Estimation dans un processus de diffusion non linéaire à retards et tests.

Abstract

In this thesis we handle problems of unidimensional and multidimensional prametric estimation of delays as well as as some hypothesis testing problems on delay parameters for nonlinear diffusion processes. We rely on the observation of a nonlinear diffusion process of continuous time, with a small diffusion coefficient and assume satisfied some regularity conditions on the trend coefficient which, otherwise, remains nondifferentiable with respect to unknown parameters. We show then that the maximum likelihood estimator of delays parameters is consistent, asymptotically normal and uniformly LAM (locally asymptotically minimax) when the diffusion coefficient tends to 0. We mainly use the techniques of the parametric estimation methods theory in asymptotic statistics due to I. Ibragimov, R. Has’ minskii, Y. Kutoyants and others, as well as the LAM bound on the risks of estimators in a more general framework which is due to J. Hajek and L. Cam. We discuss, on the other hand, the qualities of some tests of simple and composite hypothesis. It is shown that the limit distributions of these statistics are independent of the basic hypothesis, which simplifies the calculation of the threshold. We also demonstrate that these tests are consistent by studying the behavior of statistics that define them under the alternatives. To handle the basic composite hypothesis we need to know the asymptotic behavior of statistic estimators of unknown parameters. The use of the maximum likelihood estimator brings us back to build the Cramer-von Mises corresponding test and study its limit distribution.