A Result in Order Statistics, with Applications to L-moments

We show that if a linear combination of expectations of order statistics has mean zero across all random variables that have finite mean, then the linear combination is identically zero. A consequence of this result is any functional of a probability distribution can have essentially only one unbiased L-estimator (i.e., an estimator that has then form of a linear combination of order statistics): if two such linear combinations have the same expectation then they must be algebraically identical. We use this result to prove the equivalence of two statistics that have been proposed as estimators of the L-moments introduced by Hosking (1990), and to provide alternative means of computing estimators of the trimmed L-moments introduced by Elamir and Seheult (2003).