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Asymptotic analysis of extremes from autoregressive negative binomial processes

Part of: Markov processes Limit theorems Stochastic processes

Published online by Cambridge University Press:  14 July 2016

William P. McCormick  and
You Sung Park
William P. McCormick*
Affiliation:
University of Georgia
You Sung Park*
Affiliation:
University of Georgia
*
∗ Postal address: Department of Statistics, University of Georgia, Athens, GA 30602, USA.
∗ Postal address: Department of Statistics, University of Georgia, Athens, GA 30602, USA.
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Abstract

It is well known that most commonly used discrete distributions fail to belong to the domain of maximal attraction for any extreme value distribution. Despite this negative finding, C. W. Anderson showed that for a class of discrete distributions including the negative binomial class, it is possible to asymptotically bound the distribution of the maximum. In this paper we extend Anderson's result to discrete-valued processes satisfying the usual mixing conditions for extreme value results for dependent stationary processes. We apply our result to obtain bounds for the distribution of the maximum based on negative binomial autoregressive processes introduced by E. McKenzie and Al-Osh and Alzaid. A simulation study illustrates the bounds for small sample sizes.

Keywords

EXTREME VALUE LIMITING DISTRIBUTIONAUTOREGRESSIVE PROCESS

MSC classification

Primary: 60F05: Central limit and other weak theorems
Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60G70: Extreme value theory; extremal processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 29 , Issue 4 , December 1992 , pp. 904 - 920
Copyright
Copyright © Applied Probability Trust 1992 

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