The Annals of Statistics

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

Robert J. Serfling

Full-text: Open access

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.

Article information

Source
Ann. Statist. Volume 12, Number 1 (1984), 76-86.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
http://projecteuclid.org/euclid.aos/1176346393

Digital Object Identifier
doi:10.1214/aos/1176346393

Mathematical Reviews number (MathSciNet)
MR733500

Zentralblatt MATH identifier
0538.62015

JSTOR
links.jstor.org

Subjects
Primary: 62E20: Asymptotic distribution theory
Secondary: 60F05: Central limit and other weak theorems

Keywords
Order statistics $L$-statistics $M$-statistics $R$-statistics Hodges-Lehmann estimator trimmed $U$-statistics asymptotic normality

Citation

Serfling, Robert J. Generalized $L-, M-$, and $R$-Statistics. Ann. Statist. 12 (1984), no. 1, 76--86. doi:10.1214/aos/1176346393. http://projecteuclid.org/euclid.aos/1176346393.


Export citation