A Comparison of Cross-Validation Techniques in Density Estimation

In the setting of nonparametric multivariate density estimation, theorems are established which allow a comparison of the Kullback-Leibler and the least-squares cross-validation methods of smoothing parameter selection. The family of delta sequence estimators (including kemel, orthogonal series, histogram and histospline estimators) is considered. These theorems also show that either type of cross validation can be used to compare different estimators (e.g., kernel versus orthogonal series). 1. Introduction. Consider the problem of trying to estimate a d-dimensional probability density function, f(x), using a random sample, Xl,..., Xn, from f. Most proposed estimators of f depend on a "smoothing parameter," say X E DR +, whose selection is crucial to the performance of the estimator. In this paper, for the large class of delta sequence estimators, theorems are obtained which allow comparison of two smoothing parameter selectors which are known to be asymptotically optimal. An important consequence of these results is that either smoothing parameter selector may be used for a data based comparison of two density estimators, for example, kernel versus orthogonal series. Another attractive feature of these results is that they are set in a quite general framework, special cases of which provide simpler proofs of several recent asymptotic optimality results. In Sections 2 and 3 the family of delta sequence estimators and the smoothing parameter selectors are given. The theorems are stated in Section 4, with some remarks in Section 5. The rest of the paper consists of proofs.