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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