Upper Bounds for the I-MSE and max-MSE of Kernel Density Estimators

TL;DR

This paper derives finite-sample upper bounds for the mean squared error of kernel density estimators, including non-traditional kernels like sinc, providing insights into their performance beyond asymptotic analysis.

Contribution

It introduces finite-sample upper bounds for I-MSE and max-MSE of kernel density estimators, including for non-integrable kernels like sinc, under various assumptions.

Findings

Bounds are established for traditional kernels under different assumptions.

Sinc kernel estimators can outperform conventional kernels in certain scenarios.

Methods involve characteristic functions and empirical characteristic functions.

Abstract

The performance of kernel density estimators is usually studied via Taylor expansions and asymptotic approximation arguments, in which the bandwidth parameter tends to zero with increasing sample size. In contrast, this paper focusses directly on the finite-sample situation. Informative upper bounds are derived both for the integrated and the maximal mean squared error function. Results are reached for the traditional case, where the kernel is a probability density function, under various sets of assumptions on the underlying density to be estimated. Results are also derived for the important non-conventional case of the sinc kernel, which is not integrable and also takes negative values. We pin-point ways in which the sinc-based estimator performs better than the conventional kernel estimators. When proving our results we rely on methods related to characteristic and empirical characteristic functions.