Asymptotic efficiency in autoregressive processes driven by stationary Gaussian noise

In this study, almost sure asymptotic properties of the maximum likelihood estimator in an autoregressive process driven by stationary Gaussian noise are obtained. Precisely, we show that the maximum likelihood estimator is strongly consistent whereas it is well known that the least-square estimator may lack consistency, especially when the process is driven by a correlated noise. Furthermore, the local asymptotic normality of the likelihood ratio is established in order to build an asymptotically uniformly invariant most powerful procedure for testing the significance of the autoregressive parameter. Finally, we illustrate our results in a short simulation study.