# Papers

## On the uniform-in-bandwidth consistency of the general conditional U-statistics based on the copula representation

S. Bouzebda, I. Elhattab and B. Nemouchi

Statistical Papers, 2020

Stute [Ann. Probab. 19 (1991) 812-825] introduced a class of estimators called conditional $U$-statistics of $\mathbb{E}(\varphi(Y_{1},\ldots,Y_{m})\mid (X_{1},\ldots,X_{m})=\mathbf{ t}), \mbox{ for } \mathbf{ t}\in \mathbb{R}^{m}.$ In the present work, we provide a new class of estimators of conditional $U$-statistics. Read more

## The Consistency and Asymptotic Normality of the Kernel type Expectile Regression Estimator for Functional Data

M. Mohamedi, S. Bouzebda and A. Laksaci

Journal of Multivariate Analysis, 2020

The aim of this paper is to nonparametrically estimate the expectile regression in the case of a functional predictor and a scalar response. More precisely, we construct a kernel-type estimator of the expectile regression function. The main contribution of this study is the establishment of the asymptotic properties of the expectile regression estimator. Precisely, we establish the almost complete convergence with rate. Read more

## Nonparametric Recursive Method for Kernel-Type Function Estimators for Censored Data Estimators for Censored Data

S. Bouzebda and S. Slaoui

Journal of Stochastic Analysis, 2020

In the present paper, we study general kernel type estimatorsfor censored data defined by the stochastic approximation algorithm. We establish a central limit theorem for the proposed estimators. We characterize the strong pointwise convergence rate for the nonparametric recursive general kernel-type estimators under some mild conditions. Read more

## Some Asymptotic Properties of Kernel Regression Estimators of the Mode for Stationnary and Ergodic Continuous Time Processes

S. Bouzebda and S. Sidi

Revista Matemática Complutense, 2020

In the present paper, we consider the nonparametric regression model with random design based on $(\mathbf{X}_{\rm t},\mathbf{Y}_{\rm t})_{\rm t \geq 0}$ an $\mathbb{R}^{d}\times\mathbb{R}^{q}$-valued strictly stationary and ergodic continuous time process, where the regression function is given by $m(\mathbf{x},\psi) = \mathbb{E}(\psi(\mathbf{ Y}) \mid \mathbf{ X} = \mathbf{ x}))$, for a measurable function $\psi : \mathbb{R}^{q} \rightarrow \mathbb{R}$. Read more

## Exchangeably Weighted Bootstraps of Martingale Difference Arrays under the Uniformly Integrable Entropy

S. Bouzebda and N. Limnios

Journal of Stochastic Analysis, 2020

In the present work, we are mainly concerned with the uniform central limit theorem for a bootstrapped martingale-difference array of a function-indexed stochastic process under the uniformly integrable entropy condition. More precisely, we establish the consistency of the exchangeable bootstraps. Read more

## Exchangeably Weighted Bootstraps of Martingale Difference Arrays under the Uniformly Integrable Entropy

Journal of Stochastic Analysis, 2020

In the present paper, we are mainly interested in Cramér-type results for the weighted bootstrap of the $U$-statistics. The method of proof is based on the Hoeffding decomposition according to the bootstrapped Cramér transform together with the contraction technique. Read more

## Uniform Convergence Rate of the Kernel Regression Estimator Adaptive to Intrinsic Dimension in Presence of Censored Data

Journal of Nonparametric Statistics, 2020

The focus of the present paper is on the uniform in bandwidth consistency of kernel-type estimators of the regression function $\mathbb{E}(\Psi(\mathbf{Y})\mid \mathbf{{ X}}=\mathbf{ x})$ derived by modern empirical process theory, under weaker conditions on the kernel than previously used in the literature. Our theorems allow data-driven local bandwidths for these statistics. Read more

