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 be a stationary mean-zero Gaussian process with covariances satisfying:
and where D is in (0,1) and L is slowly varying at infinity. Consider the U-process defined as
where I is an interval included in and G is a symmetric function. In this paper, we provide central and non-central limit theorems for . 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.
PhD Thesis with advisors Eustasio del Barrio and Fabrice Gamboa, defended on the 16th of July, 2007, before the tribunal composed by Profesors Jean-Marc Azaïs, Bernard Bercu, Eustasio del Barrio, Fabrice Gamboa and Carlos Matrán.
This thesis is composed of three main parts. In the first part, we study some asymptotic properties of multiple integrals with respect to the empirical process. The second part is devoted to the study of the asymptotic efficiency of the Wasserstein test. The equivalence of the Wasserstein statistic with a double integral with respect to the empirical process allows us to apply the results of the first part. A simulation study is added to the study of the asymptotic power. The third part deals with large deviations for L-statistics. A large deviations principle is obtained using the topology of the Wasserstein distance on the space of measures, under conditions on the extremes.