On adaptive wavelet estimation of a class of weighted densities

TL;DR

This paper introduces an adaptive wavelet-based estimator for a class of weighted densities, achieving fast convergence rates under various risk measures, with theoretical guarantees and simulation validation.

Contribution

It presents a novel adaptive non-parametric estimator for weighted densities using wavelets, applicable to a broad class of models and risk measures.

Findings

Estimator attains fast convergence rates over Besov spaces.

Method performs well in simulations across different models.

Applicable to densities involving order statistics and extrema.

Abstract

We investigate the estimation of a weighted density taking the form $g=w(F)f$, where $f$ denotes an unknown density, $F$ the associated distribution function and $w$ is a known (non-negative) weight. Such a class encompasses many examples, including those arising in order statistics or when $g$ is related to the maximum or the minimum of $N$ (random or fixed) independent and identically distributed (\iid) random variables. We here construct a new adaptive non-parametric estimator for $g$ based on a plug-in approach and the wavelets methodology. For a wide class of models, we prove that it attains fast rates of convergence under the $\mathbb{L}_p$ risk with $p\ge 1$ (not only for $p = 2$ corresponding to the mean integrated squared error) over Besov balls. The theoretical findings are illustrated through several simulations.