Strongly consistent model selection for densities
Gérard Biau, Benoît Cadre, Luc Devroye and Laszlo Gyorfi
Test Volume 17, Number 3, pp. 531-545, 2008. ISSN 1133-0686 (Print) 1863-8260 (Online)

Abstract

Let f be an unknown multivariate density belonging to a set of densities F_k* of finite associated Vapnik-Chervonenkis dimension, where the complexity k* is unknown, and F_k subset of F_{k+1} for all k. Given an i.i.d. sample of size n drawn from f, this article presents a density estimate \hat{f}_{K_n} yielding almost sure convergence of the estimated complexity K_n to the true but unknown k* and with the property E{\int|\hat{f}_{K_n}-f|} = O(1/\sqrt{n}). The methodology is inspired by the combinatorial tools developed in Devroye and Lugosi (Combinatorial methods in density estimation. Springer, New York, 2001) and it includes a wide range of density models, such as mixture models and exponential families.

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