## Regression Analysis of Stochastic Fatigue Crack Growth Model in a Martingale Difference Framework

C. Papamichail, S. Bouzebda and N. Limnios

Journal of Statistical Theory and Practice, 2020

In the present paper, we are mainly concerned with the degradation mechanism that arises in fatigue crack growth (FCG). The crack evolution mechanism is modeled by a first order stochastic differential system, composed by a deterministic FCG equation perturbed by a stochastic process. Read more

## Strong approximations for the general bootstrap of empirical processes with applications in selected topics of nonparametric statistics

S. Bouzebda and O. El-Dakkak

Revista Matemática Complutense, 2020

The purpose of this note is to provide an approximation for the generalized bootstrapped empirical process achieving the rate in [40]. The proof is based on the same arguments used in [37]. As a consequence, we establish an approximation of the bootstrapped kernel distribution estimation. Read more

## Some Results About Kernel Estimators for Function Derivatives Based on Stationary and Ergodic Continuous Time Processes with Applications

S. Bouzebda and S. Sidi

Communications in Statistics - Theory and Methods, 2020

The derivatives of the probability density or regression functions contain important information concerning a multivariate data set, such as modal regions. Despite this importance, nonparametric estimation of higher-order derivatives of the density or regression functions have received only relatively scant attention. Read more

## Some Selected Topics for the Bootstrap of the Empirical and Quantile processes

Theory of Stochastic Processes, 2019

In the present work, we consider the asymptotic distributions of $L_p$ functionals of bootstrapped weighted uniform quantile and empirical processes. The asymptotic laws obtained are represented in terms of Gaussian integrals. We investigate the strong approximations for the bootstrapped Vervaat process and the weighted bootstrap for Bahadur-Kiefer process. Read more

## Large and moderate deviation principles for recursive kernel estimators of a regression function for spatial data defined by stochastic approximation method

S. Bouzebda and Y. Slaoui

Statistics & Probability Letters, 2019

In the present paper, we are mainly concerned with a family of kernel type estimators based upon spatial data. More precisely, we establish large and moderate deviations principles for the recursive kernel estimators of a regression function for spatial data defined by the stochastic approximation algorithm. Read more

## On general bootstrap for a multidimensional empirical estimator of a continuous-time semi-Markov kernel with applications

S. Bouzebda, C. Papamichail and N. Limnios

Journal of Nonparametric Statistics, 2018

The present paper introduces a general notion and presents results of bootstrapped empirical estimators of the semi-Markov kernels and of the conditional transition distributions for semi-Markov processes with countable state space, constructed by exchangeably weighting the sample. Our proposal provides a unification of bootstrap methods in the semi-Markov setting including, in particular, Efron’s bootstrap. Read more

## Some Asymptotic Results for the $p$-Fold Integrated Empirical Processes with Application to the Statistical Tests

S. Alvarez-Andrade, S. Bouzebda and A. Lachal

Test, 2018

The main purpose of this paper is to investigate the strong approximation of the $p$-fold integrated empirical process, p being a fixed positive integer. More precisely, we obtain the exact rate of the approximations by a sequence of weighted Brownian bridges and a weighted Kiefer process. Read more

## Nonparametric Mode Regression Estimation for Functional Stationary Ergodic Data. Asymptotic Normality and Application

S. Bouzebda, M. Chaouch and N. Laïb

Mathematical Methods of Statistics, 2016

The main purpose of the present work is to establish the functional asymptotic normality of a class of kernel conditional mode estimates whenever functional stationary ergodic data are considered. More precisely, consider a random variable $(X, Z)$ taking values in some semi-metric abstract space $E\times F$. Read more

## A semiparametric maximum likelihood ratio test for the change point in copula models

S. Bouzebda and A. Keziou

Statistical Methodology, 2013

In the present paper, a semiparametric maximum-likelihood-type test statistic is proposed and proved to have the same limit null distribution as the classical parametric likelihood one. Under some mild conditions, the limiting law of the proposed test statistic, suitably normalized and centralized, is shown to be double exponential, under the null hypothesis of no change in the parameter of copula models. Read more

