Biomedical Engineering Reference
In-Depth Information
where B
n
= f :
1
2
q
n
g. Each element of M
n
(D
n
;K
n
;m) is a
nondecreasing function because of the monotonicity constraints on
1
;:::;
q
n
.
This fact is a consequence of the variation diminishing properties of B-splines.
See, for instance, Schumaker (1981), Example 4.75 and Theorem 4.76, pages
177{178. The B-splines sieve MLE of
0
= (
0
;
0
) is the b
n
= (
b
n
;
b
n
) that
maximizes `
n
(;) over M
n
. This is equivalent to maximizing `
n
(; b
0
n
)
over B
n
. No restriction will be placed on . Thus, can be taken to be
R
d
.
The B-splines sieve MLE is easier to compute than the semiparametric
MLE considered in Huang, Rossini, and Wellner (1997). In addition, un-
der some mild regularity conditions, the sieve MLE of ,
b
n
, achieves semi-
parametric eciency as well, with the information matrix given by I(
0
) =
P(`(;
0
;
0
)
2
; where `(x;
0
;
0
) = `
1
(x;
0
;
0
) `
2
(x;
0
;
0
)(
0
) is the semi-
parametrically ecient score, where `
1
and `
2
are score functions of and ,
respectively. Here
0
is the solution to the Fredholm integral equation of the
second kind,
Z
0
(t)
K(t;x)
0
(x)dx = d(t);
(9.15)
where the form of K(t;x) and d(t) are given in the appendix at the end
of this chapter. Unfortunately, there is no explicit solution to this integral
equation. Thus, direct estimation of the information matrix is impossible for
this model. However, with the B-splines approach, the variance of
b
n
can be
readily estimated using the observed information matrix defined in Equation
(9.11) due to Theorem 9.1 and Proposition 9.3.1.
For the asymptotic normality of
b
n
and consistency of the inverse observed
information matrices, the following conditions are assumed:
(C1) (a) E(ZZ
0
) is nonsingular; (b) Z is bounded, that is, there exists z
0
> 0
such that P(kZk z
0
) = 1.
(C2) (a) There exists a positive number such that P(V U ) = 1; (b)
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