Étude asymptotique dans l’inférence statistique.

Abstract

The aim of this thesis is to present a novel mode of convergence of sequences of random variables, and to establish a better complete convergence rate's for kernel estimators of functional characteristic’s of unknown probability distribution. Complete second order moment convergence with rate is the name of this kind of convergence, which implies almost complete convergence and gives a smaller rate of convergence. Indeed, this mode is easier to obtain and gives better performances than those of the almost complete convergence in the case of the non parametric estimators with kernels of the density function, of the distribution function and of the quantile function. A great advantage of the proposed approach is that less conditions are imposed to the kernel function thanks to the use of the mean squared error expression.