Naâmane Laïb

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.

Exponential-type inequalities for martingale difference sequences. application to nonparametric regression estimation

We state large deviation inequalities for the maxima of partial sums of mar¬tingale difference sequences. Some implications of our inequalities in stating the rate of convergence in the law of large numbers and the consistency of nonparametric regression when the errors are martingale differences are given.

Generalised kernel smoothing for non-negative stationary ergodic processes

In this paper, we consider a generalised kernel smoothing estimator of the regression function with non-negative support, using gamma probability densities as kernels, which are non-negative and have naturally varying shapes. It is based on a generalisation of Hilles lemma and a perturbation idea that allows us to deal with the problem at the boundary. Its uniform consistency and asymptotic normality are obtained at interior and boundary points, under a stationary ergodic process assumption, without using traditional mixing conditions. The asymptotic mean squared error of the estimator is derived and the optimal value of smoothing parameter is also discussed. Graphical illustrations of the proposed estimator are provided for simulated as well as for real data. A simulation study is also carried out to compare our method with the competing local linear method.

Limiting distribution of weighted processes of residuals. Application to parametric nonlinear autoregressive models

Abstract We introduce methods for testing the goodness-of-fit of linear or nonlinear parametric autoregressive models of order 1, under stationarity and ergodicity assumptions. We establish two functional limit theorems for the process of deviations  n (·) between the weighted process of residuals under consideration and its parametric counterpart, under the null hypothesis H 0 . We discuss several possible tests based on these results and show that the half-sample method introduced by [10] for parametric distribution function models can be adapted to the present setting.

Nonparametric multivariate

In this paper, a nonparametric estimator is proposed for estimating the L1-median for multivariate conditional distribution when the covariates take values in an infi?nite dimensional space. The multivariate case is more appropriate to predict the components of a vector of random variables simultaneously rather than predicting each of them separately. While estimating the conditional L1-median function using the well-known Nadarya-Waston estimator, we establish the strong consistency of this estimator as well as the asymptotic normality. We also present some simulations and provide how to built conditional con?fidence ellipsoids for the multivariate L1-median regression in practice. Some numerical study in chemiometrical real data are carried out to compare the multivariate L1-median regression with the vector of marginal median regression when the covariate X is a curve as well as X is a random vector.

L_{1}

We establish a weak invariance principle for certain functionals of the regressogram estimator for regression or autoregression models where the data are strongly mixing. These functionals are constructed by cumulating the local discrepancies between the regressogram estimator and the corresponding regression function. As a byproduct, we obtain the limiting distribution of these functionals. Since the limiting process turns out to be a tractable time-changed Wiener process, we can derive from our results a family of possible nonparametric goodness-of-fit tests for the restriction to any compact interval of the regression or autoregression function. We then focus on a specially interesting test within this family. Using our preceding results, we provide estimates for the asymptotic behavior of the power of this test against both fixed and local alternatives.

-median regression estimation with functional covariates

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.