A Non-Classical Parameterization for Density Estimation Using Sample Moments

TL;DR

This paper introduces a novel non-classical density estimation method using sample moments that avoids the dependency on feasible functions, leveraging the squared Hellinger distance and convex optimization for improved performance.

Contribution

It proposes the first density estimator that matches sample moments up to any even order without assuming the true density belongs to specific function classes.

Findings

The estimator is proven to exist and be unique.

Simulation results show superior performance compared to existing methods.

The method effectively matches sample moments up to arbitrary even orders.

Abstract

Probability density estimation is a core problem of statistics and signal processing. Moment methods are an important means of density estimation, but they are generally strongly dependent on the choice of feasible functions, which severely affects the performance. In this paper, we propose a non-classical parametrization for density estimation using sample moments, which does not require the choice of such functions. The parametrization is induced by the squared Hellinger distance, and the solution of it, which is proved to exist and be unique subject to a simple prior that does not depend on data, and can be obtained by convex optimization. Statistical properties of the density estimator, together with an asymptotic error upper bound are proposed for the estimator by power moments. Applications of the proposed density estimator in signal processing tasks are given. Simulation results validate the performance of the estimator by a comparison to several prevailing methods. To the best of our knowledge, the proposed estimator is the first one in the literature for which the power moments up to an arbitrary even order exactly match the sample moments, while the true density is not assumed to fall within specific function classes.