FFT-Based Fast Bandwidth Selector for Multivariate Kernel Density Estimation

TL;DR

This paper introduces a generalized FFT-based method for fast and accurate multivariate kernel density estimation bandwidth selection, overcoming previous limitations to diagonal matrices and enabling derivative estimation with broad practical applications.

Contribution

It presents a versatile FFT-based approach that relaxes previous restrictions to diagonal bandwidth matrices, supporting general multivariate KDE and derivative computations.

Findings

Significantly reduces computational cost of bandwidth selection.

Supports non-diagonal bandwidth matrices for multivariate KDE.

Demonstrates effectiveness through extensive numerical simulations.

Abstract

The performance of multivariate kernel density estimation (KDE) depends strongly on the choice of bandwidth matrix. The high computational cost required for its estimation provides a big motivation to develop fast and accurate methods. One of such methods is based on the Fast Fourier Transform. However, the currently available implementation works very well only for the univariate KDE and it's multivariate extension suffers from a very serious limitation as it can accurately operate only with diagonal bandwidth matrices. A more general solution is presented where the above mentioned limitation is relaxed. Moreover, the presented solution can by easily adopted also for the task of efficient computation of integrated density derivative functionals involving an arbitrary derivative order. Consequently, bandwidth selection for kernel density derivative estimation is also supported. The practical usability of the new solution is demonstrated by comprehensive numerical simulations.