Problèmes d’estimation dans les processus AR aléatoires.

Abstract

We consider the class of resolvent estimators of the correlation operator ruling the functional autoregressive processes introduced by Mas. Under mild conditions on smoothing parameter, we establish exponential bounds and almost sure convergence of the resolvent estimators as well as convergence rates improving the existing results. As a consequence we derive asymptotic results on the resolvent predictors. Thereafter, we address the class of Hilbert space valued autoregressive process with random coefficients (RCARH). We derive limit theorems : strong law of great numbers, central limit theorem, compact law of the iterated logarithm and exponential inequalities and also we obtain rates of convergence. These results are crucial in the framework of Hilbert space autoregressive processes statistical analysis. We deal with resolvent estimators of the mean of random operators ruling a functional autoregressive process equation. Under mild conditions on the decay rate of a regularizing parameter, we obtain convergence in probability, exponential bounds, almost sure convergence and limiting law of the estimators and as well as results on resolvent predictors. These estimators achieve parametric rate p n (up to a logn factor). An estimator of the term variance of random operators is proposed and its convergence in probability is also shown. All these results extend and improve those of Mas in the framework of functional AR Processes with deterministic coefficients. Numerical studies and real data simulation’s are performed and adequately validate the efficiency of the resolvent predictors for Hilbertian deterministic autoregressive and random coefficient autoregressive iv Abstract models. The performance of the statistical predictor is measured by the errors forcasting and is compared to other methods existing in the literature showing competitive results.