On φ-Families of Probability Distributions

Rui F. Vigelis () and Charles C. Cavalcante ()
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Rui F. Vigelis: Federal University of Ceará
Charles C. Cavalcante: Federal University of Ceará

Journal of Theoretical Probability, 2013, vol. 26, issue 3, 870-884

Abstract: 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.

Keywords: Exponential family of probability distributions; Musielak–Orlicz spaces; Bregman divergence; 60E05 (search for similar items in EconPapers)
Date: 2013
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DOI: 10.1007/s10959-011-0400-5

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