Generalized $L-, M-$, and $R$-Statistics

Abstract

A class of statistics generalizing U-statistics and L-statistics, and containing other varieties of statistic as well, such as trimmed U-statistics, is studied. Using the differentiable statistical function approach, differential approximations are obtained and the influence curves of these generalized L-statistics are derived. These results are employed to establish asymptotic normality for such statistics. Parallel generalizations of M- and R-statistics are noted. Strong convergence, Berry-Esseen rates, and computational aspects are discussed.