FFT-Based Fast Bandwidth Selector for Multivariate Kernel Density Estimation

There are two main computational problems related to the kernel density estimation (KDE): (a) fast evaluating of the kernel density estimates, (b) fast estimating of the optimal bandwidth. Progress towards the latter problem has been relatively slow. The high computational cost required for direct bandwidth estimation provides great motivation for development of fast and accurate methods. One of such method is based on the Fast Fourier Transform and works very well for the univariate KDE. Unfortunately, its multivariate extension suffers a very serious limitation as it can accurately operate only with the constrained (that is diagonal) bandwidth matrices. In this paper we present a complete solution where the above mentioned limitation is rectified. The practical usability of our method is demonstrated by comprehensive numerical simulations.