Semiparametric mixtures of generalized exponential families

A semiparametric mixture model is characterized by a non-parametric mixing distribution Q (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 Q. However, this can lead to inconsistent estimation of both Q 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 Q 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.