Convergence properties of an empirical error criterion for multivariate density estimation (Q1821451)

For the purpose of comparing different nonparametric density estimators, \textit{E. J. Wegman} [J. Stat. Comput. Simul. 1, 225-245 (1972; Zbl 0243.62029)] introduced an empirical error criterion. In a recent paper by \textit{P. Hall} [Stochastic Processes Appl. 13, 11-25 (1982; Zbl 0486.60022)] 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 \textit{J. Komlós, P. Major} and \textit{G. Tusnády} [Z. Wahrscheinlichkeitstheor. Verw. Geb. 32, 111-131 (1975; Zbl 0308.60029)] does not always give the strongest results possible.