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A multivariate modified histogram density estimate depending on a reference density g and a partition P has been proved to have good consistency properties according to several information theoretic criteria. Given an i.i.d. sample, we show how to select automatically both g and P so that the expected L 1 error of the corresponding selected estimate is within a given constant multiple of the best possible error plus an additive term which tends to zero under mild assumptions. Our method is inspired by the combinatorial tools developed by Devroye and Lugosi [Devroye, L. and Lugosi, G., 2001, Combinatorial Methods in Density Estimation (New York, NY: Springer–Verlag)] and it includes a wide range of reference density and partition models. Results of simulations are also presented. [ABSTRACT FROM AUTHOR]

The convergence of measure estimates in the sense of Kullback-Leibler divergence is required in many applications in decision and information theory. Recently, modified histograms have been shown to have good properties with respect to information divergences. For these estimates deterministic optimal bandwidths have been given, but no automatic smoothing procedure has been shown to be asymptotically optimal. In the present article, we consider the Kullback-Leibler cross-validation method for selecting the bin width of modified histograms. We analyze the behavior of the Kullback-Leibler divergence and of its expectation and prove that the cross-validated estimate is asymptotically optimal with respect to the Kullback-Leibler divergence. [ABSTRACT FROM AUTHOR]