Biomedical Engineering Reference
In-Depth Information
the proposed method can be implemented with a least-squares nonparametric
regression fitting program.
Our proposed approach is different from the method of Murphy and van der
Vaart (1999) and Murphy and van der Vaart (2000) in two important as-
pects. First, their approach requires computation of the profile likelihood,
while our proposed approach directly uses the likelihood or an approximation
of the likelihood. Second, their approach requires construction of least favor-
able submodels with certain properties, while our approach requires solving a
nonparametric regression problem. Computationally, our proposed approach
is much easier to implement, as there is no need to profile an often complicated
likelihood function.
In a class of sieve MLEs using a linear approximation space, the proposed
estimator of the information matrix is shown to be the same as the inverse of
the observed information matrix for the sieve MLE. This equivalence is useful
both theoretically and computationally. First, it enables a simple consistency
proof of the observed information matrix in the semiparametric setting. Sec-
ond, it provides two ways of computing the observed information matrix: one
can either directly compute the observed information matrix or fit a least-
squares nonparametric regression. Because of its numerical convenience and
good theoretical properties, the class of sieve MLEs using polynomial splines is
utilized in our numerical demonstration and is recommended for applications
of general semiparametric estimation.
The chapter is organized as follows. Section 9.2 describes the motivation
and the least-squares approach. Section 9.3 specializes the general approach to
a class of sieve MLEs. Section 9.4 applies the proposed method along with the
spline-based sieve MLE to two models, the Cox proportional hazards model
with interval-censored data and the semiparametric Poisson mean model for
panel count data, studied in Huang and Wellner (1997) and Wellner and Zhang
(2007), respectively. Section 9.5 renders numerical results via simulations and
applications in real-life examples for the models discussed in Section 9.4. Sec-
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