The following classical characterization of the exponential distribution is well known. Let X 1 ,X 2 , . . . X n be independent and identically distributed random variables. Their common distribution is exponential if and only if random variables X 1 and n min(X 1 , . . .,X n ) have the same distribution. In this note we show that the characterization can be substantially simplified if the exponentiality is characterized within a broad family of distributions that includes, in particular, gamma, Weibull and generalized exponential distributions. Then the necessary and sufficient condition is the equality only expectations of these variables. A similar characterization holds for the maximum.
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References
T.A. Azlarov and N.A.Volodin, Characterization Problems Associated with the Exponential Distribution, Springer-Verlag (1986).
M.Ahsanullah and G.Hamedani, Exponential Distribution — Theory and Methods, Nova Publishers, New York (2010).
R.D. Gupta and D.Kundu, “Generalized exponential distributions,” Austr. New Zeland J. Statist., 41, No. 2, 173–188 (1999).
M.Abramovitz and I.Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, National Bureau of Standards, Applied Mathematics Series 55 (1964).
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*This research is supported by the Russian Scientific Foundation, project 14–11–00364
Proceedings of the XXXII International Seminar on Stability Problems for Stochastic Models, Trondheim, Norway, June 16–21, 2014.
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Ushakov, N.G., Ushakov, V.G. A Note on Characterizations of the Exponential Distribution*. J Math Sci 214, 132–138 (2016). https://doi.org/10.1007/s10958-016-2763-8
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DOI: https://doi.org/10.1007/s10958-016-2763-8