The Annals of Statistics
- Ann. Statist.
- Volume 31, Number 2 (2003), 586-612.
Valid asymptotic expansions for the maximum likelihood estimator of the parameter of a stationary, Gaussian, strongly dependent process
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.
Ann. Statist., Volume 31, Number 2 (2003), 586-612.
First available in Project Euclid: 22 April 2003
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Lieberman, Offer; Rousseau, Judith; Zucker, David M. Valid asymptotic expansions for the maximum likelihood estimator of the parameter of a stationary, Gaussian, strongly dependent process. Ann. Statist. 31 (2003), no. 2, 586--612. doi:10.1214/aos/1051027882. https://projecteuclid.org/euclid.aos/1051027882