The Annals of Probability

On the Influence of Extremes on the Rate of Convergence in the Central Limit Theorem

Peter Hall

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Abstract

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.

Article information

Source
Ann. Probab., Volume 12, Number 1 (1984), 154-172.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176993380

Digital Object Identifier
doi:10.1214/aop/1176993380

Mathematical Reviews number (MathSciNet)
MR723736

Zentralblatt MATH identifier
0534.62030

JSTOR
links.jstor.org

Subjects
Primary: 60F05: Central limit and other weak theorems
Secondary: 62G30: Order statistics; empirical distribution functions 60G50: Sums of independent random variables; random walks

Keywords
Central limit theorem extremes order statistics rate of convergence sums of independent random variables

Citation

Hall, Peter. On the Influence of Extremes on the Rate of Convergence in the Central Limit Theorem. Ann. Probab. 12 (1984), no. 1, 154--172. doi:10.1214/aop/1176993380. https://projecteuclid.org/euclid.aop/1176993380


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