Biomedical Engineering Reference
In-Depth Information
estingly, this estimator is exactly the observed information matrix if we treat
the semiparametric sieve MLE as a parametric MLE problem.
In addition to the expression of information matrix given in Equa-
tion (9.7), we note that it can be expressed through the second deriva-
`
11
(x; )
@
`
1
(x; ),
@
tives. Consider the case when d =
1. Denote
=
`
12
(x; )
@s
`
1
(;
(s)
; x)j
s=0
,
`
21
(x; )(h)
@
`
2
(x; )(h), and letting
@
@
=
=
(@=@s)
1(s)
j
s=0
= h
1
, `
22
(x; )(h;h
1
) =
@s
`
2
(;
1(s)
; x)(h): Then the infor-
@
mation matrix can be written as
I(
0
) = E
h
`
11
(
0
; X) 2`
12
(
0
; X)(
0
) + `
22
(
0
; X)(
0
;
0
)
i
:
This expression leads to an alternative estimator of I(
0
), given by
h
`
11
(X
i
; b
n
) 2`
12
(X
i
; b
n
)(
~
n
) + `
22
(X
i
; b
n
)(
~
n
;
~
n
)
i
;
X
n
1
(9.19)
i=1
~
n
is the minimizer of
where
h
`
11
(X
i
; b
n
) 2`
12
(X
i
; b
n
)(h) + `
22
(X
i
; b
n
)(h;h)
i
X
~
n
(h; b
n
) = n
1
i=1
over H
n
. However, further work is needed to show the consistency of ~
n
(
~
n
; b
n
),
and it also would be interesting to investigate how this estimator behaves
compared to the one proposed in this chapter.
As implied in Example 2, this semiparametric inference procedure is not
robust against model misspecification. If the true model is not the working
model for forming the likelihood, the inference may be invalid. Wellner and
Zhang (2007) developed a theorem for semiparametric M-estimators, general-
izing the result of Huang (1996). Lu et al. (2009) extend the semiparametric
M-estimators inference to the spline-based sieve semiparametric estimation
problems. It remains an interesting research problem on how to generalize
the proposed least-squares method to estimate the variance of the regression
parameter estimates in order to make robust semiparametric inference.
Search WWH ::
Custom Search