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April 1997 Asymptotic expansion of M-estimators with long-memory errors
Hira L. Koul, Donatas Surgailis
Ann. Statist. 25(2): 818-850 (April 1997). DOI: 10.1214/aos/1031833675

Abstract

This paper obtains a higher-order asymptotic expansion of a class of M-estimators of the one-sample location parameter when the errors form a long-memory moving average. A suitably standardized difference between an M-estimator and the sample mean is shown to have a limiting distribution. The nature of the limiting distribution depends on the range of the dependence parameter $\theta$. If, for example, $1/3 < \theta < 1$, then a suitably standardized difference between the sample median and the sample mean converges weakly to a normal distribution provided the common error distribution is symmetric. If $0 < \theta < 1/3$, then the corresponding limiting distribution is nonnormal. This paper thus goes beyond that of Beran who observed, in the case of long-memory Gaussian errors, that M-estimators $T_n$ of the one-sample location parameter are asymptotically equivalent to the sample mean in the sense that $\Var(T_n)/\Var(\bar{X}_n) \to 1$ and $T_n = \bar{X}_n + o_P(\sqrt{\Var(\bar{X}_n)}).$

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Hira L. Koul. Donatas Surgailis. "Asymptotic expansion of M-estimators with long-memory errors." Ann. Statist. 25 (2) 818 - 850, April 1997. https://doi.org/10.1214/aos/1031833675

Information

Published: April 1997
First available in Project Euclid: 12 September 2002

zbMATH: 0885.62101
MathSciNet: MR1439325
Digital Object Identifier: 10.1214/aos/1031833675

Subjects:
Primary: 62M10
Secondary: 65G30

Rights: Copyright © 1997 Institute of Mathematical Statistics

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Vol.25 • No. 2 • April 1997
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