Estimation, validation and identification of multivariate weak ARMA models.

Summary In this thesis we extend the scope of vector ARMA (AutoRegressive Moving-Average) models by considering error terms that are uncorrelated but may contain nonlinear dependencies. These models are called vector weak ARMAs and allow to deal with processes that may have very general nonlinear dynamics. In contrast, we call strong ARMAs the models typically used in the literature in which the error term is assumed to be iid noise. Since weak ARMA models are in particular dense in the set of regular stationary processes, they are much more general than strong ARMA models. The problem we will be concerned with is the statistical analysis of vector weak ARMA models. More precisely, we study the estimation and validation problems. First, we study the asymptotic properties of the near-maximum likelihood estimator and the least squares estimator. The asymptotic variance matrix of these estimators is of the "sandwich" form, and can be very different from the asymptotic variance obtained in the strong case. Then, we pay particular attention to the validation problems. First, we propose modified versions of the Wald, Lagrange multiplier and likelihood ratio tests to test linear restrictions on the parameters of weak vector ARMA models. Second, we are interested in residual-based tests, which aim at verifying that the residuals of the estimated models are indeed white noise estimates. In particular, we are interested in portmanteau tests, also called autocorrelation tests. We show that the asymptotic distribution of the residual autocorrelations is normally distributed with a covariance matrix different from the strong case (i.e. under idd assumptions on the noise). We derive the asymptotic behavior of the port-mantle statistics. In the standard strong ARMA framework, it is known that the asymptotic distribution of portmanteau tests is correctly approximated by a chi-square. In the general case, we show that this asymptotic distribution is that of a weighted sum of chi-squares. This distribution can be very different from the usual chi-deux approximation of the strong case. We therefore propose modified portmanteau tests to test the adequacy of weak vector ARMA models. Finally, we are interested in the choice of weak vector ARMA models based on the minimization of an information criterion, namely the one introduced by Akaike (AIC). With this criterion, we try to approximate the distance (often called Kullback-Leibler information) between the true law of the observations (unknown) and the law of the estimated model. We will see that the corrected criterion (AICc) in the context of weak vector ARMA models can, again, be very different from the strong case.