The central limit theorem for sums of trimmed variables with heavy tails |
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Authors: | Istvá n Berkes,Lajos Horvá th |
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Affiliation: | a Institute of Statistics, Graz University of Technology, Münzgrabenstrasse 11, Graz, Austriab Department of Mathematics, University of Utah, Salt Lake City, UT 84112-0090, USA |
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Abstract: | Trimming is a standard method to decrease the effect of large sample elements in statistical procedures, used, e.g., for constructing robust estimators and tests. Trimming also provides a profound insight into the partial sum behavior of i.i.d. sequences. There is a wide and nearly complete asymptotic theory of trimming, with one remarkable gap: no satisfactory criteria for the central limit theorem for modulus trimmed sums have been found, except for symmetric random variables. In this paper we investigate this problem in the case when the variables are in the domain of attraction of a stable law. Our results show that for modulus trimmed sums the validity of the central limit theorem depends sensitively on the behavior of the tail ratio P(X>t)/P(|X|>t) of the underlying variable X as t→∞ and paradoxically, increasing the number of trimmed elements does not generally improve partial sum behavior. |
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Keywords: | 60F05 60E07 60G50 62G20 62G30 |
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