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Semiparametric Mixtures of Generalized Exponential Families
Author(s) -
CHARNIGO RICHARD,
PILLA RAMANI S.
Publication year - 2007
Publication title -
scandinavian journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.359
H-Index - 65
eISSN - 1467-9469
pISSN - 0303-6898
DOI - 10.1111/j.1467-9469.2006.00532.x
Subject(s) - identifiability , mathematics , exponential family , parametric statistics , semiparametric model , mixing (physics) , mixture model , inference , nesting (process) , parametric model , class (philosophy) , econometrics , exponential function , statistics , mathematical analysis , computer science , artificial intelligence , physics , materials science , quantum mechanics , metallurgy
. A semiparametric mixture model is characterized by a non‐parametric mixing distribution (with respect to a parameter θ ) and a structural parameter β common to all components. Much of the literature on mixture models has focused on fixing β and estimating . However, this can lead to inconsistent estimation of both 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 and β 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.