Robust estimation of mixing measures in finite mixture models

TL;DR

This paper introduces robust estimators for mixing measures in finite mixture models that remain stable and accurate even when the kernel functions are misspecified, achieving optimal convergence rates.

Contribution

It proposes flexible estimators inspired by Hellinger distance and model selection, ensuring consistency and optimal convergence under kernel misspecification.

Findings

Estimators consistently recover the true number of components.

Achieve optimal convergence rates in both well- and mis-specified kernel settings.

Validated through simulations with synthetic and real data.

Abstract

In finite mixture models, apart from underlying mixing measure, true kernel density function of each subpopulation in the data is, in many scenarios, unknown. Perhaps the most popular approach is to choose some kernel functions that we empirically believe our data are generated from and use these kernels to fit our models. Nevertheless, as long as the chosen kernel and the true kernel are different, statistical inference of mixing measure under this setting will be highly unstable. To overcome this challenge, we propose flexible and efficient robust estimators of the mixing measure in these models, which are inspired by the idea of minimum Hellinger distance estimator, model selection criteria, and superefficiency phenomenon. We demonstrate that our estimators consistently recover the true number of components and achieve the optimal convergence rates of parameter estimation under both the well- and mis-specified kernel settings for any fixed bandwidth. These desirable asymptotic properties are illustrated via careful simulation studies with both synthetic and real data.