ARMA-Order estimation via matrix perturbation theory

Allmost all methods for estimating the model order in system identification -either involve the maximization of likelihood functions, a time-consuming task, -or rely upon the determination of the rank of (covariance) matrices, a task generally achieved by means of heuristic tests. We propose a scheme belonging to this second class of methods for which, however we theoretically justify the test. Using the asymptotic properties of sample serial covariances and some results from matrix perturbation theory, we obtain the statistical distribution of the "smallest" eigenvalue of -say- the Hankel matrix build upon the estimated covariances, under the hypothesis that the corresponding exact Hankel matrix possesses one single zero eigenvalue. This allows us to develop and justify a test which moreover only requires the knowledge of the "smallest" eigenvalue and an associated eigenvector. A complete eigendecomposition is thus not necessary further limiting the computations. The new order determination scheme is compared on simulated examples to the more time consuming approaches based on likelihood maximizations, the performance appear to be comparable.