# Consistent Kernel Density Estimation with Non-Vanishing Bandwidth

## TL;DR

This paper introduces a fixed-bandwidth kernel density estimator (fbKDE) that remains consistent for estimating continuous densities, challenging the traditional requirement of bandwidth shrinking with sample size, and demonstrates its effectiveness through theoretical analysis and experiments.

## Contribution

The paper proposes the fbKDE, a novel fixed-bandwidth estimator that guarantees consistency and provides convergence rates, expanding the scope of kernel density estimation methods.

## Key findings

- fbKDE is consistent for continuous square-integrable densities.
- Provides convergence rates for radial and box kernels.
- Experimental results favor fbKDE over standard and variable bandwidth KDEs.

## Abstract

Consistency of the kernel density estimator requires that the kernel bandwidth tends to zero as the sample size grows. In this paper we investigate the question of whether consistency is possible when the bandwidth is fixed, if we consider a more general class of weighted KDEs. To answer this question in the affirmative, we introduce the fixed-bandwidth KDE (fbKDE), obtained by solving a quadratic program, and prove that it consistently estimates any continuous square-integrable density. We also establish rates of convergence for the fbKDE with radial kernels and the box kernel under appropriate smoothness assumptions. Furthermore, in an experimental study we demonstrate that the fbKDE compares favorably to the standard KDE and the previously proposed variable bandwidth KDE.