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Generalized maximum likelihood estimates for exponential families. (English) Zbl 1133.62039
Summary: For a standard full exponential family on \(\mathbb R^d\), or its canonically convex subfamily, the generalized maximum likelihood estimator is an extension of the mapping that assigns to the mean \(a\in\mathbb R^d\) of a sample for which a maximizer \(\vartheta^*\) of the corresponding likelihood function exists, the member of the family parameterized by \(\vartheta^*\). This extension assigns to each \(a\in\mathbb R^d\) with the likelihood function bounded above a member of the closure of the family in variation distance. Its detailed description, complete characterization of domain and range, and additional results are presented, not imposing any regularity assumptions. In addition to basic convex analysis tools, the authors’ prior results on convex cores of measures and closures of exponential families are used.

MSC:
62H12 Estimation in multivariate analysis
62H05 Characterization and structure theory for multivariate probability distributions; copulas
60A10 Probabilistic measure theory
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[1] Barndorff-Nielsen O. (1978). Information and Exponential Families in Statistical Theory. Wiley, New York · Zbl 0387.62011
[2] Brown, L.D.: Fundamentals of Statistical Exponential Families. Inst. of Math. Statist. Lecture Notes–Monograph Series, Vol. 9 (1986) · Zbl 0685.62002
[3] Chentsov, N.N.: Statistical Decision Rules and Optimal Inference. Translations of Mathematical Monographs, Amer. Math. Soc., Providence–Rhode Island, 1982 (Russian original: Nauka, Moscow, 1972)
[4] Csiszár I. and Matúš F. (2001). Convex cores of measures on \(\mathbb R^d\) . Studia Sci. Math. Hungar. 38: 177–190 · Zbl 0997.28002
[5] Csiszár, I., Matúš, F.: Information closure of exponential families and generalized maximum likelihood estimates. In: Proc. 2002 IEEE Int. Symp. Inform. Theory, p. 434 (2002)
[6] 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
[7] 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
[8] Csiszár I. and Matúš F. (2005). Closures of exponential families. Ann. Probab. 33: 582–600 · Zbl 1068.60008 · doi:10.1214/009117904000000766
[9] Csiszár, I., Matúš, F.: Generalized maximum likelihood estimates for infinite dimensional exponential families. In: Proceedings Prague Stochastics 2006. Prague, Czech Republic, pp. 288–297 (2006)
[10] Eriksson N., Fienberg S.E., Rinaldo A. and Sullivant S. (2006). Polyhedral conditions for the nonexistence of the MLE for hierarchical log-linear models. J. Symbol. Comput. 41: 222–233 · Zbl 1120.62015 · doi:10.1016/j.jsc.2005.04.003
[11] Letac, G.: Lectures on Natural Exponential Families and their Variance Functions. Monografias de Matemática 50. Instituto de Matemática Pura e Aplicada, Rio de Janeiro (1992) · Zbl 0983.62501
[12] Lauritzen S.L. (1996). Graphical Models. Clarendon, Oxford · Zbl 0907.62001
[13] Rinaldo, A.: On maximum likelihood estimation in log-linear models. Technical Report 833, Department of Statistics, Carnegie Mellon University (2006)
[14] Rinaldo, A.: Computing maximum likelihood estimates in log-linear models. Technical Report 835. Department of Statistics, Carnegie Mellon University (2006)
[15] Rockafellar R.T. (1970). Convex Analysis. Princeton University Press, Princeton · Zbl 0193.18401
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