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The mean radius of curvature of an exponential family. (English) Zbl 0987.62005

Summary: Let \(F=F(\mu)\) be the natural exponential family (NEF) on the Euclidean space \(\mathbb{R}^d\) generated by the measure \(\mu\) and let \(k_\mu= \log L_\mu\) be the log-Laplace transform of \(\mu\). A notion of mean radius of curvature function for the NEF \(F\) is introduced using the mean radius of curvature function of the convex epigraph of the function \(k_\mu\). An algebraic property of the variance function of \(F\) is deduced and some characteristic properties for the family \(F\) related to the mean radius of curvature function are discussed. The results are illustrated by some examples.

MSC:

62E10 Characterization and structure theory of statistical distributions
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[1] Casalis, M., The \(2d+4\) simple quadratic natural exponential families on \(R^d\), Ann. Statist., 24, 1828-1854 (1994) · Zbl 0867.62042
[2] Firey, W.J., 1967. The determination of convex bodies from their mean radius of curvature functions. Mathematika 14 (Part 1) (27), 1-13.; Firey, W.J., 1967. The determination of convex bodies from their mean radius of curvature functions. Mathematika 14 (Part 1) (27), 1-13. · Zbl 0161.19302
[3] Hassairi, A., Les \((d+3)\) G-orbites de la classe de Morris-Mora des familles exponentielles sur \(R^d\), C.R. Acad. Sci. Paris Sér. 1, 217, 887-890 (1993) · Zbl 0799.62016
[4] Hassairi, A., Lajmi, S., 2001. Riesz exponential families on symmetric cones. J. Theoret. Probab. 14 (4), 927-948.; Hassairi, A., Lajmi, S., 2001. Riesz exponential families on symmetric cones. J. Theoret. Probab. 14 (4), 927-948. · Zbl 0984.60022
[5] Letac, G., A characterization of the Wishart exponential families by an invariance property, J. Theoret. Probab., 2, 1, 71-86 (1989) · Zbl 0672.62061
[6] Letac, G., 1992. Lectures on Natural Exponential Families and their Variance Functions. IMPA, Rio de Janeiro.; Letac, G., 1992. Lectures on Natural Exponential Families and their Variance Functions. IMPA, Rio de Janeiro. · Zbl 0983.62501
[7] Letac, G.; Mora, M., Natural real exponential families with cubic variance functions, Ann. Statist., 18, 1-37 (1990) · Zbl 0714.62010
[8] Morris, C. N., Natural exponential families with quadratic variance functions, Ann. Statist., 10, 65-80 (1982) · Zbl 0498.62015
[9] Webster, R., Convexity. (1994), Oxford University Press: Oxford University Press Oxford
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