## New Entropy Estimator with an Application to Test of Normality

S. Bouzebda, I. Elhattab, A. Keziou and T. Lounis

Communications in Statistics, Theory and Methods, 2013

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. Read more

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

S. Bouzebda and N. Limnios

Journal of Multivariate Analysis, 2013

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. Read more

## Strong approximations for weighted bootstrap of empirical and quantile processes with applications

Statistical Methodology, 2013

The main purpose of this paper is to investigate the strong approximation of the weighed bootstrap of empirical and quantile processes. The bootstrap idea is to reweight the original empirical distribution by stochastic weights. Our results are applied in two concrete statistical problems: the Q–Q processes as well as the kernel-type density estimator. Read more

## Exchangeably weighted bootstraps of empirical estimator for semi-Markov kernel

S. Bouzebda and N. Limnios

Comptes Rendus de l Academie des Sciences, Series I, Mathematics, 2013

A general notion of bootstrapped empirical estimators, of the semi-Markov kernels and of the conditional transition probabilities for semi-Markov processes with countable state space, constructed by exchangeably weighting sample, is introduced. Read more

## A Strong Invariance Theorem of the Tail Empirical Copula Processes

S. Bouzebda and T. Zari

Communications in Statistics - Theory and Methods, 2013

We study the behavior of bivariate empirical copula process $\mathbb{G}_n(\cdot,\cdot)$ on pavements $[0,k_n/n]^2$ of $[0,1]^2,$ where $k_n$ is a sequence of positive constants fulfilling some conditions.. Read more

## Dual Divergences Estimators of the Tail Index

S. Bouzebda and M. Cherfi

International Scholarly Research Network, ISRN Probability and Statistics, 2012

The main purpose of the present paper is to propose a new estimator of the tail index using $\phi$-divergences and the duality technique. These estimators are explored with respect to robustness through the influence function approach. The empirical performances of the proposed estimators are illustrated by simulation. Read more

## On the Strong Approximation of General Bootstrap of Empirical Copula Processes with Applications

S. Bouzebda

Mathematical Methods of Statistics, 2012

The purpose of the present paper is to provide a strong invariance principle for the generalized bootstrapped empirical copula process with the rate of the approximation for multivariate empirical processes. As a by-product, we obtain a uniform-in-bandwidth consistency result for kernel-type estimators of copula derivatives, which is of its own interest.. Read more

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

S. Bouzebda and N.E. El Faouzi

Statistics, 2012

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. Read more

## General bootstrap for dual phi-divergence estimates

S. Bouzebda and M. Cherfi

Journal of Probability and Statistics, 2012

A general notion of bootstrapped $\phi$-divergence estimates constructed by exchangeably weighting sample is introduced. Asymptotic properties of these generalized bootstrapped $\phi$-divergence estimates are obtained, by mean of the empirical process theory, which are applied to construct the bootstrap confidence set with asymptotically correct coverage probability. Read more

## Test of Symmetry Based on Copula Function

S. Bouzebda and M. Cherfi.

Journal of Statistical Planning and Inference, 2012

We introduce some new nonparametric statistical tests of symmetry. The limiting behaviors of the proposed statistics are established under the null hypothesis. Emphasis is placed on explanation of the strong approximation methodology. Read more

## Some New Multivariate Tests of Independence

S. Bouzebda

Mathematical Methods of Statistics, 2011

We introduce some new nonparametic tests of independence which are functionals of the modified multivariate empirical copula process. In this work we extend the modified empirical process of Nikitin and Sporysheva (2009), which is defined by using a family of Gaussian processes of Deheuvels (2007b) to the multivariate case. Read more

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

S. Bouzebda and I. Elhattab

Electronic Journal of Statistics, 2011

We establish uniform-in-bandwidth consistency for kernel-type estimators of the differential entropy. We consider two kernel-type estima- tors of Shannon’s entropy. As a consequence, an asymptotic $100\%$ confi- dence interval of entropy is provided. Read more

