A Moment Problem for Order Statistics

Abstract

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.