The Annals of Statistics

Edgeworth expansions for studentized statistics under weak dependence

S. N. Lahiri

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In this paper, we derive valid Edgeworth expansions for studentized versions of a large class of statistics when the data are generated by a strongly mixing process. Under dependence, the asymptotic variance of such a statistic is given by an infinite series of lag-covariances, and therefore, studentizing factors (i.e., estimators of the asymptotic standard error) typically involve an increasing number, say, of lag-covariance estimators, which are themselves quadratic functions of the observations. The unboundedness of the dimension of these quadratic functions makes the derivation and the form of the expansions nonstandard. It is shown that in contrast to the case of the studentized means under independence, the derived Edgeworth expansion is a superposition of three distinct series, respectively, given by one in powers of n−1/2, one in powers of [n/]−1/2 (resulting from the standard error of the studentizing factor) and one in powers of the bias of the studentizing factor, where n denotes the sample size.

Article information

Ann. Statist., Volume 38, Number 1 (2010), 388-434.

First available in Project Euclid: 31 December 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems
Secondary: 62E20: Asymptotic distribution theory 62M10: Time series, auto-correlation, regression, etc. [See also 91B84]

Cramer’s condition linear process M-estimators smooth function model spectral density estimator strong mixing


Lahiri, S. N. Edgeworth expansions for studentized statistics under weak dependence. Ann. Statist. 38 (2010), no. 1, 388--434. doi:10.1214/09-AOS722.

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