On the usefulness of Meyer wavelets for deconvolution and density estimation

TL;DR

This paper demonstrates that Meyer wavelets are highly effective for density estimation and deconvolution, offering computational simplicity and near-optimal convergence rates, with practical benefits shown through simulations.

Contribution

It introduces a wavelet-based estimator using Meyer wavelets that simplifies computation and achieves near-minimax rates for density estimation and deconvolution.

Findings

Estimator achieves near-minimax convergence rates.

Computational simplicity due to Fourier transform and band-limited properties.

Good finite sample performance demonstrated through simulations.

Abstract

The aim of this paper is to show the usefulness of Meyer wavelets for the classical problem of density estimation and for density deconvolution from noisy observations. By using such wavelets, the computation of the empirical wavelet coefficients relies on the fast Fourier transform of the data and on the fact that Meyer wavelets are band-limited functions. This makes such estimators very simple to compute and this avoids the problem of evaluating wavelets at non-dyadic points which is the main drawback of classical wavelet-based density estimators. Our approach is based on term-by-term thresholding of the empirical wavelet coefficients with random thresholds depending on an estimation of the variance of each coefficient. Such estimators are shown to achieve the same performances of an oracle estimator up to a logarithmic term. These estimators also achieve near-minimax rates of convergence over a large class of Besov spaces. A simulation study is proposed to show the good finite sample performances of the estimator for both problems of direct density estimation and density deconvolution.