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April 2003 Valid asymptotic expansions for the maximum likelihood estimator of the parameter of a stationary, Gaussian, strongly dependent process
Offer Lieberman, Judith Rousseau, David M. Zucker
Ann. Statist. 31(2): 586-612 (April 2003). DOI: 10.1214/aos/1051027882

Abstract

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.

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Offer Lieberman. Judith Rousseau. David M. Zucker. "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

Information

Published: April 2003
First available in Project Euclid: 22 April 2003

zbMATH: 1067.62021
MathSciNet: MR1983543
Digital Object Identifier: 10.1214/aos/1051027882

Subjects:
Primary: 62E17, 62M10

Rights: Copyright © 2003 Institute of Mathematical Statistics

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Vol.31 • No. 2 • April 2003
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