Central Limit Theorem of the Smoothed Empirical Distribution Functions for Asymptotically Stationary Absolutely Regular Stochastic Processes

Abstract

Let F n be an estimator obtained by integrating a kernel type density estimator based on a random sample of size n. A central limit theorem is established for the target statistic F n ξ n , where the underlying random vector forms an asymptotically stationary absolutely regular stochastic process, and ξ n is an estimator of a multivariate parameter ξ by using a vector of U-statistics. The results obtained extend or generalize previous results from the stationary univariate case to the asymptotically stationary multivariate case. An example of asymptotically stationary absolutely regular multivariate ARMA process and an example of a useful estimation of F ξ are given in the applications.

Robust analysis of the martingale hypothesis

“…The analysis of the asymptotic property of such an estimatorρ s T , with possibly an interval A T increasing with T , will require other tools than the functional Central Limit Theorem from Karlsen, Tjostheim (2001), such as limit theorems for U -statistics [see e.g. Dedecker, Prieur (2007), Elharfaoui, Harel (2008)].…”

“…A recurrent process admits an invariant nonnegative density, but that density does not necessarily sum up to one. In other words, the invariant measure is not necessarily a probability 18. The nonparametric confidence intervals are computed locally without taking into account the fact that they are constant.…”

A simple variance inequality for -statistics of a Markov chain with applications

“…Several authors have studied U-statistics for stationary sequences of dependent random variables under different dependence conditions: see Arcones (1998), Borovkova et al (2001), Dehling (2006) and the references therein. Much less efforts have been spent on the behavior of U-statistics for non-stationary and asymptotically stationary processes; see Harel and Puri (1990) and Elharfaoui and Harel (2008). In this letter, we establish a variance inequality for U-statistics whose underlying sequence is an ergodic Markov Chain (which is not assumed to be stationary).…”

Comportement asymptotique de lʼestimateur non paramétrique de la fonction de renouvellement associée à des variables aléatoires positives indépendantes et non stationnaires

“…En considérant le cas d'un processus non stationnaire et absolument régulier, Elharfaoui et Harel [2] avaient utilisé la U -statistique pour déterminer un estimateur de la fonction de répartition empirique et avaient étudié les comportements asymptotiques de cet estimateur.…”

“…Définition 1.1. Une U -statistique [2] pour une fonctionnelle régulière θ(F ) de degré k est définie par :…”