Biomedical Engineering Reference
In-Depth Information
Fisher information. The asymptotic normality and eciency results provide
insight into the theoretical properties of maximum likelihood estimators. Un-
fortunately, in many semiparametric models studied in the aforementioned
articles, the ecient Fisher information is either very complicated or may not
have an explicit expression.This makes it dicult to estimate the variance of
the MLE in semiparametric models. Therefore, it is imperative to develop com-
putationally ecient methods for constructing consistent variance estimators
of the MLE in order to apply semiparametric theory to statistical inference in
practice.
In this chapter, we consider consistent variance estimation in a class of
semiparametric models that are parameterized in terms of a finite-dimensional
parameter and a parameter in a general space. Hence, is often called
an innite-dimensional parameter. Two important examples are Cox's (1972)
proportional hazards model for interval-censored data (Finkelstein and Wolfe
(1985); Huang et al. (1997)) and the proportional mean model for panel count
data (Sun and Wei (2000); Wellner and Zhang (2007)). In these two examples,
is the nite-dimensional regression coecient, is the logarithm of the
baseline hazard function or the logarithm of the baseline mean function.
An existing method for estimating the variance of the MLE in semipara-
metric models is to use the second derivative of the prole likelihood of at
the maximum likelihood estimate. For a xed value of , the prole likelihood
is the maximum of the likelihood with respect to . Because the prole likeli-
hood often can only be computed numerically, discretized versions of its second
derivative must be used. Murphy and van der Vaart (1999, 2000) showed that
the discretized version of the second derivative of the profile likelihood pro-
vides consistent variance estimators in a class of semiparametric models under
appropriate conditions. However, to apply the results of Murphy and van der
Vaart (1999, 2000), certain least favorable sub-models with the right proper-
ties must be constructed, which may be dicult. The result of the discretized
version of the second derivative of the profile likelihood may be sensitive to
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