Biomedical Engineering Reference
In-Depth Information
be shown that
n
1=2
^
n
0
!
d
N
0;I
1
(
0
)
;
(9.2)
where I(
0
) is the ecient Fisher information matrix for
0
, adjusted for the
presence of nuisance parameter
0
. The definition of I(
0
) is given below.
This holds for models cited in the previous section and for the examples in
Section 9.4. Thus, estimation of the asymptotic variance of
^
n
is equivalent to
estimation of I(
0
) provided I(
0
) is nonsingular. Of course, for the problem
of estimating I(
0
) to be meaningful, we need to first establish (9.2).
The calculation of I(
0
) and its central role in asymptotic eciency the-
ory for semiparametric models have been systematically studied by Begun
et al. (1983), van der Vaart (1991), and Bickel et al. (1993) and the references
therein. In the following, we first briefly describe how the information I(
0
) is
defined in a class of semiparamtric models in order to motivate the proposed
information estimator.
Let `(x; ;) = log p(x; ;) be the log-likelihood for a sample of size 1.
Consider a parametric smooth sub-model in fp
;
: (;) 2 g with
parameter (;
(s)
), where
(0)
= and
s=0
@
(s)
@s
= h:
Let H be the class of functions h defined by this equation. Suppose H is
equipped with a norm kk
H
. The score operator for is
s=0
`
2
(x; )(h) =
@
@s
`(x; ;
(s)
)
:
(9.3)
Observe that `
2
is a linear operator mapping H to L
2
(P
;
). So for constants
c
1
, c
2
and h
1
;h
2
2H,
`
2
(x; )(c
1
h
1
+ c
2
h
2
) = c
1
`
2
(x; )(h
1
) + c
2
`
2
(x; )(h
2
):
(9.4)
The linearity of `
2
is crucial to the proposed method and will be used in
Section 9.3. For a d-dimensional ,
`
1
(x; ) is the vector of partial derivatives
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