Statistique asymptotique dans les modèles bilinéaires À changement de régime Markovien

Abstract

This thesis investigates some probabilistic properties and statistical applications of general Markovswitching bilinear processes (MS-BL) that offers remarkably rich dynamics and complex behavior to model non Gaussian data with structural changes. In these models, the parameters are allowed to depend on unobservable time-homogeneous and stationary Markov chain with finite state space. So, some basic issues concerning this class of models including necessary and sufficient conditions ensuring the existence of ergodic stationary (in some sense) solutions, existence of finite moments of any order and β-mixing are studied. As a consequence, we observe that the local stationarity of the underlying process is neither sufficient nor necessary to obtain the global stationarity. Also, the covariance functions of the process and its power are evaluated and it is shown that the second (resp. higher)-order structure is similar to a some linear processes, and hence admit ARMA representation. We establish also sufficient conditions for the MS-BL model to be β-mixing and geometrically ergodic. We then use these results to give sufficient conditions for β-mixing of a family of MS-GARCH(1,1) processes. A number of illustrative examples are given to clarify the theory and the variety of applications. Secondary, we illustrate the fundamental problems linked with MS-BL models, i.e., parameters estimation by considering a maximum likelihood (ML) approach. So, we provide the detail on the asymptotic properties of ML,in particular, we discuss conditions for its strong consistency. Finally, we used another approach for illustrate the fundamental problems linked with MS-BL models, i.e., parameters estimation by a minimum L₂-distance estimator (MDE). So, we provide the detail on the asymptotic properties of MDE, in particular, we discuss conditions for its consistency and asymptotic normality. Numerical experiments on simulated data sets are presented to highlight the theoretical results.