by Peter Green (Bristol, UK); article to appear in the Encyclopaedia of Statistical Sciences, update volume.
Penalized likelihood is discussed in two rather different settings:
function estimation and model choice. The connection is that, in
both cases, a penalty term is added to the log-likelihood of the data,
before it is used in some standard inferential
criterion, in order to circumvent difficulties related to
dimensionality. In function estimation,
the problem is that the object of inference is infinite-dimensional,
while in model choice, it is a general difficulty with comparison
between models of differing dimension.
Penalized maximum likelihood estimation in nonparametric regression
and density estimation were reviewed
by Silverman in volume 6 of the Encyclopaedia of
this is an update and extension of that article;
there is additional material especially in respect of aspects of
penalized likelihood in nonlinear regression of various kinds,
in inverse problems, and in model choice.
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