Inference for mixtures of symmetric distributions

By David R. Hunter, Shaoli Wang, Thomas P. Hettmansperger. 2007

126 citations recorded by the source corpus.

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 semiparametric 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.

Subfields: Artificial Intelligence, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/5ab16225a2597b93a5c7ef9863d6bf2a6fcb48d2

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Citations in the corpus

mixtools: An R Package for Analyzing Mixture Models (2009, methodology): “Both Hunter et al. (2007) and Bordes, Mottelet, and Vandekerkhove (2006) study a particular case of model (1) in which x is univariate and where φ ( · ) is a density that is assumed to be completely unspeciﬁed except that it is symmetric about zero.”
SEMIPARAMETRIC ESTIMATION OF A TWO-COMPONENT MIXTURE MODEL (2006, background): “There is an extensive literature on this subject, including the monographs of Everitt and Hand [16], Titterington, Smith and Makov [40], McLachlan and Basford [28] and McLachlan and Peel [29].”
Compared to what? Variation in the impacts of early childhood education by alternative care type (2016, result): “Similar results hold for a broad class of parametric models (Frühwirth-Schnatter, 2006) and for distributions with shape restrictions, such as symmetry (Bordes et al., 2006; Hunter et al., 2007).”
Semi-Parametric Estimation for Conditional Independence Multivariate Finite Mixture Models (2015, methodology): “[25] simultaneously propose a location-shift semiparametric mixture model gθ(x) = m”
Estimation of a two‐component mixture model with applications to multiple testing (2012, background): “as been discussed in many papers, but generally with parametric assumptions on the model. Genovese and Wasserman (2004) discuss identiﬁability when Fb is the uniform distribution and F has a density. Hunter et al. (2007) and Bordes et al. (2006) discuss identiﬁability for location shift mixtures of ”
Semiparametric mixtures of regressions (2012, background): “It builds on work on location mixtures of an unspecified symmetric distribution, as introduced by Bordes, Mottelet, and Vanderkerkhove (2006) and Hunter, Wang, and Hettmansperger (2007).”