Duality for real and multivariate exponential families

Gérard Letac

Journal of Multivariate Analysis, 2022, vol. 188, issue C

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).

Keywords: Dilogarithm distribution; Landau distribution; Large deviations; Quadratic and cubic real exponential families; Tweedie scale; Wishart distributions (search for similar items in EconPapers)
Date: 2022
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Citations: View citations in EconPapers (2)

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DOI: 10.1016/j.jmva.2021.104811

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