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A clustering law for some discrete order statistics

Part of: Nonparametric inference

Published online by Cambridge University Press:  14 July 2016

Sunder Sethuraman
Sunder Sethuraman*
Affiliation:
Iowa State University
*
∗ Postal address: 400 Carver Hall, Department of Mathematics, Iowa State University, Ames, IA 50011, USA. Email address: [email protected]
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Abstract

Let X1, X2, …, Xn be a sequence of independent, identically distributed positive integer random variables with distribution function F. Anderson (1970) proved a variant of the law of large numbers by showing that the sample maximum moves asymptotically on two values if and only if F satisfies a ‘clustering’ condition, In this article, we generalize Anderson's result and show that it is robust by proving that, for any r ≥ 0, the sample maximum and other extremes asymptotically cluster on r + 2 values if and only if Together with previous work which considered other asymptotic properties of these sample extremes, a more detailed asymptotic clustering structure for discrete order statistics is presented.

Keywords

Order statisticslaw of large numbersextremesasymptoticsclustering

MSC classification

Primary: 62G32: Statistics of extreme values; tail inference
Type
Research Papers
Information
Journal of Applied Probability , Volume 40 , Issue 1 , March 2003 , pp. 226 - 241
Copyright
Copyright © Applied Probability Trust 2003 

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Footnotes

Research supported in part by NSF grant DMS-00711504.

References

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