Cross-validation in density estimation

Abstract

Considerable interest has been shown recently in the use of cross-validatory procedures. Much of this attention is due to the efforts of Stone (1974a, b; 1977a, b), who has provided a general discussion of the method of cross-validatory choice. Dawid (1974) suggested that, in certain circumstances, cross-validation might lead to inconsistency, and some of these fears were confirmed by Stone (1977b). In the present paper we examine a popular cross-validatory argument for selecting the smoothing parameter, or window size, in the nonparametric estimation of a density. We derive results describing the order of magnitude of the cross-validatory window, and show that the resulting estimator will perform suboptimally. Our results are confirmed by Monte Carlo trials. Alternative techniques, such as that proposed by Silverman (1978a), seem preferable. Habbema, Hermans & van den Broek (1974) suggested a method for discriminant analysis based on nonparametric density estimation. They considered kernel-type estimates, and in the univariate case these usually have the form