Fast nonparametric near-maximum likelihood estimation of a mixing density

TL;DR

This paper proves the convergence of a new algorithm for estimating smooth mixing densities in mixture models, providing a data-driven stopping rule and demonstrating its empirical effectiveness.

Contribution

It offers a rigorous proof for the convergence of a smoothing algorithm to the nonparametric maximum likelihood estimator and introduces a practical stopping rule.

Findings

The algorithm converges to the nonparametric MLE.

The proposed stopping rule produces smooth estimates.

Simulations show strong empirical performance.

Abstract

Mixture models are regularly used in density estimation applications, but the problem of estimating the mixing distribution remains a challenge. Nonparametric maximum likelihood produce estimates of the mixing distribution that are discrete, and these may be hard to interpret when the true mixing distribution is believed to have a smooth density. In this paper, we investigate an algorithm that produces a sequence of smooth estimates that has been conjectured to converge to the nonparametric maximum likelihood estimator. Here we give a rigorous proof of this conjecture, and propose a new data-driven stopping rule that produces smooth near-maximum likelihood estimates of the mixing density, and simulations demonstrate the quality empirical performance of this estimator.