Cross Validation Bandwidth Selection for Derivatives of Multidimensional Densities

Little attention has been given to the effect of higher order kernels for bandwidth selection for multidimensional derivatives of densities. This paper investigates the extension of cross validation methods to higher dimensions for the derivative of an unconditional joint density. I present and derive different cross validation criteria for arbitrary kernel order and density dimension, and show consistency of the estimator. Doing a Monte Carlo simulation study for various orders of kernels in the Gaussian family and additionally comparing a weighted integrated square error criterion, I find that higher order kernels become increasingly important as the dimension of the distribution increases. I find that standard cross validation selectors generally outperform the weighted integrated square error cross validation criteria. Using the infinite order Dirichlet kernel tends to have the best results.