# Averaging of density kernel estimators

## TL;DR

This paper introduces a new averaging method for density kernel estimators that combines multiple estimators to improve accuracy, achieving asymptotic efficiency and outperforming existing procedures in numerical tests.

## Contribution

It proposes a computationally efficient averaging procedure for kernel density estimators that asymptotically matches the best possible combination (oracle) and enhances performance.

## Key findings

- The averaging method improves mean integrated square errors in numerical experiments.
- The estimator is asymptotically as efficient as the oracle.
- The approach outperforms existing procedures in simulations.

## Abstract

Averaging provides an alternative to bandwidth selection for density kernel estimation. We propose a procedure to combine linearly several kernel estimators of a density obtained from different, possibly data-driven, bandwidths. The method relies on minimizing an easily tractable approximation of the integrated square error of the combination. It provides, at a small computational cost, a final solution that improves on the initial estimators in most cases. The average estimator is proved to be asymptotically as efficient as the best possible combination (the oracle), with an error term that decreases faster than the minimax rate obtained with separated learning and validation samples. The performances are tested numerically, with results that compare favorably to other existing procedures in terms of mean integrated square errors.