Djamal Louani

Nonparametric kernel regression estimation for functional stationary ergodic data: Asymptotic properties

The aim of this paper is to study asymptotic properties of the kernel regression estimate whenever functional stationary ergodic data are considered. More precisely, in the ergodic data setting, we consider the regression of a real random variable Y over an explanatory random variable X taking values in some semi-metric abstract space. While estimating the regression function using the well-known Nadaraya-Watson estimator, we establish the consistency in probability, with a rate, as well as the asymptotic normality which induces a confidence interval for the regression function usable in practice since it does not depend on any unknown quantity. We also give the explicit form of the conditional bias term. Note that the ergodic framework is more convenient in practice since it does not need the verification of any condition as in the mixing case for example.

Asymptotic normality of kernel estimators of the conditional mode under strong mixing hypothesis

Let (X n ,Y n ) n ≤1 be a R d ×R valued stationary process. Define the estimator of the conditional mode of Y 1 given X 1=x as the random variable θ n (x) that maximizes a kernel estimator of the conditional density of Y 1 given X 1 = x. We establish asymptotic normality of θ n (x) when the process (X n ,Y n ) n ≤1 is assumed to be strongly mixing. We derive from our results asymptotic normality of a predictor and propose a confidence bands for the conditional mode function. A simulation study shows how good the normality of the conditional mode function estimator is when dealing with samples of finite sizes.

On the asymptotic normality of the kernel estimators of the density function and its derivatives under censoring

In this paper, we study asymptotic normality of the kernel estimators of the density function and its derivatives as well as the mode in the randomly right censorship model. The mode estimator is defined as the random variable that maximizes the kernel density estimator. Our results are stated under some suitable conditions upon the kernel function, the smoothing parameter and both distributions functions that appear in this model. Here, the Kaplan–Meier estimator of the distribution function is used to build the estimates. We carry out a simulation study which shows how good the normality works.

Principes de grandes déviations pour l’estimateur à noyau de la densité de probabilité

Resume Nous etablissons dans cette note un principe de grandes deviations ponctuel et un principe de grandes deviations uniforme pour l’estimateur, par la methode du noyau, de la densite de probabilite. L’estimation est ici construite a partir de suites de variables aleatoires reelles independantes et identiquement distribuees.

UniformL 1-distance large deviations in nonparametric density estimation

In this paper, we are concerned with uniform large deviations probabilities for theL1-error in kernel density estimation. Several results are stated taking into account uniformity over classes of density functions as well as over families of kernels and intervals of smoothing parameters. Some limits are well-identified and are universal in the sense that they do not depend neither on the distribution of the observations nor on the estimation kernel. Our results allow to compare performances of the test based on theL1-deviation of the kernel density estimator to that of the Kolmogorov-Smirnov test when testing goodness-of-fit of composite hypotheses.

Efficiency of some tests when testing symmetry hypothesis

In this article, we introduce two new tests to test the symmetry of a distribution. Based on the L1-distance and the Kernel and histogram density estimation methods, these tests are compared via their Bahadur relative efficiency to several tests available in the literature. It arises that our tests reach better performances than a number of usual tests among whom we cite the sign test and the Kolmogorov–Smirnov test. Beforehand, large deviations results are stated for the associated statistics. The local asymptotic optimality relative to these tests is also studied.

ON THE CONDITIONAL HOMOSCEDASTICITY TEST IN AUTOREGRESSIVE MODEL WITH ARCH ERROR

ABSTRACT In this paper, we study the functional limiting law of the cumulative residual process associated to autoregression models with ARCH error when the data are assumed to be stationary and ergodic. Under homoscedasticity hypothesis of the model, it is stated that the limiting process is a time changed Wiener process plus a Gaussian random variable. On the basis of the law of the limiting process, we propose a chi-square type test to test the homoscedasticity hypothesis. A numerical comparisons of performances of our test, the Kolmogorov-Smirnov type test proposed by Chen and An[1] and the Lagrange multiplier test are carried out.

Un théorème de grandes déviations pour l’estimateur de la densité par la méthode de projection

Resume L’objet de cette Note est d’etablir un principe de grandes deviations ponctuel pour l’estimateur de la densite de probabilite par la methode de projection. Un resultat general est obtenu pour une base reguliere quelconque et des corollaires avec des fonctions de taux explicites sont deduits pour la base de Haar et la base trigonometrique. L’estimation est ici faite a partir de suites de variables aleatoires independantes et identiquement distribuees.

Large deviation results for the nonparametric regression function estimator on functional data

This paper is devoted to the study of large deviation behavior in the setting of the estimation of the regression function on functional data. A large deviation principle is stated for a process Zn, defined below, allowing to derive a pointwise large deviation principle for the Nadaraya- Watson-type l-indexed regression function estimator as a by-product. Moreover, a uniform over VC-classes Chernoff type large deviation result is stated for the deviation of the l-indexed regression estimator.

Fractal Dimensions for Some Increments of the Uniform Empirical Process

In this paper, we investigate the limiting behavior of increments of the uniform empirical process. More precisely, we are concerned by sets of exceptional oscillation points related to large and small increments. We prove that these sets are random fractals and evaluate their Hausdorff dimensions. This work is a complement to the previous investigations carried out by Deheuvels and Mason(6) where Csörgő–Révész–Stute-type increments are studied.