Open Access
December 1996 Incomplete generalized L-statistics
Ola Hössjer
Ann. Statist. 24(6): 2631-2654 (December 1996). DOI: 10.1214/aos/1032181173


Given data $X_1, \dots, X_n$ and a kernel h with m arguments, Serfling introduced the class of generalized L-statistics (GL-statistics), which is defined by taking linear combinations of the ordered $h(X_{i_1}, \dots, X_{i_m})$ where $(i_1, \dots, i_m)$ ranges over all $n!/(n - m)!$ distinct m-tuples of $(1, \dots, n)$. In this paper we derive a class of incomplete generalized L-statistics (IGL-statistics) by taking linear combinations of the ordered elements from a subset of ${h(X_{i_1}, \dots, X_{i_m})}$ with size $N(n)$. A special case is the class of incomplete U-statistics, introduced by Blom. Under very general conditions, the IGL-statistic is asymptotically equivalent to the GL-statistic as soon as $N(n)/n \to \infty \as n \to \infty$, which makes the IGL much more computationally feasible. We also discuss various ways of selecting the subset of ${h(X_{i_1}, \dots, X_{i_m})}$. Several examples are discussed. In particular, some new estimates of the scale parameter in nonparametric regression are introduced. It is shown that these estimates are asymptotically equivalent to an IGL-statistic. Some extensions, for example, functionals other than L and multivariate kernels, are also addressed.


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Ola Hössjer. "Incomplete generalized L-statistics." Ann. Statist. 24 (6) 2631 - 2654, December 1996.


Published: December 1996
First available in Project Euclid: 16 September 2002

zbMATH: 0868.62043
MathSciNet: MR1425972
Digital Object Identifier: 10.1214/aos/1032181173

Primary: 62G30
Secondary: 62F35 , 62G20

Keywords: $L$-statistics , $U$-statistics , asymptotic normality , incomplete $U$-statistics , invariance principle , Nonparametric regression , order statistics , scale estimation

Rights: Copyright © 1996 Institute of Mathematical Statistics

Vol.24 • No. 6 • December 1996
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