U-statistics

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.

Central limit theorem for multiple integrals with respect to the empirical process

Joint work with Eustasio del Barrio. Article publicated in Statistics and Probability Letters, vol. 79(2), p. 188-195, January 2009.
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In that article, we give some results of weak convergence of multiple integrals with respect to the empirical process. We consider objects of type
$J_{n,m}(h)=\int'h(x_1, \dots, x_m)d\mathbb{G}_n(x_1)\dots \mathbb{G}_n(x_m),$
where h is a symmetric real-valued square integrable function of m variables. $X_1, \dots, X_n$ is a P-distributed i.i.d. sample, and $\mathbb{P}_n=\frac{1}{n}\sum_{i=1}^n\delta_{X_i}$ and $\mathbb{G}_n=\sqrt{n}(\mathbb{P}_n-P)$ are respectively the associated empirical measure and the empirical process. $\int'$ is the integral where the integration on the diagonal has been omitted. We include the case of non degenerate kernels with respect to the underlying distribution. Our results are related to earlier results on U-statistics. We introduce a stochastic integral with respect to the Brownian bridge which allows us to express the limit in a unified way in the degenerate and non degenerate cases. Using the multiple integral with respect to the empirical process has an advantage with respect to using U-statistics: the Central Limit Theorem we obtain is simpler. It does not involve the degeneracy of the kernel and the limit is expressed in a precise way.

PhD Thesis: asymptotic efficiency of tests related with the Wasserstein statistic

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.