Revisiting consistency of a recursive estimator of mixing distributions

TL;DR

This paper improves the theoretical understanding of predictive recursion (PR) for estimating mixing distributions, demonstrating its consistency in scale mixture models of uniforms and showing its practical effectiveness in monotone density estimation.

Contribution

The paper establishes new, weaker consistency conditions for PR and proves its effectiveness for scale mixture of uniforms, enhancing its theoretical foundation and practical applicability.

Findings

PR is consistent for scale mixtures of uniforms.

PR mixture density estimator performs well in monotone density estimation.

New weaker conditions for PR consistency are developed.

Abstract

Estimation of the mixing distribution under a general mixture model is a very difficult problem, especially when the mixing distribution is assumed to have a density. Predictive recursion (PR) is a fast, recursive algorithm for nonparametric estimation of a mixing distribution/density in general mixture models. However, the existing PR consistency results make rather strong assumptions, some of which fail for a class of mixture models relevant for monotone density estimation, namely, scale mixtures of uniform kernels. In this paper, we develop new consistency results for PR under weaker conditions. Armed with this new theory, we prove that PR is consistent for the scale mixture of uniforms problem, and we show that the corresponding PR mixture density estimator has very good practical performance compared to several existing methods for monotone density estimation.