Biomedical Engineering Reference
In-Depth Information
With a random sample X
1
;:::;X
n
and a consistent estimator b
n
of
0
, we
can estimate I() as follows. First, we nd the
b
n
that minimizes
X
k`
1
(X
i
; b
n
) `
2
(X
i
; b
n
)(h)k
2
n
(h; b
n
) n
1
(9.9)
i=1
over H
n
. Here we assume that such a minimizer exists. Because `
2
is a linear
operator, this minimization problem is essentially a least-squares nonpara-
metric regression problem and it can be solved for each component separately.
Then a natural estimator of I(
0
) is
X
I
n
= n
1
[ `
1
(X
i
; b
n
) `
2
(X
i
; b
n
)(
b
n
)]
2
:
(9.10)
i=1
Let a
jk
(x; ;h) be the (j;k)-th element in the d d matrix [ `
1
(x; )
`
2
(x; )(h)]
2
. Denote A
jk
= fa
jk
(x; ;h) : 2T;h 2H
n
g; 1 ;j;k d. We
use the linear functional notation for integrals. So for any probability measure
Q, Qf =
R
fdQ as long as the integral is well-defined. Let P = P
0
;
0
. We
make the following assumptions.
(A1) The classes of functions A
jk
; 1 j;k d, are Glivenko-Cantelli.
(A2) For any f
n
g H
n
satisfying k
n
0
k
H
! 0, P[ `
2
(;
0
)(
n
)
`
2
(;
0
)(
0
)]
2
!
p
0: In addition, if b
n
!
p
0
, then P[ `
1
(; b
n
) `
1
(;
0
)]
2
!
p
0 and sup
h2H
n
P[ `
2
(; b
n
)(h) `
2
(;
0
)(h)]
2
!
p
0;
Theorem 9.1: Suppose H
n
!H and
b
n
argmin
H
n
n
(h; b
n
) exists. Suppose
(9.6), (A1), and (A2) hold. Then for any sequence fb
n
g that converges to
0
in probability, we have I
n
!
p
I(
0
), in the sense that each element of I
n
converges in probability to its corresponding element in I(
0
).
The proof of this theorem is given in the appendix at the end of this
chapter. This theorem provides sucient conditions ensuring consistency of
the estimated least favorable direction and I
n
as an estimator of I(
0
). This
result appears useful in a large class of models and is easy to apply. In the
next section, we show that, in a class of sieve models, when b
n
is the MLE,
the I
n
is actually the observed information based on the outer product of the
first derivatives of the log-likelihood.
Search WWH ::
Custom Search