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Characterizations of the exponential distribution by order statistics

Published online by Cambridge University Press:  14 July 2016

J. S. Huang
J. S. Huang*
Affiliation:
University of Guelph, Ontario
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Abstract

Let X1,n ≦ … ≦ Xn, n be the order statistics of a sample of size n from a distribution function F. Desu (1971) showed that if for all n ≧ 2, nX1,n is identically distributed as X1, 1, then F is the exponential distribution (or else F degenerates). The purpose of this note is to point out that special cases of known characterization theorems already constitute an improvement over this result. We show that the characterization is preserved if “identically distributed” is weakened to “having identical (finite) expectation”, and “for all n ≧ 2” is weakened to “for a sequence of n's with divergent sum of reciprocals”.

Keywords

CHARACTERIZATIONEXPONENTIAL DISTRIBUTIONORDER STATISTICSEXPECTED VALUEMÜNTZ'S THEOREM
Type
Short Communications
Information
Journal of Applied Probability , Volume 11 , Issue 3 , September 1974 , pp. 605 - 608
Copyright
Copyright © Applied Probability Trust 1974 

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References

[1] Arnold, B. C. (1971) Two characterizations of the exponential distribution using order statistics. Technical report, Iowa State University.Google Scholar
[2] Chan, L. K. (1967) On a characterization of distributions by expected values of extreme order statistics. Amer. Math. Monthly 74, 950–951.CrossRefGoogle Scholar
[3] Desu, M. M. (1971) A characterization of the exponential distribution by order statistics Ann. Math. Statist. 42, 837–838.CrossRefGoogle Scholar
[4] Epstein, B. and Sobel, M. (1953) Life testing. J. Amer. Statist. Assoc. 48, 486–502.Google Scholar
[5] Huang, J. S. (1975) Characterization of distributions by the expected values of the order statistics. Ann. Math. Statist. To appear.CrossRefGoogle Scholar
[6] Konheim, A. G. (1971) A note on order statistics. Amer. Math. Monthly 78, 524.CrossRefGoogle Scholar
[7] Pollak, M. (1973) On equal distributions. Ann. Statist. 1, 180–182.Google Scholar
[8] Puri, P. S. and Rubin, H. (1970) A characterization based on the absolute difference of two i.i.d. random variables. Ann. Math. Statist. 41, 2113–2122.Google Scholar
[9] Rényi, A. (1953) On the theory of order statistics. Acta Math. Acad. Sci. Hung. 4, 191–231.CrossRefGoogle Scholar