On Estimating the Order of an ARMA Process

Abstract Estimating the structure of an adequate model is a crucial step in system identification. Different criteria have been proposed to determine the orders of a scalar autoregressive-moving average model. Most of them rely upon the maximization of likelihood functions, a complex, time-consuming task. We propose a fully different approach based upon the determination of the rank of estimated matrices. Numerous heuristic tests have already been derived around the relations existing between the rank of covariance matrices and ARMA orders. Using some results from matrix perturbation theory and the asymptotic properties of sample serial covariances, we present a test allowing to decide how many eigenvalues of an estimated covariance matrix should be declared equal to zero. To verify the usefulness of these asymptotic developments, some simulations are proposed. The results are in agreement with the predictions for quite small sample sizes and the performances are satisfactory.