On the Characterization of Distributions by Their L-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, . . .. 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 rth L-moment of a distribution is a linear combination of the expectations of the largest (or smallest) order statistics from samples of size 1, 2, . . . , r. We show that a wide range of distributions can be characterized by their L-moments with no redundancy: a set that contains all of the L-moments except one no longer suffices to characterize the distribution.