On limiting towards the boundaries of exponential families

František Matúš

Kybernetika (2015)

  • Volume: 51, Issue: 5, page 725-738
  • ISSN: 0023-5954

Abstract

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This work studies the standard exponential families of probability measures on Euclidean spaces that have finite supports. In such a family parameterized by means, the mean is supposed to move along a segment inside the convex support towards an endpoint on the boundary of the support. Limit behavior of several quantities related to the exponential family is described explicitly. In particular, the variance functions and information divergences are studied around the boundary.

How to cite

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Matúš, František. "On limiting towards the boundaries of exponential families." Kybernetika 51.5 (2015): 725-738. <http://eudml.org/doc/276062>.

@article{Matúš2015,
abstract = {This work studies the standard exponential families of probability measures on Euclidean spaces that have finite supports. In such a family parameterized by means, the mean is supposed to move along a segment inside the convex support towards an endpoint on the boundary of the support. Limit behavior of several quantities related to the exponential family is described explicitly. In particular, the variance functions and information divergences are studied around the boundary.},
author = {Matúš, František},
journal = {Kybernetika},
keywords = {exponential family; variance function; Kullback–Leibler divergence; relative entropy; information divergence; mean parametrization; convex support},
language = {eng},
number = {5},
pages = {725-738},
publisher = {Institute of Information Theory and Automation AS CR},
title = {On limiting towards the boundaries of exponential families},
url = {http://eudml.org/doc/276062},
volume = {51},
year = {2015},
}

TY - JOUR
AU - Matúš, František
TI - On limiting towards the boundaries of exponential families
JO - Kybernetika
PY - 2015
PB - Institute of Information Theory and Automation AS CR
VL - 51
IS - 5
SP - 725
EP - 738
AB - This work studies the standard exponential families of probability measures on Euclidean spaces that have finite supports. In such a family parameterized by means, the mean is supposed to move along a segment inside the convex support towards an endpoint on the boundary of the support. Limit behavior of several quantities related to the exponential family is described explicitly. In particular, the variance functions and information divergences are studied around the boundary.
LA - eng
KW - exponential family; variance function; Kullback–Leibler divergence; relative entropy; information divergence; mean parametrization; convex support
UR - http://eudml.org/doc/276062
ER -

References

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  1. Ay, N., 10.1214/aop/1020107773, The Annals of Probability 30 (2002), 416-436. Zbl1010.62007MR1894113DOI10.1214/aop/1020107773
  2. Barndorff-Nielsen, O., Information and Exponential Families in Statistical Theory., Wiley, New York 1978. Zbl1288.62007MR0489333
  3. Brown, L. D., Fundamentals of Statistical Exponential Families., Inst. of Math. Statist. Lecture Notes - Monograph Series 9 (1986). Zbl0685.62002MR0882001
  4. 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). Zbl0484.62008MR0645898
  5. Csiszár, I., Matúš, F., 10.1214/009117904000000766, The Annals of Probability 33 (2005), 582-600. Zbl1068.60008MR2123202DOI10.1214/009117904000000766
  6. Csiszár, I., Matúš, F., 10.1007/s00440-007-0084-z, Probability Theory and Related Fields 141 (2008), 213-246. Zbl1133.62039MR2372970DOI10.1007/s00440-007-0084-z
  7. Graham, R., Knuth, D., Patashnik, O., Concrete Mathematics. Second edition., Addison-Wesley, Reading, Massachusetts 1994, p. 446. MR1397498
  8. 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. Zbl0983.62501MR1182991
  9. Matúš, F., Ay, N., On maximization of the information divergence from an exponential family., In: Proc. WUPES'03 (J. Vejnarová, ed.), University of Economics, Prague 2003, pp. 99-204. 
  10. Matúš, F., Optimality conditions for maximizers of the divergence from an EF., Kybernetika 43 (2007), 731-746. MR2376334
  11. Matúš, F., 10.1109/tit.2009.2032806, IEEE Trans. Inform. Theory 55 (2009), 5375-5381. MR2597169DOI10.1109/tit.2009.2032806
  12. F., F.Matúš, Rauh, J., 10.1109/isit.2011.6034269, In: Proc. IEEE ISIT 2011, St. Petersburg 2011, pp. 809-813. DOI10.1109/isit.2011.6034269
  13. Montúfar, G., J., J. Rauh, Ay, N., 10.1007/978-3-642-40020-9_85, In: Proc. GSI 2013, Paris 2013, Lecture Notes in Computer Science 8085 (2013), 759-766. Zbl1322.62060DOI10.1007/978-3-642-40020-9_85
  14. Rauh, J., 10.1109/tit.2011.2136230, IEEE Trans. Inform. Theory 57 (2011), 3236-3247. MR2817016DOI10.1109/tit.2011.2136230
  15. Rockafellar, R. T., 10.1017/s0013091500010142, Princeton University Press, 1970. Zbl1011.49013MR0274683DOI10.1017/s0013091500010142

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