +




∏
∏
␪
␪
"
(A.4)
because Y 1 , . . . , Y N are i.i.d. and there are N!/(N − n)! ordered samples
of size n without replacement from a fi nite population of N units. Note
that with x N fi xed and letting N − n → ∞, the joint density shown in
Equation A.4 approaches
1
1
w ( ) (
)
E[w ( ) ]
n
i
i
i
x f x
Y
=
∏
␪
␪
(A.5)