On the estimation of smooth densities by strict probability densities at optimal rates in sup-norm

TL;DR

This paper demonstrates that certain variable bandwidth density estimators can achieve optimal convergence rates in the supremum norm for smooth densities, ensuring they are strict probability densities.

Contribution

It proves that specific variable bandwidth estimators attain minimax rates in sup-norm for smooth densities, extending previous results to strict probability densities.

Findings

Estimators achieve minimax rates in sup-norm for densities in $C^4$ and $C^6$.

Estimates are strict probability densities.

Results apply over bounded sets where preliminary estimates are bounded away from zero.

Abstract

It is shown that the variable bandwidth density estimator proposed by McKay (1993a and b) following earlier findings by Abramson (1982) approximates density functions in $C^4(\mathbb R^d)$ at the minimax rate in the supremum norm over bounded sets where the preliminary density estimates on which they are based are bounded away from zero. A somewhat more complicated estimator proposed by Jones McKay and Hu (1994) to approximate densities in $C^6(\mathbb R)$ is also shown to attain minimax rates in sup norm over the same kind of sets. These estimators are strict probability densities.