## On the Multivariate Two-sample Problem using Strong Approximations of Empirical Copula Processes

S. Bouzebda, N.E. El Faouzi and T. Zari

Communications in Statistics, Theory and Methods, 2011

In this article, we establish optimal rates for the strong approximation of empirical copula processes in $\mathbb R^2$ by sequences of Gaussian processes. These results are applied to investigate Cramer–von Mises-type statistics. Read more

## Motorway travel time prediction based on to data and weather effect integration

N.E. El Faouzi, R. Billot and S. Bouzebda

IET Intelligent Transport Systems A, 2010

This study reports the main findings of the Travel time Prediction based on electronic Toll collection (ETC) data with wEather effect integration on mOtorways (TPTEO) project aiming at developing and implementing a route planner tool and travel time prediction system on the interurban motorway network managed by French motorway AREA Company. Read more

## Uniform in bandwidth consistency of the kernel-type estimator of the Shannon’s entropy

S. Bouzebda and I. Elhattab

Comptes Rendus Academie des Sciences, Series I, Mathematics, 2010

We establish uniform-in-bandwidth consistency for kernel-type estimators of the differential entropy. We consider two kernel-type estimators of Shannon’s entropy. As a consequence, an asymptotic $100\%$ confidence interval of entropy is provided.. Read more

## Strong Approximation of the Smoothed Q-Q Processes

S. Bouzebda

Far East Journal of Theoretical Statistics, 2010

In this paper, we describe the limiting behavior of the Q-Q plot kerneltype- estimators. The bootstrapped version of the Q-Q processes is discussed. The latter is applied to construct the confidence band. Read more

## New estimates and tests of independence in semiparametric copula models

S. Bouzebda and A. Keziou

Kybernetika, 2010

We introduce new estimates and tests of independence in copula models with unknown margins using $\Phi$-divergences and the duality technique. The asymptotic laws of the estimates and the test statistics are established both when the parameter is an interior or a boundary value of the parameter space. Read more

## A new test procedure of independence in Copula models via chi-square-divergence

S. Bouzebda and A. Keziou

Communications in Statistics, Theory and Methods, 2010

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 $\chi_2$-divergence on signed finite measures. Read more

## New estimates and tests of independence in semiparametric copula models

S. Bouzebda and A. Keziou

Comptes Rendus de l Academie des Sciences, Series I, Mathematics, 2010

We propose a semiparametric test of independence in copula models for bivariate survival censored data. We give the limit laws of the estimate of the parameter and the proposed test statistic under the null hypothesis of independence. Read more

## Bootstrap of the Hill estimator: limit theorems

S. Bouzebda

Annales ISUP, 2010

We develop a bootstrap method for estimating the Pareto index of an extreme value distribution. We begin by considering a sequence $X_1,\ldots,X_n$ of i.i.d. random variables with distribution function $F(\cdot)$ satisfying $\lim_{x\rightarrow \infty}\frac{1-F(xt)}{1-F(x)}=t^{-\frac{1}{c}}~~\mbox{for all}~~ t>0.$ Read more

## Uniform-in-Bandwidth Consistency for a Nonparametric Estimate of the Entropy under Random Censorship

S. Bouzebda and I. Elhattab

Electronic Journal of Statistics, 2009

We establish uniform-in-bandwidth consistency for kernel-type estimators of the differential entropy. We consider two kernel-type estimators of Shannon’s entropy. As a consequence, an asymptotic $100\%$ confidence interval of entropy is provided. Read more

## Estimation and tests of independence in copula models via divergences

S. Bouzebda and A. Keziou

Comptes Rendus de l Academie des Sciences, Series I, Mathematics, 2009

We introduce new estimates and tests of independence in copula models with unknown margins using $\varphi$-divergences and the duality technique. The asymptotic laws of the estimates and the test statistics are established both when the parameter is an interior point or not. Read more

## Some Results About Kernel Estimators for Function Derivatives Based on Stationary and Ergodic Continuous Time Processes with Applications

S. Bouzebda and A. Keziou

Mathematical Methods of Statistics, 2008

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 concerned with the case where the parameter lies on the boundary of the admissible domain. Read more