Asymptotic properties of U-processes under long-range dependence

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

Article published in Annals of Statistics, vol. 39, n. 3, p. 1399-1426, 2011. Download a pdf version.

Let $$(X_i)_{i\geq1}$$ be a stationary mean-zero Gaussian process with covariances $$\rho(k) = E(X_1X_{k+1})$$ satisfying:
$$\rho(0=1$$ and $$\rho(k)=k^{-D}L(k)$$ where D is in (0,1) and L is slowly varying at infinity. Consider the U-process $$\{U_n(r), r \in I\}$$ defined as
$$U_n(r)=\frac{1}{n(n-1)}\sum_{1\leq i\neq j \leq n}\mathbb{1}_{\{G(X_i,X_j\leq r\}},$$
where I is an interval included in $$\mathbb{R}$$ 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, 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.