Inference for mixtures of symmetric distributions

TL;DR

This paper addresses parameter estimation in finite mixtures of symmetric distributions, introducing a new identifiability concept and a distance-based estimator that outperforms traditional methods, especially with heavy-tailed components.

Contribution

It develops the concept of k-identifiability for symmetric mixture models and proposes a novel, consistent, and asymptotically normal distance-based estimation method.

Findings

The estimator generalizes the Hodges--Lehmann estimator.

It outperforms maximum likelihood in heavy-tailed scenarios.

Provides conditions for k-identifiability when k=2 or 3.

Abstract

This article discusses the problem of estimation of parameters in finite mixtures when the mixture components are assumed to be symmetric and to come from the same location family. We refer to these mixtures as semi-parametric because no additional assumptions other than symmetry are made regarding the parametric form of the component distributions. Because the class of symmetric distributions is so broad, identifiability of parameters is a major issue in these mixtures. We develop a notion of identifiability of finite mixture models, which we call k-identifiability, where k denotes the number of components in the mixture. We give sufficient conditions for k-identifiability of location mixtures of symmetric components when k=2 or 3. We propose a novel distance-based method for estimating the (location and mixing) parameters from a k-identifiable model and establish the strong consistency and asymptotic normality of the estimator. In the specific case of L_2-distance, we show that our estimator generalizes the Hodges--Lehmann estimator. We discuss the numerical implementation of these procedures, along with an empirical estimate of the component distribution, in the two-component case. In comparisons with maximum likelihood estimation assuming normal components, our method produces somewhat higher standard error estimates in the case where the components are truly normal, but dramatically outperforms the normal method when the components are heavy-tailed.