Lower Bounds on PositiveL-Statistics

We consider samples of independent random variables with identical possibly continuous distributions, for which we establish optimal positive lower bounds on the expectations of linear combinations of order statistics in units generated by the central absolute moments of various orders, and describe the distributions for which the bounds are attained. We show that the bounds are zero with a possible exception of ones expressed in terms of mean absolute deviation units. We also specify the general results for important examples of the sample maximum, the sample range and Gini mean differences.