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January, 1973 On Equal Distributions
Moshe Pollak
Ann. Statist. 1(1): 180-182 (January, 1973). DOI: 10.1214/aos/1193342397


It is shown that two distributions both of which have a finite expectation are equal if and only if for every $n \geqq 1$ there exists $1 \leqq k \leqq n$ such that the $k$th order statistics from samples of size $n$ of each distribution have equal expectations. Similarly, it is shown that a distribution with finite expectation is symmetric about zero if and only if for every $n \geqq 0$ there exists $0 \leqq k \leqq 2n + 1$ such that the sum of the expectations of the $k$th smallest and the $k$th largest observations in a sample of size $2n + 1$ is zero.


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Moshe Pollak. "On Equal Distributions." Ann. Statist. 1 (1) 180 - 182, January, 1973.


Published: January, 1973
First available in Project Euclid: 25 October 2007

zbMATH: 0263.62008
MathSciNet: MR331582
Digital Object Identifier: 10.1214/aos/1193342397

Rights: Copyright © 1973 Institute of Mathematical Statistics


Vol.1 • No. 1 • January, 1973
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