Biomedical Engineering Reference
In-Depth Information
class not only has good approximation power, but is also computationally
convenient. We will use this approximation space in the next section.
Let
n
be an approximation space for both and H. Suppose it has a
set of basis functions b
n
= (b
1
;:::;b
q
n
)
0
, such that every 2
n
can be
represented as =
P
q
n
j=1
j
b
j
b
0
n
, where = (
1
;:::;
q
n
)
0
2 B
n
R
q
n
is a vector of real numbers. So, every 2
n
can be identified with a vector
2 B
n
. Here the dimension q
n
is a positive integer depending on sample
size n. To ensure consistency of b
n
, we need q
n
! 1 as n ! 1. In general,
for
b
n
to be asymptotically normal, we need to control the growth rate of q
n
appropriately.
The sieve MLE of
0
= (
0
;
0
) is defined to be the (
b
n
;
b
n
) that maximizes
the log-likelihood `
n
(;) over
n
. Equivalently, one can find (
b
n
;
b
n
) that
maximizes `
n
(; b
0
n
) over B
n
. Then,
b
n
= b
0
n
b
n
.
Now consider estimation of I(
0
). First we introduce some notation. Let
`
2
(x; b
n
)(b
n
) = ( `
2
(x; b
n
)(b
1
);:::; `
2
(x; b
n
)(b
q
n
))
T
;
and
`
1
(X
i
; b
n
)
2
X
X
`
1
(X
i
; b
n
) `
T
A
11
= n
1
; A
12
= n
1
2
(X
i
; b
n
)(b
n
);
i=1
i=1
l
2
(X
i
; b
n
)(b
n
)
2
X
A
21
= A
12
; A
22
= n
1
:
i=1
The outer product version of the observed information matrix for is given
by
O
n
= A
11
A
12
A
22
A
21
:
(9.11)
Here for any matrix A, A
denotes its generalized inverse. Although A
22
may
not be unique, A
12
A
22
A
21
is unique by results on the generalized inverse; see,
for example, Rao (1973), Chapter 1, result 1b.5 (vii), page 26. The use of
the generalized inverse in Equation (9.11) is for generality of these formulas.
When the nonparametric component is a smooth function, then for any xed
sample size, the sieve MLE is obtained over a finite-dimensional approximation
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