Valid Edgeworth Expansions for the Whittle Maximum Likelihood Estimator for Stationary Long-memory Gaussian Time Series

In this paper, we prove the validity of an Edgeworth expansion to the distribution of the Whittle maximum likelihood estimator for stationary long-memory Gaussian models with unknown parameter theta in Theta subset R^{d_{theta}} . The error of the (s-2)-order expansion is shown to be o(n^{(s-2)/2}) -- the usual iid rate -- for a wide range of models, including the popular ARFIMA(p,d,q) models. The expansion is valid under mild assumptions on the behavior of spectral density and its derivatives in the neighborhood of the origin. As a by-product, we generalize a Theorem by Fox and Taqqu (1987) concerning the asymptotic behavior of Toeplitz matrices. Lieberman, Rousseau, and Zucker (2002) (LRZ) establish a valid Edgeworth expansion for the maximum likelihood estimator for stationary long-memory Gaussian models. For a significant class of models, their expansion is shown to have an error of o(n-1). The results given here improve upon those of LRZ in that the results provide an Edgeworth expansion for an asymptotically efficient estimator, as LRZ do, but the error of the expansion is shown to be o(n^{-(s-2)/2}), not o(n^{-1}), for a broad range of models.