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On φ-Families of Probability Distributions

Author

Listed:
  • Rui F. Vigelis

    (Federal University of Ceará)

  • Charles C. Cavalcante

    (Federal University of Ceará)

Abstract

We generalize the exponential family of probability distributions. In our approach, the exponential function is replaced by a φ-function, resulting in a φ-family of probability distributions. We show how φ-families are constructed. In a φ-family, the analogue of the cumulant-generating function is a normalizing function. We define the φ-divergence as the Bregman divergence associated to the normalizing function, providing a generalization of the Kullback–Leibler divergence. A formula for the φ-divergence where the φ-function is the Kaniadakis κ-exponential function is derived.

Suggested Citation

  • Rui F. Vigelis & Charles C. Cavalcante, 2013. "On φ-Families of Probability Distributions," Journal of Theoretical Probability, Springer, vol. 26(3), pages 870-884, September.
    • Handle: RePEc:spr:jotpro:v:26:y:2013:i:3:d:10.1007_s10959-011-0400-5
    DOI: 10.1007/s10959-011-0400-5
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    References listed on IDEAS

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    1. Alberto Cena & Giovanni Pistone, 2007. "Exponential statistical manifold," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 59(1), pages 27-56, March.
    2. M. Grasselli, 2010. "Dual connections in nonparametric classical information geometry," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 62(5), pages 873-896, October.
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