Convergence properties of an empirical error criterion for multivariate density estimation

For the purpose of comparing different nonparametric density estimators, Wegman (J. Statist. Comput. Simulation 1 225-245) introduced an empirical error criterion. In a recent paper by Hall (Stochastic Process. Appl. 13 11-25) it is shown that this empirical error criterion converges to the mean integrated square error. Here, in the case of kernel estimation, the results of Hall are improved in several ways, most notably multivariate densities are treated and the range of allowable bandwidths is extended. The techniques used here are quite different from those of Hall, which demonstrates that the elegant Brownian Bridge approximation of Komlós, Major, and Tusnády (Z. Warsch. Verw. Gebrete 32 111-131) does not always give the strongest results possible.