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Incomplete generalized \(L\)-statistics. (English) Zbl 0868.62043

Summary: Given data \(X_1,\dots,X_n\), and a kernel \(h\) with \(m\) arguments, R. J. Serfling [ibid. 12, 76-86 (1984; Zbl 0538.62015)] 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 \((1,\dots,n)\). 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 G. Blom [Biometrika 63, 573-580 (1976; Zbl 0352.62034)]. 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.

MSC:

62G30 Order statistics; empirical distribution functions
62G20 Asymptotic properties of nonparametric inference
60F17 Functional limit theorems; invariance principles
62G07 Density estimation
62F35 Robustness and adaptive procedures (parametric inference)
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