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Duality for real and multivariate exponential families

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  • Letac, Gérard

Abstract

Consider a measure μ on Rn generating a natural exponential family F(μ) with variance function VF(μ)(m) and Laplace transform exp(ℓμ(s))=∫Rnexp(−〈x,s〉μ(dx)).A dual measure μ∗ satisfies −ℓμ∗′(−ℓμ′(s))=s. Such a dual measure does not always exist. One important property is ℓμ∗′′(m)=(VF(μ)(m))−1, leading to the notion of duality among exponential families (or rather among the extended notion of T exponential families TF obtained by considering all translations of a given exponential family F).

Suggested Citation

  • Letac, Gérard, 2022. "Duality for real and multivariate exponential families," Journal of Multivariate Analysis, Elsevier, vol. 188(C).
    • Handle: RePEc:eee:jmvana:v:188:y:2022:i:c:s0047259x21000890
    DOI: 10.1016/j.jmva.2021.104811
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    References listed on IDEAS

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    1. M. Marucho & C. A. Garcia Canal & Huner Fanchiotti, 2006. "The Landau Distribution For Charged Particles Traversing Thin Films," International Journal of Modern Physics C (IJMPC), World Scientific Publishing Co. Pte. Ltd., vol. 17(10), pages 1461-1476.
    2. Vladimir Vladimirovich Vinogradov & Richard Bruce Paris, 2021. "On two extensions of the canonical Feller–Spitzer distribution," Journal of Statistical Distributions and Applications, Springer, vol. 8(1), pages 1-25, December.
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    Cited by:

    1. Shaul K. Bar-Lev & Gérard Letac & Ad Ridder, 2024. "A delineation of new classes of exponential dispersion models supported on the set of nonnegative integers," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 76(4), pages 679-709, August.

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