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Uniform approximation to distributions of extreme order statistics

Published online by Cambridge University Press:  01 July 2016

R.-D. Reiss
R.-D. Reiss*
Affiliation:
University of Siegen
*
∗Postal address: Department of Mathematics, University of Siegen, Hölderlinstr. 3, 59 Siegen 21, West Germany.
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Abstract

This paper deals with asymptotic expansions of the distribution of the kth-largest order statistic Zn–k+1:n for the sample size n. These expansions establish higher-order approximations which hold uniformly over all Borel sets. In the particular case of the distribution of Zn–k+1:n under the uniform distribution and the exponential distribution, the approximating measures are linear combinations of ‘negative’ gamma distributions and linear combinations of extreme-value distributions. These results can be extended to the case of the joint distribution of the k largest order statistics. A numerical comparison to a different asymptotic expansion is given where the normal distribution is the leading term.

Keywords

EXTREME ORDER STATISTICSCOMPARISON OF APPROXIMATIONHIGHER-ORDER APPROXIMATIONS
Type
Research Article
Information
Advances in Applied Probability , Volume 13 , Issue 3 , September 1981 , pp. 533 - 547
Copyright
Copyright © Applied Probability Trust 1981 

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