Biomedical Engineering Reference
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of `(x; ) with respect to . For each component of `
1
, a score operator for
is dened as in Equation (9.3). So the score operator for corresponding to
`
1
is defined as
`
2
(x; )(h) ( `
2
(x; )(h
1
);:::; `
2
(x; )(h
d
))
0
;
(9.5)
where h (h
1
;:::;h
d
)
0
with h
k
2H; 1 k d.
Let P
2
be the closed linear span of f`
2
(h) : h 2Hg. Then P
2
L
2
(P
;
).
The ecient score function for the k-th component of is `
1;k
( `
1;k
j P
2
),
where `
1;k
is the k-th component of `
1
(x; ) and ( `
1;k
j P
2
) is the projection of
`
1;k
onto P
2
. Equivalently, ( `
1;k
j P
2
) is the minimizer of the squared residual
E[ `
1;k
(X;
0
)
k
]
2
over
k
2 P
2
. See, for example, van der Vaart (1991),
Section 6, and Bickel et al. (1993), Theorem 1, page 70. We assume that
k
is
a score operator for , that is, there exists an
0k
2H such that
`
2
(x; )(
0k
); 1 k d:
k
=
(9.6)
Let
0
= (
01
;
02
;:::;
0d
)
0
. The ecient score for is `(x; ) `
1
(x; )
`
2
(x; )(
0
). The
0
is often called the least favorable direction. Under Equa-
tion (9.6),
0
= arg min
h2H
d
(h;
0
), where
(h; ) Ek`
1
(X; ) `
2
(X; )(h)k
2
:
(9.7)
The information matrix for is
I() = E[`(X; )]
2
= E[ `
1
(X; ) `
2
(X; )(
0
)]
2
;
(9.8)
where a
2
= aa
0
for any a 2 R
d
.
Therefore, to estimate I(), it is natural to consider minimizing an em-
pirical version of (9.7). As in nonparametric regression, if the space H is too
large, minimization over this space may not yield consistent estimators of
0
and I(
0
). We use an approximation space H
n
(a sieve) that is smaller than
H and H
n
!H, in the sense that, for any h 2H, there exists h
n
2H
n
such
that kh
n
hk
H
! 0 as n !1.
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