Strongly consistent model selection for densities ## AbstractLet 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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