Gaussian maximum likelihood estimation for ARMA models. I. Time series

Abstract

We provide a direct proof for consistency and asymptotic normality of Gaussian maximum likelihood estimators for causal and invertible autoregressive moving-average (ARMA) time series models, which were initially established by Hannan [Journal of Applied Probability (1973) vol. 10, pp. 130-145] via the asymptotic properties of a Whittle's estimator. This also paves the way to establish similar results for spatial processes presented in the follow-up article by Yao and Brockwell [Bernoulli (2006) in press]. © 2006 The Authors.