Asymptotic Properties of U-Processes under Long-Range Dependence and Applications
Celine Levy-Leduc1, Murad S. Taqqu2, Eric Moulines1, Helene Boistard3, Valderio A. Reisen4
1CNRS/LTCI/Telecom ParisTech, Paris, France; 2Department of Mathematics, Boston University, United States; 3Toulouse School of Economics - GREMAQ, Universite Toulouse 1, Toulouse, France; 4Departamento de Estatistica, Universidade Federal do Espirito Santo, Vitoria, Brazil

Let (Xi) be a stationary mean-zero Gaussian process with correlations ρ(k) satisfying: ρ(k) = k−D L(k), where D is in (0,1) and L is slowly varying at infinity. Consider the U-process {Un(r), r in I} defined as: Un(r) = (n(n−1))−1 Σ1≤i≠j≤n1{G(Xi,Xj)≤r}, where I is an interval of the real line and G is a symmetric function. We provide central and non-central 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 robust location estimator, the Shamos-Bickel scale estimator and the Rousseeuw-Croux scale estimator, which are robust scale estimators. These robust estimators are shown to have the same asymptotic distribution as the classical location and scale estimators in some cases. We also include computer simulation in order to examine how well the estimators perform at finite sample sizes.

Keywords: Long-range dependence; U-process; Hodges-Lehmann estimator; Shamos-Bickel estimator

Biography: Celine Levy-Leduc is a CNRS research scientist (CR) at the LTCI which is a joint lab with TELECOM ParisTech. Her research interests include: estimation under long-range dependence, robust statistics, changepoint detection, semiparametric statistics and anomaly detection in the Internet traffic.