Open Access
February 2007 Inference for mixtures of symmetric distributions
David R. Hunter, Shaoli Wang, Thomas P. Hettmansperger
Ann. Statist. 35(1): 224-251 (February 2007). DOI: 10.1214/009053606000001118


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 L2-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.


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David R. Hunter. Shaoli Wang. Thomas P. Hettmansperger. "Inference for mixtures of symmetric distributions." Ann. Statist. 35 (1) 224 - 251, February 2007.


Published: February 2007
First available in Project Euclid: 6 June 2007

zbMATH: 1114.62035
MathSciNet: MR2332275
Digital Object Identifier: 10.1214/009053606000001118

Primary: 62G05

Keywords: Deconvolution , Hodges–Lehmann estimator , Identifiability , semi-parametric mixtures

Rights: Copyright © 2007 Institute of Mathematical Statistics

Vol.35 • No. 1 • February 2007
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