FFT-Based Algorithms for Kernel Density Estimation and Bandwidth Selection

Abstract

This chapter is devoted to presentation of the results of author’s own research on fast computation of kernel density estimators and bandwidth selection. The developed methods are based on the fast Fourier transform (FFT) algorithm that relies on a preliminary data transformation known as data binning. We start with the description of binning rules, both for the univariate and multivariate cases, and provide the reader with several numerical examples. The main aim of this chapter is to describe a complete derivation of the FFT-based method for KDE bandwidth selection. First, some important limitations of the existing solution are emphasized and crucial modifications are proposed. Then, the next step shows how the FFT-based method can be used for very fast KDE computations, as well as for bandwidth selection. Following that, we present a detailed derivation of the least square cross validation bandwidth selector using the FFT-based method. We also discuss the use of this method for plug-in and smoothed cross validation bandwidth selectors. The final part is devoted to an overview of extended computer simulations confirming high performance and accuracy levels of the FFT-based method for KDE and bandwidth selection.