Inequalities for variances of order statistics originating from urn models

Abstract

We consider the drawing with and without replacement models from numerical populations. For order statistics based on the above drawing schemes, we provide sharp upper bounds for their variances, expressed in the single observation variance units. We also characterize the populations for which the bounds are attained.

Optimal Moment Inequalities for Order Statistics From Nonnegative Random Variables

“…Beyond the well-developed theory on expectation bounds for order statistics and records, the corresponding theory to other moments does not seem to have received much attention. Of course, some exceptions exist concerning variances; see, e.g., Papadatos [19]; Jasiński and Rychlik [13,14]; Rychlik [32,33]. The purpose of the present work is to obtain tight upper bounds for the moments of a single order statistic from a nonnegative population.…”