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Closures of exponential families. (English) Zbl 1068.60008
The paper considers exponential families E of probability measures (PMs) having both discrete and continuous components as a convex set of canonical parameters. The main purpose of this paper is to investigate, in the general case when all common regularity conditions are absent, the properties of three important closures for exponential families E: the variation (distance) closure (V-closure), the information (divergence) closure (I-closure), and reverse I-closure (rI-closure). The essential tools used are the newly defined notions of the convex core of a measure, the accessible faces of a convex set, and the extension of an exponential family E.
The paper is organized as follows: Section 2 introduces the necessary notions and preliminary results, Section 3 develops in detail the main properties of the three closures of E families, while Section 4 is devoted to the weak convergence in exponential families. The final Sections 5 and 6 expose comprehensive proofs of previously enclosed technical results. The investigated closures are of major relevance in large deviation theory, for a generalized maximum likelihood estimate, in canonical statistics etc.

MSC:
60A10 Probabilistic measure theory
60B10 Convergence of probability measures
62B10 Statistical aspects of information-theoretic topics
52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces)
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