Kernel density estimators for discrete multivariate data are investigated, using the notation framework of contingency tables. We derive large sample properties of kernel estimators and the least-squares cross-validation method for choosing the bandwidth, including the asymptotic bias, the mean summed squared error, the actual summed squared error, and the asymptotic distribution of the resulting non-parametric estimator. We show that the least-squares cross-validation procedure is superior to Kullback-Leibler cross-validation in terms of mean summed squared error, but that the least-squares cross-validation is still sub-optimal concerning actual summed squared error.
