Statistical minimax approach of the Hausdorff moment problem

TL;DR

This paper introduces a statistical method for estimating probability densities from noisy moment observations, achieving optimal convergence rates in the Hausdorff moment problem.

Contribution

It provides the first minimax optimal estimator for the Hausdorff moment problem under noisy observations, with proven bounds on convergence rates.

Findings

Estimator attains minimax rate of convergence.

Provides upper and lower bounds on estimation error.

Validates the approach through theoretical analysis.

Abstract

The purpose of this paper is to study the problem of estimating a compactly supported density of probability from noisy observations of its moments. In fact, we provide a statistical approach to the famous Hausdorff classical moment problem. We prove an upper bound and a lower bound on the rate of convergence of the mean squared error showing that the considered estimator attains minimax rate over the corresponding smoothness classes.