A Note on Characterizations of the Exponential Distribution

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 distri...

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Bibliographic Details
Published inJournal of mathematical sciences (New York, N.Y.) Vol. 214; no. 1; pp. 132 - 138
Main Authors Ushakov, N. G., Ushakov, V. G.
Format Journal Article
LanguageEnglish
Published New York Springer US 01.04.2016
Springer
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Summary: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.
ISSN:1072-3374
1573-8795
DOI:10.1007/s10958-016-2763-8