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April, 1971 A Moment Problem for Order Statistics
Joseph B. Kadane
Ann. Math. Statist. 42(2): 745-751 (April, 1971). DOI: 10.1214/aoms/1177693423


Necessary and sufficient conditions are given for a triangular array of numbers to be expectations of order statistics of some nonnegative random variable. Using well-known recurrence relations, the expectations of all order statistics of the largest sample size, $n$, in the triangular array, or the expectations of the smallest of every sample size up to and including $n$ are sufficient to determine the whole array. The former are reduced to a Stieltjes moment problem, the latter to a Hausdorff moment problem. These results are applied to show that for every sample size, there is a positive random variable with geometrically increasing expectations of order statistics with arbitrary ratio and expectation of smallest order statistic. However, only the degenerate distributions have geometrically increasing expectations of order statistics for more than one sample size, even when the ratio and mean of the smallest order statistic can depend on the sample size. These results were required for a study of participation in discussion groups.


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Joseph B. Kadane. "A Moment Problem for Order Statistics." Ann. Math. Statist. 42 (2) 745 - 751, April, 1971.


Published: April, 1971
First available in Project Euclid: 27 April 2007

zbMATH: 0218.62046
MathSciNet: MR292241
Digital Object Identifier: 10.1214/aoms/1177693423

Rights: Copyright © 1971 Institute of Mathematical Statistics

Vol.42 • No. 2 • April, 1971
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