Biomedical Engineering Reference
In-Depth Information
9.5.2
Applications ::::::::::::::::::::::::::::::::::::::::::::: 256
9.6
Discussion :::::::::::::::::::::::::::::::::::::::::::::::::::::::: 258
Acknowledgments :::::::::::::::::::::::::::::::::::::::::::::::: 260
Appendix ::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 260
Bibliography :::::::::::::::::::::::::::::::::::::::::::::::::::::: 264
9.1
Introduction
In a regular parametric model, the maximum likelihood estimator (MLE) is
asymptotically normal with variance equal to the inverse of the Fisher infor-
mation, and the Fisher information can be estimated by the observed infor-
mation. This result provides large sample justification for the use of normal
approximation to the distribution of MLE. An important factor making this
approximation useful in statistical inference is that the observed information
can be readily computed and is consistent. In many situations, consistency of
the observed information follows directly from the law of large numbers and
consistency of MLE.
Asymptotic normality of the MLE of regular parameters continues to hold
in many semiparametric and nonparametric models. See, for example, Chen
(1988), Chen (1995), Geskus and Groeneboom (1996), Gill (1989), Wong and
Severini (1991), Groeneboom and Wellner (1992), Severini and Wong (1992),
Gu and Zhang (1993), Murphy (1995), Murphy et al. (1997), van der Vaart
(1996), Huang (1996), Huang and Rossini (1997), and Wellner and Zhang
(2007), among many others. The general semiparametric theory and many im-
portant models are systematically studied in the topic by Bickel et al. (1993).
Two recent reviews of the state of the art in semiparametric inference can be
found in Fan and Li (2006) and Wellner et al. (2006).
In many of these examples, the MLE or a smooth functional of the MLE
is asymptotically normal with variance equal to the inverse of the ecient
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