Incomplete generalized L-statistics

• 成果类型: Article
• 署名作者: Hossjer, O
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/aos/1032181173
• 发表日期: 1996
• 页码: 2631-2654
• 关键词: U-STATISTICS; LIMIT-THEOREMS; random-variables; slope selection; s-estimators; regression; algorithm

摘要

Given data X(1),..., 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(i1),..., X(im)), where (i(1),...,i(m)) ranges over all n!/(n - m)! distinct m-tuples of (1,..., 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(i1),..., X(im))} 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 --> infinity as n --> infinity, which makes the IGL much more computationally feasible. We also discuss various ways of selecting the subset of {h(X(i1),..., X(im))}. 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.