Distribution estimation consistent in total variation and in twotypes of information divergence
Barron, A.R.; Gyorfi, L.; van der Meulen, E.C.
Information Theory, IEEE Transactions on
Volume 38, Issue 5, Sep 1992 Page(s):1437 - 1454
Digital Object Identifier 10.1109/18.149496
Summary:The problem of the nonparametric estimation of a probability
distribution is considered from three viewpoints: the consistency in
total variation, the consistency in information divergence, and
consistency in reversed-order information divergence. These types of
consistencies are relatively strong criteria of convergence, and a
probability distribution cannot be consistently estimated in either type
of convergence without any restrictions on the class of probability
distributions allowed. Histogram-based estimators of distribution are
presented which, under certain conditions, converge in total variation,
in information divergence, and in reversed-order information divergence
to the unknown probability distribution. Some a priori information about
the true probability distribution is assumed in each case. As the
concept of consistency in information divergence is stronger than that
of convergence in total variation, additional assumptions are imposed in
the cases of informational divergences
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