Annals of Statistics

Asymptotic properties of U-processes under long-range dependence

C. Lévy-Leduc, H. Boistard, E. Moulines, M. S. Taqqu, and V. A. Reisen

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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.

Article information

Ann. Statist., Volume 39, Number 3 (2011), 1399-1426.

First available in Project Euclid: 13 May 2011

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F17: Functional limit theorems; invariance principles 62M10: Time series, auto-correlation, regression, etc. [See also 91B84] 62G30: Order statistics; empirical distribution functions 62G20: Asymptotic properties

Long-range dependence U-process Hodges–Lehmann estimator Wilcoxon-signed rank test sample correlation integral


Lévy-Leduc, C.; Boistard, H.; Moulines, E.; Taqqu, M. S.; Reisen, V. A. Asymptotic properties of U -processes under long-range dependence. Ann. Statist. 39 (2011), no. 3, 1399--1426. doi:10.1214/10-AOS867.

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Supplemental materials

  • Supplementary material: Proofs of Lemmas 9, 10, 12, 13, 14, 15, 16, 17 and 18 and some numerical experiments. This supplement contains proofs of Lemmas 9, 10, 12, 13, 14, 15, 16, 17 and 18 and a section containing numerical experiments illustrating some results of Section 4.