The Annals of Statistics

Asymptotic expansion of M-estimators with long-memory errors

Hira L. Koul and Donatas Surgailis

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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 $T_n = \bar{X}_n + o_P(\sqrt{\Var(\bar{X}_n)}).$

Article information

Source
Ann. Statist., Volume 25, Number 2 (1997), 818-850.

Dates
First available in Project Euclid: 12 September 2002

Permanent link to this document
https://projecteuclid.org/euclid.aos/1031833675

Digital Object Identifier
doi:10.1214/aos/1031833675

Mathematical Reviews number (MathSciNet)
MR1439325

Zentralblatt MATH identifier
0885.62101

Subjects
Primary: 62M10: Time series, auto-correlation, regression, etc. [See also 91B84]
Secondary: 65G30: Interval and finite arithmetic

Keywords
Moving average errors Appell polynomials second order efficiency

Citation

Koul, Hira L.; Surgailis, Donatas. Asymptotic expansion of M -estimators with long-memory errors. Ann. Statist. 25 (1997), no. 2, 818--850. doi:10.1214/aos/1031833675. https://projecteuclid.org/euclid.aos/1031833675


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  • EAST LANSING, MICHIGAN 48824-1027 LITHUANIA E-MAIL: koul@assist.stt.msu.edu