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June 2011 Asymptotic properties of U-processes under long-range dependence
C. Lévy-Leduc, H. Boistard, E. Moulines, M. S. Taqqu, V. A. Reisen
Ann. Statist. 39(3): 1399-1426 (June 2011). DOI: 10.1214/10-AOS867


Let (Xi)i≥1 be a stationary mean-zero Gaussian process with covariances $\rho(k)=\mathbb {E}(X_{1}X_{k+1})$ satisfying ρ(0) = 1 and ρ(k) = kDL(k), where D is in (0, 1), and L is slowly varying at infinity. Consider the U-process {Un(r), rI} defined as $$U_n(r) =\frac{1}{n(n−1)} \sum_{1≤i≠j≤n}1_{\{G(X_i, X_j)≤r\}},$$ 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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C. Lévy-Leduc. H. Boistard. E. Moulines. M. S. Taqqu. V. A. Reisen. "Asymptotic properties of U-processes under long-range dependence." Ann. Statist. 39 (3) 1399 - 1426, June 2011.


Published: June 2011
First available in Project Euclid: 13 May 2011

zbMATH: 1242.62100
MathSciNet: MR2850207
Digital Object Identifier: 10.1214/10-AOS867

Primary: 60F17 , 62G20 , 62G30 , 62M10

Keywords: Hodges–Lehmann estimator , long-range dependence , sample correlation integral , U-process , Wilcoxon-signed rank test

Rights: Copyright © 2011 Institute of Mathematical Statistics


Vol.39 • No. 3 • June 2011
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