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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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##### References:
 [1] Barndorff-Nielsen, O. (1978). Information and Exponential Families in Statistical Theory. Wiley, New York. · Zbl 0387.62011 [2] Brown, L. D. (1986). Fundamentals of Statistical Exponential Families. IMS Lecture Notes Monogr. Ser. 9 . Hayward, CA. · Zbl 0685.62002 [3] Chentsov, N. N. (1982). Statistical Decision Rules and Optimal Inference. Amer. Math. Soc., Providence, RI. · Zbl 0484.62008 [4] Csiszár, I. (1984). Sanov property, generalized \emphI-projections, and a conditional limit theorem. Ann. Probab. 12 768–793. JSTOR: · Zbl 0544.60011 · doi:10.1214/aop/1176993227 · links.jstor.org [5] Csiszár, I. and Matúš, F. (2000). Information projections revisited. In Proc. 2000 IEEE International Symposium on Information Theory 490. IEEE, New York. · Zbl 1063.94016 [6] Csiszár, I. and Matúš, F. (2001). Convex cores of measures on $$\R^d$$. Studia Sci. Math. Hungar. 38 177–190. · Zbl 0997.28002 · doi:10.1556/SScMath.38.2001.1-4.12 [7] Csiszár, I. and Matúš, F. (2003). Information projections revisited. IEEE Trans. Inform. Theory 49 1474–1490. · Zbl 1063.94016 · doi:10.1109/TIT.2003.810633 [8] Csiszár, I. and Matúš, F. (2004). On information closures of exponential families: A counterexample. IEEE Trans. Inform. Theory 50 922–924. · Zbl 1284.94029 · doi:10.1109/TIT.2004.826661 [9] Harremoës, P. (2002). The information topology. In Proc. 2002 IEEE International Symposium on Information Theory 431. IEEE, New York. [10] Hiriart-Urruty, J.-B. and Lemaréchal, C. (2001). Fundamentals of Convex Analysis. Springer, Berlin. · Zbl 0998.49001 [11] Letac, G. (1992). Lectures on Natural Exponential Families and Their Variance Functions. Monografias de Matemática 50 . Instituto de Matemática Pura e Aplicada, Rio de Janeiro, Brazil. · Zbl 0983.62501 [12] Rockafellar, R. T. (1970). Convex Analysis. Princeton Univ. Press. · Zbl 0193.18401
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