Semiparametric Mixtures of Generalized Exponential Families
A semiparametric mixture model is characterized by a non-parametric mixing distribution <b><scriptface>Q</scriptface></b> (with respect to a parameter <b>"θ"</b>) and a structural parameter <b>"β"</b> common to all components. Much of the literature on mixture models has focused on fixing <b>"β"</b> and estimating <scriptface>Q</scriptface>. However, this can lead to inconsistent estimation of both <scriptface>Q</scriptface> and the order of the model "m". Creating a framework for consistent estimation remains an open problem and is the focus of this article. We formulate a class of generalized exponential family (GEF) models and establish sufficient conditions for the identifiability of finite mixtures formed from a GEF along with sufficient conditions for a nesting structure. Finite identifiability and nesting structure lead to the central result that semiparametric maximum likelihood estimation of <scriptface>Q</scriptface> and <b>"β"</b> fails. However, consistent estimation is possible if we restrict the class of mixing distributions and employ an information-theoretic approach. This article provides a foundation for inference in semiparametric mixture models, in which GEFs and their structural properties play an instrumental role. Copyright 2007 Board of the Foundation of the Scandinavian Journal of Statistics..
Year of publication: |
2007
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Authors: | CHARNIGO, RICHARD ; PILLA, RAMANI S. |
Published in: |
Scandinavian Journal of Statistics. - Danish Society for Theoretical Statistics, ISSN 0303-6898. - Vol. 34.2007, 3, p. 535-551
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Publisher: |
Danish Society for Theoretical Statistics Finnish Statistical Society Norwegian Statistical Association Swedish Statistical Association |
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