Cross-validated density estimates based on Kullback–Leibler information

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.