In this paper we investigate an optimal property of maximum likelihood and quasi-maximum likelihood estimators of Gaussian autoregressive moving average processes by the second order approximation of the sampling distribution. It is shown that appropriate modifications of these estimators for Gaussian ARMA processes are second order asymptotically efficient if efficiency is measured by the degree of concentration of the sampling distribution up to second order. This concept of efficiency was introduced by Akahira and Takeuchi (1981).
"On the Second Order Asymptotic Efficiency of Estimators of Gaussian ARMA Processes." Ann. Statist. 11 (1) 157 - 169, March, 1983. https://doi.org/10.1214/aos/1176346066