We establish the validity of an Edgeworth expansion to the distribution of the maximum likelihood estimator of the parameter of a stationary, Gaussian, strongly dependent process. The result covers ARFIMA-type models, including fractional Gaussian noise. The method of proof consists of three main ingredients: (i) verification of a suitably modified version of Durbin's general conditions for the validity of the Edgeworth expansion to the joint density of the log-likelihood derivatives; (ii) appeal to a simple result of Skovgaard to obtain from this an Edgeworth expansion for the joint distribution of the log-likelihood derivatives; (iii) appeal to and extension of arguments of Bhattacharya and Ghosh to accomplish the passage from the result on the log-likelihood derivatives to the result for the maximum likelihood estimators. We develop and make extensive use of a uniform version of a theorem of Dahlhaus on products of Toeplitz matrices; the extension of Dahlhaus' result is of interest in its own right. A small numerical study of the efficacy of the Edgeworth expansion is presented for the case of fractional Gaussian noise.
"Valid asymptotic expansions for the maximum likelihood estimator of the parameter of a stationary, Gaussian, strongly dependent process." Ann. Statist. 31 (2) 586 - 612, April 2003. https://doi.org/10.1214/aos/1051027882