ASYMPTOTIC PROPERTIES OF U-PROCESSES UNDER LONG-RANGE DEPENDENCE

• Published in: The Annals of statistics Vol. 39; no. 3; pp. 1399 - 1426
• Main Authors: Lévy-Leduc, C, Boistard, H, Moulines, E, Taqqu, M. S., Reisen, V. A.
• Format: Journal Article
• Language: English
• Published: Cleveland, OH Institute of Mathematical Statistics 01-06-2011; The Institute of Mathematical Statistics
• Subjects: 60F17; 62G20; 62G30; 62M10; Algebraic combinatorics; Asymptotic properties; Combinatorics; Combinatorics. Ordered structures; Covariance; Covariance matrices; Estimators; Exact sciences and technology; General topics; Hodges–Lehmann estimator; Infinity; Limit theorems; Long-range dependence; Mathematical functions; Mathematics; Normal distribution; Probability and statistics; Probability theory and stochastic processes; Random variables; sample correlation integral; Sciences and techniques of general use; Statistics; Stochastic processes; Studies; Theorems; Topological theorems; U-process; Umklapp process; Wilcoxon-signed rank test; Rank statistic; Wilcoxon test; Statistical distribution; Asymptotic behavior; Estimator robustness; Symmetric function; Mean estimation; Wiener Itô integral; Long range dependence; Parametric method; Statistical method; Selection problem; Gaussian process; Central limit theorem; Covariance; Limit distribution; Gaussian processes; Multiple integral; Stationary process
• ISSN: 0090-5364; 2168-8966
• DOI: 10.1214/10-AOS867

Let (X i ) i≥1 be a stationary mean-zero Gaussian process with covariances ρ(k) = 𝔼(X 1 X k+1 ) satisfying ρ(0) = 1 and ρ(k) = k− D L(k), where D is in (0, 1), and L is slowly varying at infinity. Consider the U-process {U n (r), r ∈ I} defined as $U_{n}(r)=\frac{1}{n(n-1)}\sum_{1\leq i\neq j\leq n}1_{\{G(X_{i},X_{j})\leq r\}}$ , 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 U n . 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 Weiner—Itô integrals.