An Asymptotically Optimal Window Selection Rule for Kernel Density Estimates

Abstract

Kernel estimates of an unknown multivariate density are investigated, with mild restrictions being placed on the kernel. A window selection rule is considered, which can be interpreted in terms of cross-validation. Under the mild assumption that the unknown density and its one-dimensional marginals are bounded, the rule is shown to be asymptotically optimal. This strengthens recent results of Peter Hall.