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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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

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

First available in Project Euclid: 19 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


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

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


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.

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