hal-00442874, version 1
Asymptotic properties of U-processes under long-range dependence
(2009-12-23)
Abstract: Let $(X_i)_{i\geq 1}$ be a stationary mean-zero Gaussian process with covariances $\rho(k)=\PE(X_{1}X_{k+1})$ 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 $$ U_n(r)=\frac{1}{n(n-1)}\sum_{1\leq i\neq j\leq n}\1_{\{G(X_i,X_j)\leq r\}}\; , $$ 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.
- 1:
- Télécom ParisTech – CNRS : UMR5141
- 2:
- CNRS : UMR5604 – Université des Sciences Sociales - Toulouse I – École des Hautes Études en Sciences Sociales [EHESS] – Institut national de la recherche agronomique (INRA) : UMR
- 3:
- Boston University
- 4:
- Universade Federal Do Espirito Santo
- Domain : Mathematics/Statistics
Statistics/Statistics Theory - Keywords : Long-range dependence – $U$-process – Hodges-Lehmann estimator – Wilcoxon-signed rank test – sample correlation integral
- Available versions : v1 (2009-12-23) v2 (2010-12-03)
- hal-00442874, version 1
- http://hal.archives-ouvertes.fr/hal-00442874
- oai:hal.archives-ouvertes.fr:hal-00442874
- From:
- Submitted on: Wednesday, 23 December 2009 11:55:01
- Updated on: Wednesday, 23 December 2009 17:53:14
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