On the characterization of distributions by their LL-moments

A distribution with finite mean is uniquely determined by the set of expectations of the largest (or smallest) order statistics from samples of size 1,2,…1,2,…. However, this characterization contains some redundancy; some of the expectations can be dropped from the set and the remaining elements of the set still suffice to characterize the distribution. The r  th LL-moment of a distribution is a linear combination of the expectations of the largest (or smallest) order statistics from samples of size 1,2,…,r1,2,…,r. We show that a wide range of distributions can be characterized by their LL-moments with no redundancy; a set that contains all of the LL-moments except one no longer suffices to characterize the distribution.