Asymptotic properties of U-processes under long-range dependence

2009/12/23 by Céline Lévy‐Leduc, Céline Lévy-Leduc, Lévy-Leduc, Céline +9
Computer Science · Economics, Econometrics and Finance · Mathematics · #Bayesian Methods and Mixture Models #Financial Risk and Volatility Modeling #Statistical Methods and Inference #math.ST #stat.TH

paper · pdf · doi:10.48550/arxiv.0912.4688

arxiv created 2010/12/03 · arxiv updated 2010/12/08

Abstract

Let (X_i)_{i≥ 1} be a stationary mean-zero Gaussian process with covariances ρ(k)=\PE(X₁X_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)=(1)/(n(n-1))∑_{1≤ i≠ j≤ n}\1_\G(X_i,X_j)≤ r\ , where I is an interval included in \rset and G is a symmetric function. In this paper, we provide central and non-central limit theorems for U_n. They are used to derive the asymptotic behavior of the Hodges-Lehmann estimator, the Wilcoxon-signed rank statistic, the sample correlation integral and an associated scale estimator. The limiting distributions are expressed through multiple Wiener-Itô integrals.

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