Data-driven aggregation in non-parametric density estimation on the real line

TL;DR

This paper introduces a data-driven aggregation method for non-parametric density estimation on the real line, improving adaptivity and optimality by combining model selection with aggregation techniques.

Contribution

The paper proposes a novel data-driven aggregation approach that unifies model selection and aggregation, achieving oracle- and minimax-optimal density estimation.

Findings

The aggregated estimator is shown to be oracle- and minimax-optimal.

Simulation results demonstrate improved finite sample performance over traditional model selection.

The method adapts to unknown smoothness of the density.

Abstract

We study non-parametric estimation of an unknown density with support in R (respectively R+). The proposed estimation procedure is based on the projection on finite dimensional subspaces spanned by the Hermite (respectively the Laguerre) functions. The focus of this paper is to introduce a data-driven aggregation approach in order to deal with the upcoming bias-variance trade-off. Our novel procedure integrates the usual model selection method as a limit case. We show the oracle- and the minimax-optimality of the data-driven aggregated density estimator and hence its adaptivity. We present results of a simulation study which allow to compare the finite sample performance of the data-driven estimators using model selection compared to the new aggregation.