Mohamed El Machkouri

Asymptotic normality of the Parzen–Rosenblatt density estimator for strongly mixing random fields

We prove the asymptotic normality of the kernel density estimator (introduced by Rosenblatt, Proc Natl Acad Sci USA 42:43–47, 1956 and Parzen, Ann Math Stat 33:1965–1976, 1962) in the context of stationary strongly mixing random fields. Our approach is based on the Lindeberg’s method rather than on Bernstein’s small-block-large-block technique and coupling arguments widely used in previous works on nonparametric estimation for spatial processes. Our method allows us to consider only minimal conditions on the bandwidth parameter and provides a simple criterion on the strong mixing coefficients which do not depend on the bandwidth.

Asymptotic normality of kernel estimates in a regression model for random fields

We establish the asymptotic normality of the regression estimator in a fixed-design setting when the errors are given by a field of dependent random variables. The result applies to martingale-difference or strongly mixing random fields. On this basis, a statistical test that can be applied to image analysis is also presented.

Kahane–Khintchine inequalities and functional central limit theorem for stationary random fields

We establish new Kahane-Khintchine inequalities in Orlicz spaces induced by exponential Young functions for stationary real random fields which are bounded or satisfy some finite exponential moment condition. Next, we give sufficient conditions for partial sum processes indexed by classes of sets satisfying some metric entropy condition to converge in distribution to a set-indexed Brownian motion. Moreover, the class of random fields that we study includes [phi]-mixing and martingale difference random fields.

Orthomartingale-coboundary decomposition for stationary random fields

We provide a new projective condition for a stationary real random field indexed by the lattice

On local linear regression for strongly mixing random fields

\Z^d

On the asymptotic normality of frequency polygons for strongly mixing spatial processes

to be well approximated by an orthomartingale in the sense of Cairoli (1969). Ourmain result can be viewed as a multidimensional version of the martingale-coboundary decomposition method which the idea goes back to Gordin (1969). It is a powerfull tool for proving limit theorems or large deviations inequalities for stationary random fields when the corresponding result is valid for orthomartingales.

ON THE CENTRAL AND LOCAL LIMIT THEOREM FOR MARTINGALE DIFFERENCE SEQUENCES

We investigate the local linear kernel estimator of the regression function g of a stationary and strongly mixing real random field observed over a general subset of the lattice Zd. Assuming that g is differentiable with derivative g, we provide a new criterion on the mixing coefficients for the consistency and the asymptotic normality of the estimators of g and g under mild conditions on the bandwidth parameter. Our results improve the work of Hallin etal. (2004) in several directions.

A central limit theorem for stationary random fields

This paper establishes the asymptotic normality of frequency polygons in the context of stationary strongly mixing random fields indexed by

Nonparametric Regression Estimation for Random Fields in a Fixed-Design

\mathbb {Z}^d