198
Appendix A
Let « (l|x n , ␪, ␤) and h ( y | ␪, ␤) be defi ned in the same manner as
Equations A.19 and A.22 respectively. The score function with respect
to b l is therefore
w l
l
j l
l
=
∞
∂


=
−


∂


∂
∂
+
−
+


∂
∂


=
∑
∫
1
0
log
w
log
,
log G
, ,
1, 2, ...,
( )
( )
(
)
(
)
(
)
( )
n
lj
lj
j
l
f
n
n
l
l
L
A
N n
d
d
l
k
x
b
b
b
␤
␤
␪ ␤
␤
␪ ␤
(A.78)
where
(
)
( )
log
,
E w ,
, ,
f
l
Y
w l
l
l
∂
= − 



∂b
␪ ␤
␤
␪ ␤
(A.79)
and
( )
log G n
l
d
l
∂
∂b
␤ can be approximated in the same way as
log dG n (l|␤) (see Eqs. A.15 and A.17). The score function with respect
to ␪ is the same as Equation A.18. Maximum-likelihood estimates
(␪ ˆ , ␤ ˆ ) are solutions to
1
1
log ,...,
log ,
log ,...,
log
m
k
L
L
L
L
u


∂
∂
∂
∂
=


∂
∂
∂
∂


0
u
b
b
(A.80)
A Newton–Raphson approach to the calculation of (␪ ˆ , ␤ ˆ ) would
require second derivatives of log L and would be a mess—and computationally expensive.
When w( y, ␤) is parameterized by only one parameter, such as
w( y, ␤) 5 y
␤
, the maximum-likelihood estimates may be obtained by
solving the equation
log
0,
1, 2, ...,
r
L
r
m
∂
=
=
∂u
by the EM algorithm to fi nd ␪ ˆ (␤) for each fi xed ␤. Then ␤ ˆ is determined
from the log-likelihood profi le




ˆ
log
( ),
L ␪ ␤ ␤ either graphically or via
a one-dimensional, gradient-free maximization algorithm. The logmaximized relative-likelihood function


=
−


max
ˆ
ˆˆ
( ) log
( ),
log ( , )
R
L
L
␤
␪␤ ␤
␪␤
(A.81)