Abstract
In a recent article Pillai (1990,Ann. Inst. Statist. Math.,42, 157–161) showed that the distribution 1−E α(−x α), 0<α≤1; 0≤x, whereE α(x) is the Mittag-Leffler function, is infinitely divisible and geometrically infinitely divisible. He also clarified the relation between this distribution and a stable distribution. In the present paper, we generalize his results by using Bernstein functions. In statistics, this generalization is important, because it gives a new characterization of geometrically infinitely divisible distributions with support in (0, ∞).
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References
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Pillai, R. N. (1990). On Mittag-Leffler functions and related distributions,Ann. Inst. Statist. Math.,42, 157–161.
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Fujita, Y. A generalization of the results of Pillai. Ann Inst Stat Math 45, 361–365 (1993). https://doi.org/10.1007/BF00775821
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DOI: https://doi.org/10.1007/BF00775821