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.
"An Asymptotically Optimal Window Selection Rule for Kernel Density Estimates." Ann. Statist. 12 (4) 1285 - 1297, December, 1984. https://doi.org/10.1214/aos/1176346792