Generalized maximum likelihood estimation of the mean of parameters of mixtures, with applications to sampling

TL;DR

This paper develops a generalized maximum likelihood estimation approach for the mean of parameters in mixture models, providing theoretical insights, connections to empirical Bayes, and demonstrating practical applications and performance in sampling problems.

Contribution

It introduces a new perspective on GMLE for mixture models, linking it to weak convergence and empirical Bayes, with applications to sampling and real data analysis.

Findings

GMLE has desirable properties and representations.

The approach relates to empirical Bayes estimation.

Demonstrated effectiveness through simulations and real data.

Abstract

Let $f(y|\theta), \; \theta \in \Omega$ be a parametric family, $\eta(\theta)$ a given function, and $G$ an unknown mixing distribution. It is desired to estimate $E_G (\eta(\theta))\equiv \eta_G$ based on independent observations $Y_1,...,Y_n$, where $Y_i \sim f(y|\theta_i)$, and $\theta_i \sim G$ are iid. We explore the Generalized Maximum Likelihood Estimators (GMLE) for this problem. Some basic properties and representations of those estimators are shown. In particular we suggest a new perspective, of the weak convergence result by Kiefer and Wolfowitz (1956), with implications to a corresponding setup in which $\theta_1,...,\theta_n$ are {\it fixed} parameters. We also relate the above problem, of estimating $\eta_G$, to non-parametric empirical Bayes estimation under a squared loss. Applications of GMLE to sampling problems are presented. The performance of the GMLE is demonstrated both in simulations and through a real data example.