Gauss-Newton and M-estimation for ARMA processes with infinite variance

We consider two estimation procedures, Gauss-Newton and M-estimation, for the parameters of an ARMA (p,q) process when the innovations belong to the domain of attraction of a nonnormal stable distribution. The Gauss-Newton or iterative least squares estimate is shown to have the same limiting distribution as the maximum likelihood and Whittle estimates. The latter was derived recently by Mikosch et al. (1995). We also establish the weak convergence for a class of M-estimates, including the case of least absolute deviation, and show that, asymptotically, the M-estimate dominates both the Gauss-Newton and Whittle estimates. A brief simulation is carried out comparing the performance of M-estimation with iterative and ordinary least squares. As suggested by the asymptotic theory, M-estimation, using least absolute deviation for the loss function, outperforms the other two procedures.