On the Bias and Variance of Finite Element Discretizations for FFT-Based Kernel Density Estimation

A computationally-efficient procedure for kernel density estimation using the FFT algorithm has been given by Silverman (\cite{silverman}), with extensions by Jones and Lotwick (\cite{jones2}). This procedure requires the empirical characteristic function to be interpolated on a regular mesh which leads to high-frequency approximation errors in this step. In the case of density estimation, these high-frequency errors are subsequently damped by the smoothing effect of the kernel multiplier. Nevertheless, these small errors can lead to a significant loss of accuracy whenever the kernel density estimates or its derivatives are used as a part of some larger statistical procedure. In this paper, we describe systematic finite element discretization procedures for improving the accuracy of the FFT-based algorithms. We derive the bias and variance of the kernel density estimates for the FFT-based algorithms, and this analysis suggests modifications to the computational procedure to obtain estimates free from interpolation bias. Simulation studies that verify the results of the analysis are presented. Finally, an application that motivated the study described in this paper is discussed.