Salim Bouzebda

A test of independence in some copula models

The purpose of this paper is to provide limit laws for semiparametric estimators of copulas. Some statistical tests of independence are introduced as a consequence of this methodology. We are primarily concernedwith the case where the parameter lies on the boundary of the admissible domain.

Uniform-in-bandwidth consistency for kernel-type estimators of Shannon’s entropy

We establish uniform-in-bandwidth consistency for kernel-type estimators of the differential entropy. We consider two kernel-type estimators of Shannons entropy. As a consequence, an asymptotic 100% confidence interval of entropy is provided.

New two-sample tests based on the integrated empirical copula processes

We introduce a new test of equality between two dependence structures. The new statistics are functionals of a suitably integrated two-sample empirical copula process. The limiting behaviours of the proposed statistics are established under the null hypothesis. Emphasis is placed on the explanation of the strong approximation methodology.

A New Test Procedure of Independence in Copula Models via χ2-Divergence

We introduce a new test procedure of independence in the framework of parametric copulas with unknown marginals. The method is based essentially on the dual representation of χ2-divergence on signed finite measures. The asymptotic properties of the proposed estimate and the test statistic are studied under the null and alternative hypotheses, with simple and standard limit distributions both when the parameter is an interior point or not.

On the Multivariate Two-Sample Problem Using Strong Approximations of Empirical Copula Processes

In this article, we establish optimal rates for the strong approximation of empirical copula processes in ℝ2 by sequences of Gaussian processes. These results are applied to investigate Cramér–von Mises-type statistics.

On general bootstrap of empirical estimator of a semi-Markov kernel with applications

The aim of this paper is to introduce a general bootstrap by exchangeable weight random variables for empirical estimators of the semi-Markov kernels and of the conditional transition probabilities for semi-Markov processes with countable state space. Asymptotic properties of these generalized bootstrapped empirical distributions are obtained by a martingale approach. We show how to apply our results to the construction of confidence intervals and change point problem where the limiting distribution of the proposed statistic is derived under the null hypothesis.

Some Nonparametric Tests for Change-Point Detection Based on the ℙ-ℙ and ℚ-ℚ Plot Processes

Abstract We propose nonparametric procedures for testing change-point by using the ℙ-ℙ and ℚ-ℚ plots processes. The limiting distributions of the proposed statistics are characterized under the null hypothesis of no change and also under contiguous alternatives. We give an estimator of the change-point coefficient and obtain its strong consistency. We introduce the bootstrapped version of ℙ-ℙ and ℚ-ℚ processes, requiring the estimation of quantile density, and obtain their limiting laws. Finally, we propose and investigate the exchangeable bootstrap of the empirical ℙ-ℙ plot and ℚ-ℚ plot processes which avoids the problem of the estimation of quantile density, which is of its own interest. These results are used for calculating p-values of the proposed test statistics. Emphasis is placed on the explanation of the strong approximation methodology.

New Entropy Estimator with an Application to Test of Normality

In the present article, we propose a new estimator of entropy based on smooth estimators of quantile density. The consistency and asymptotic distribution of the proposed estimates are obtained. As a consequence, a new test of normality is proposed. A small power comparison is provided. A simulation study for the comparison, in terms of mean squared error, of all estimators under study is performed.

Multivariate wavelet density and regression estimators for stationary and ergodic discrete time processes: Asymptotic results

ABSTRACT In the present paper, we are mainly concerned with the non parametric estimation of the density as well as the regression function by using orthonormal wavelet bases. We provide the strong uniform consistency properties with rates of these estimators, over compact subsets of , under a general ergodic condition on the underlying processes. We characterize the asymptotic normality of considered wavelet-based estimators, under easily verifiable conditions. The asymptotic properties of these estimators are obtained, by means of the martingale approach.

Some asymptotic results for the integrated empirical process with applications to statistical tests

ABSTRACT The main purpose of this paper is to investigate the strong approximation of the integrated empirical process. More precisely, we obtain the exact rate of the approximations by a sequence of weighted Brownian bridges and a weighted Kiefer process. Our arguments are based in part on the Komlós et al. (1975)s results. Applications include the two-sample testing procedures together with the change-point problems. We also consider the strong approximation of the integrated empirical process when the parameters are estimated. Finally, we study the behavior of the self-intersection local time of the partial-sum process representation of the integrated empirical process.