Mixing Strategies for Density Estimation

General results on adaptive density estimation are obtained with respect to any countable collection of estimation strategies under Kullback-Leibler and squared L₂ losses. It is shown that without knowing which strategy works best for the underlying density, a single strategy can be constructed by mixing the proposed ones to be adaptive in terms of statistical risks. A consequence is that under some mild conditions, an asymptotically minimax rate adaptive estimator exists for a given countable collection of density classes; that is, a single estimator can be constructed to be simultaneously minim ax-rate optimal for all the function classes being considered. A demonstration is given for high-dimensional density estimation on [0, 1]^d where the constructed estimator adapts to smoothness and interaction-order over some piecewise Besov classes and is consistent for all the densities with finite entropy.