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 evaluation of the kernel density estimates, and (b) fast estimation of the optimal bandwidth. However, progress towards the latter problem has been rather relatively slow. The high computational cost required by direct bandwidth estimation provides a big motivation to develop fast and accurate methods. One of such methods is based on the Fast Fourier Transform and works very well for the univariate KDE. Unfortunately, its multivariate extension suffers from a very serious limitation as it can accurately operate only with the very specific (i.e., diagonal) bandwidth matrices. In this paper we present a more general solution where the above mentioned limitation is relaxed. The practical usability of our method is demonstrated by comprehensive numerical simulations.