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General exponential models on the unit simplex and related multivariate inverse Gaussian distributions. (English) Zbl 0758.62027

Summary: O. E. Barndorff-Nielsen and B. Jørgensen [J. Multivariate Anal. 39, 106-116 (1991; Zbl 0739.62015)] have introduced some parametric models on the unit simplex. The distributions associated with these models have been obtained by conditioning on the sum of \(d\) independent generalized inverse Gaussian random variables. We use a constructive approach to derive some of these models by first mapping the inverse Gaussian law on \((0,1)\) and formally extending it on the unit simplex. This technique is then applied to a mixture-inverse Gaussian distribution studied recently by B. Jørgensen, V. Seshadri and G. A. Whitmore [Scand. J. Stat. 18, No. 1, 77-89 (1991; Zbl 0723.62006)]. The distributions are then retransformed to yield two versions of a multidimensional inverse Gaussian distribution.

MSC:

62H05 Characterization and structure theory for multivariate probability distributions; copulas
62E10 Characterization and structure theory of statistical distributions
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References:

[1] Barndorff-Nielsen, O. E.; Jørgensen, B., Some parametric models on the simplex, J. Multivariate Anal., 39, 106-116 (1989) · Zbl 0739.62015
[2] Jørgensen, B., Statistical Properties of the Generalized Inverse Gaussian Distribution (1982), Springer: Springer New York, Lecture Notes in Statistics No. 9 · Zbl 0486.62022
[3] Jørgensen, B.; Seshadri, V.; Whitmore, G. A., On the mixture of the inverse Gaussian distribution with its complementary reciprocal, Scand. J. Statist., 18, 1, 77-89 (1991) · Zbl 0723.62006
[4] Morris, C. N., Natural exponential families with quadratic variance functions, Ann. Statist., 10, 65-80 (1982) · Zbl 0498.62015
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