Adaptation to inhomogeneous smoothness for densities with irregularities

TL;DR

This paper develops an adaptive kernel density estimator with variable bandwidth to effectively estimate densities with irregularities, achieving faster convergence rates and automatic bandwidth selection without prior knowledge of irregularity points.

Contribution

It introduces a novel adaptive kernel estimator with non-constant bandwidth near irregularities, improving estimation rates and providing a data-driven bandwidth selection method.

Findings

Estimator attains faster rates than fixed bandwidth methods.

Lower bound results confirm the optimality of the rates.

Numerical illustrations demonstrate practical effectiveness.

Abstract

We estimate on a compact interval densities with isolated irregularities, such as discontinuities or discontinuities in some derivatives. From independent and identically distributed observations we construct a kernel estimator with non-constant bandwidth, in particular in the vicinity of irregularities. It attains faster rates, for the risk $L_{p}, p\geq 1$, than usual estimators with a fixed global bandwidth. Optimality of the rate found is established by a lower bound result. We then propose an adaptive method inspired by Lepski's method for automatically selecting the variable bandwidth, without any knowledge of the regularity of the density nor of the points where the regularity breaks down. The procedure is illustrated numerically on examples.