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February, 1984 On the Influence of Extremes on the Rate of Convergence in the Central Limit Theorem
Peter Hall
Ann. Probab. 12(1): 154-172 (February, 1984). DOI: 10.1214/aop/1176993380


Let $\bar{X}$ be the mean of a random sample from a distribution which is symmetric about its unknown mean $\mu$ and has known variance $\sigma^2$. The classical method of constructing a hypothesis test or confidence interval for $\mu$ is to use the normal approximation to $n^{\frac{1}{2}}(\bar{X} - \mu)/\sigma$. In order to make this procedure more robust, we might lightly trim the mean by removing extremes from the sample. It is shown that this procedure can greatly improve the rate of convergence in the central limit theorem, but only if the new mean is rescaled in a rather complicated way. From a practical point of view, the removal of extreme values does not make the test or confidence interval more robust.


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Peter Hall. "On the Influence of Extremes on the Rate of Convergence in the Central Limit Theorem." Ann. Probab. 12 (1) 154 - 172, February, 1984.


Published: February, 1984
First available in Project Euclid: 19 April 2007

zbMATH: 0534.62030
MathSciNet: MR723736
Digital Object Identifier: 10.1214/aop/1176993380

Primary: 60F05
Secondary: 60G50 , 62G30

Keywords: central limit theorem , Extremes , order statistics , rate of convergence , Sums of independent random variables

Rights: Copyright © 1984 Institute of Mathematical Statistics

Vol.12 • No. 1 • February, 1984
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