Asymptotic properties of U-processes under long-range dependence
C. Lévy-Leduc, H. Boistard, E. Moulines, M. S. Taqqu, and V. A. Reisen
Source: Ann. Statist. Volume 39, Number 3
(2011), 1399-1426.
Abstract
Let (Xi)i≥1 be a stationary mean-zero Gaussian process with covariances satisfying ρ(0) = 1 and ρ(k) = k−DL(k), where D is in (0, 1), and L is slowly varying at infinity. Consider the U-process {Un(r), r ∈ I} defined as
Un(r) = 1/n(n−1) ∑1≤i≠j≤n1{G(Xi, Xj)≤r},
where I is an interval included in ℝ, and G is a symmetric function. In this paper, we provide central and noncentral limit theorems for Un. They are used to derive, in the long-range dependence setting, new properties of many well-known estimators such as the Hodges–Lehmann estimator, which is a well-known robust location estimator, the Wilcoxon-signed rank statistic, the sample correlation integral and an associated robust scale estimator. These robust estimators are shown to have the same asymptotic distribution as the classical location and scale estimators. The limiting distributions are expressed through multiple Wiener–Itô integrals.
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Keywords: Long-range dependence; U-process; Hodges–Lehmann estimator; Wilcoxon-signed rank test; sample correlation integral
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Permanent link to this document: http://projecteuclid.org/euclid.aos/1305292041
Digital Object Identifier: doi:10.1214/10-AOS867
Zentralblatt MATH identifier: 05947537
Mathematical Reviews number (MathSciNet): MR2850207
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