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Improved Berry-Esseen-Chebyshev Bounds with Statisical Applications

Published online by Cambridge University Press:  18 October 2010

Jean-Marie Dufour  and
Marc Hallin
Jean-Marie Dufour
Affiliation:
Université de Montréal
Marc Hallin
Affiliation:
Université Libre de Bruxelles
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Abstract

A Sharpening Of Nonuniform bounds of the Berry-Esseen type initially obtained by Esseen and later generalized by Kolodjažnyĭ–who also proved that they are, in some sense, optimal–is proposed. Further, the corresponding inequalities are shown to provide uniformly improved Chebyshev bounds for the tail probabilities of the distribution functions to be approximated. In contrast with most results on Berry–Esseen bounds, which emphasize rates of convergence to normality, the bounds proposed are sufficiently explicit to allow the computation of numerical bounds on a distribution function. For example, they can be applied to the sum of a small number of independent random variables. The bounds are easy to compute and can be used in confidence estimation as well as in testing problems. Applications include signed-rank tests, permutation tests, and the chi-square approximation to Bartlett's test statistic for the homogeneity of several variances.

Type
Articles
Information
Econometric Theory , Volume 8 , Issue 2 , June 1992 , pp. 223 - 240
Copyright
Copyright © Cambridge University Press 1992

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