Cross -Validation of Multivariate Densities

In recent years, the focus of study in smoothing parameter selection for kernel density estimation has been on the univariate case, while multivariate kernel density estimation has been largely neglected. In part, this may be due to the perception that calibrating multivariate densities is substantially more diicult. In this paper, we explicitly derive and compare multivariate versions of the bootstrap method of Taylor (1989), the least-squares cross-validation method developed by Bowman (1984) and Rudemo (1982), and a biased cross-validation method similar to that of Scott and Terrell (1987) for mul-tivariate kernel estimation using the product kernel estimator. The theoretical behavior of these cross-validation algorithms is shown to improve (surprisingly) as the dimension increases, approaching the best rate of O(n ?1=2). Simulation studies suggest that the new biased cross-validation method performs quite well and with reasonable variability as compared to the other two methods. Bivariate examples with heart disease and ozone data are given to illustrate the behavior of these algorithms.