The Annals of Statistics

Asymptotic expansion of M-estimators with long-memory errors

Hira L. Koul and Donatas Surgailis

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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

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

First available in Project Euclid: 12 September 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Moving average errors Appell polynomials second order efficiency


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.

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