The Annals of Statistics

Incomplete generalized L-statistics

Ola Hössjer

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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.

Article information

Ann. Statist., Volume 24, Number 6 (1996), 2631-2654.

First available in Project Euclid: 16 September 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G30: Order statistics; empirical distribution functions
Secondary: 62G20: Asymptotic properties 62F35: Robustness and adaptive procedures

Asymptotic normality incomplete $U$-statistics invariance principle order statistics nonparametric regression $L$-statistics scale estimation $U$-statistics


Hössjer, Ola. Incomplete generalized L -statistics. Ann. Statist. 24 (1996), no. 6, 2631--2654. doi:10.1214/aos/1032181173.

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