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March, 1984 Generalized $L-, M-$, and $R$-Statistics
Robert J. Serfling
Ann. Statist. 12(1): 76-86 (March, 1984). DOI: 10.1214/aos/1176346393


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.


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Robert J. Serfling. "Generalized $L-, M-$, and $R$-Statistics." Ann. Statist. 12 (1) 76 - 86, March, 1984.


Published: March, 1984
First available in Project Euclid: 12 April 2007

zbMATH: 0538.62015
MathSciNet: MR733500
Digital Object Identifier: 10.1214/aos/1176346393

Primary: 62E20
Secondary: 60F05

Keywords: $L$-statistics , $M$-statistics , $R$-statistics , asymptotic normality , Hodges-Lehmann estimator , order statistics , trimmed $U$-statistics

Rights: Copyright © 1984 Institute of Mathematical Statistics

Vol.12 • No. 1 • March, 1984
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