A theoretical comparison between moments and L-moments

Despite its popularity in applied statistics, standard measures of shape have long been recognized to be unsatisfactory, due to their extreme sensitivity to outliers and poor sample efficiency. These difficulties seem to be largely overcome by a new system: the L-moments. During the last decade several authors have established the superior performance of L-moments over classical moments based on heuristic studies, but until present no formal explanation has been provided. We address these issues from a theoretical viewpoint. Our comparative programme is focussed on two aspects, which highlight the statistical performance of a descriptive measure: qualitative robustness and global efficiency. L-moments are treated as members of a general class of descriptive measures that are shown to outperform conventional moments based on these criteria. Consequently, they may be considered as appealing substitutes as L-moments to replace the standard measures. Since the results obtained hold for rather large nonparametric sets of distribution functions, they unify previous heuristic studies.