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 kernel, 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).