A note on kernel density estimation at a parametric rate

TL;DR

This paper characterizes kernels that achieve parametric convergence rates in kernel density estimation and proposes an asymptotic bandwidth selection method to optimize estimator performance.

Contribution

It provides a characterization of kernels that attain the parametric rate and introduces a bandwidth choice for consistent estimation at that rate.

Findings

Superkernel estimator outperforms traditional methods with proper bandwidth.

Asymptotic bandwidth selection achieves the $n^{-1}$ mean integrated squared error rate.

Theoretical characterization guides kernel choice for optimal density estimation.

Abstract

In the context of kernel density estimation, we give a characterization of the kernels for which the parametric mean integrated squared error rate $n^{-1}$ may be obtained, where $n$ is the sample size. Also, for the cases where this rate is attainable, we give an asymptotic bandwidth choice that makes the kernel estimator consistent in mean integrated squared error at that rate and a numerical example showing the superior performance of the superkernel estimator when the bandwidth is properly chosen.