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Generalized Maximum Likelihood Estimates for Exponential Families

Tuesday, March 6, 2007 - 10:30am - 11:20am
EE/CS 3-180
František Matúš (Czech Academy of Sciences (AVČR))
Exponential families underpin numerous models of statistics
and information geometry that have significant applications.
For a standard full exponential family, or its canonically convex
subfamily, if the corresponding likelihood function from a sample
has a maximizer t* then, by the maximum likelihood principle, the
data are judged to be generated by the probability measure P* from
the family that is parameterized by t*. Since the likelihood depends
on data only through their mean, in this way the mean is mapped to
P*. In a joint work with Imre Csiszar, Budapest, we study an
extension of this mapping, the generalized maximum likelihood
estimator. It assigns to each point of the space at which the
likelihood function is bounded above, a probability measure from
the closure of the family in variation distance. A detailed
description, complete characterization of domain and range, and
additional results will be presented, not imposing any regularity
assumptions.