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
Subjects
Mathematics
Mathematics and Statistics
Probability Density Function
Exponential Distribution
Cumulative Distribution Function
Gamma Distribution
Hazard Rate
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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