Bandwidth Selection for Multivariate Kernel Density Estimation Using MCMC

Kernel density estimation for multivariate data is an important technique that has a wide range of applications in econometrics and finance. However, it has received significantly less attention than its univariate counterpart. The lower level of interest in multivariate kernel density estimation is mainly due to the increased difficulty in deriving an optimal data-driven bandwidth as the dimension of data increases. We provide Markov chain Monte Carlo (MCMC) algorithms for estimating optimal bandwidth matrices for multivariate kernel density estimation. Our approach is based on treating the elements of the bandwidth matrix as parameters whose posterior density can be obtained through the likelihood cross-validation criterion. Numerical studies for bivariate data show that the MCMC algorithm generally performs better than the plug-in algorithm under the Kullback-Leibler information criterion, and is as good as the plug-in algorithm under the mean integrated squared errors (MISE) criterion. Numerical studies for 5 dimensional data show that our algorithm is superior to the normal reference rule. Our MCMC algorithm is the first data-driven bandwidth selector for kernel density estimation with more than two variables, and the sampling algorithm involves no increased difficulty as the dimension of data increase