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Lieberman, Offer; Rousseau, Judith; Zucker, David M.

Valid asymptotic expansions for the maximum likelihood estimators of the parameter of a stationary, Gaussian, strongly dependent process. (English) Zbl 1067.62021

Ann. Stat. 31, No. 2, 586-612 (2003).
Summary: 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 J. Durbin’s [Biometrika 67, 311–333 (1980; Zbl 0436.62020)] 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 Ib M. Skovgaard [Int Stat. Rev. 54, 169–186 (1986; Zbl 0615.62018); Scand. J. Stat., Theory Appl. 8, 227–236 (1981; Zbl 0477.62009)] to obtain from this an Edgeworth expansion for the joint distribution of the log-likelihood derivatives; (iii) appeal to and extension of arguments of R. N. Bhattacharya and J. K. Ghosh [Ann. Stat. 6, 434–451 (1978; Zbl 0396.62010)] 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 R. Dahlhaus [ibid. 17, No. 4, 1749–1766 (1989; Zbl 0703.62091)] on products of Toeplitz matrices; the extension of Dahlhaus’ results 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.

Cited in 2 Reviews
Cited in 19 Documents

MSC:

62E20 Asymptotic distribution theory in statistics
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62E17 Approximations to statistical distributions (nonasymptotic)

Keywords:

long memory processes; ARFIMA models

Citations:

Zbl 0615.62018; Zbl 0477.62009; Zbl 0436.62020; Zbl 0396.62010; Zbl 0703.62091
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Full Text: DOI

References:

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