Statistical analysis of extremes is often conducted for predicting large return period events. In this paper, LH moments, a generalization of L moments, that are based on linear combinations of higher-order statistics are introduced for characterizing the upper part of distributions and larger events in data. Using LH moments reduces undesirable influences that small sample events may have on the estimation of large return period events. The method of LH moments is formulated for the generalized extreme value (GEV) distribution along with two detailed examples. Monte Carlo simulations are conducted to illustrate the performance of using the method of LH moments to fit the GEV distribution to both GEV and non-GEV samples. LH moment diagrams are constructed using annual maximum flow data for 107 streams worldwide to examine the ability of the GEV distribution to model those series.