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Asymptotic analysis of extremes from autoregressive negative binomial processes

Part of: Stochastic processes Limit theorems Markov 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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References

Al-Osh, M. A. and Alzaid, A. A. (1987) First order integer-valued autoregressive (INAR(1)) process. J. Time Series Anal. 8, 261–275.CrossRefGoogle Scholar
Aldous, D. (1989) Probability Approximations via the Poisson Clumping Heuristic. Springer-Verlag, New York.CrossRefGoogle Scholar
Alzaid, A. A. and Al-Osh, M. (1990) An integer-valued pth-order autoregressive structure (INAR(p)) process. J. Appl. Prob. 27, 314–324.CrossRefGoogle Scholar
Anderson, C. W. (1970) Extreme value theory for a class of discrete distributions with applications to some stochastic processes. J. Appl. Prob. 7, 99–113.CrossRefGoogle Scholar
Asmussen, S. (1987) Applied Probability and Queues. Wiley, New York.Google Scholar
Athreya, K. B. (1988) On the maximum sequence in a critical branching process. Ann. Prob. 16, 502–507.CrossRefGoogle Scholar
Berman, S. M. (1986) Extreme sojourns for random walks and birth-and-death processes. Stoch. Models 2, 393–408.CrossRefGoogle Scholar
Chernick, M. R., Hsing, T. and Mccormick, W. P. (1991) Calculating the extremal index for a class of stationary sequences. Adv. Appl. Prob. 23, 835–850.CrossRefGoogle Scholar
Iglehart, D. L. (1972) Extreme values in the GI/G/1 queue. Ann. Math. Statist. 43, 627–635.CrossRefGoogle Scholar
Leadbetter, M. R., Lindgren, G. and Rootzen, H. (1983) Extremes and Related Properties of Random Sequences and Processes. Springer-Verlag, New York.CrossRefGoogle Scholar
Mccormick, W. P. and Park, Y. S. (1992) Approximating the distribution of the maximum queue length for M/M/s queues. To appear in Queueing Theory and Related Processes , ed. Basawa, I. V. and Bhat, U. N., Oxford University Press.Google Scholar
Mckenzie, E. (1986) Autoregressive-moving-average processes with negative binomial and geometric marginal distributions. Adv. Appl. Prob. 18, 679–705.Google Scholar
Mckenzie, E. (1988a) Some ARMA models for dependent sequences of Poisson counts. Adv. Appl. Prob. 20, 822–835.CrossRefGoogle Scholar
Mckenzie, E. (1988b) The distributional structure of finite moving-average processes. J. Appl. Prob. 25, 313–321.CrossRefGoogle Scholar
Rootzen, H. (1986) Extreme value theory for moving average processes. Ann. Prob. 14, 612–652.CrossRefGoogle Scholar
Rootzen, H. (1988) Maxima and exceedances of stationary Markov chains. Adv. Appl. Prob. 20, 371–390.CrossRefGoogle Scholar
Serfozo, R. F. (1988a) Extreme values of birth and death processes and queues. Stoch. Proc. Appl. 27, 291–306.CrossRefGoogle Scholar
Serfozo, R. F. (1988b) Extreme values of queue lengths in M/G/1 and GI/M/1 systems. Math. Operat. Res. 13, 349–357.CrossRefGoogle Scholar
Steutel, F. W. and Van Harn, K. (1979) Discrete analogues of self-decomposability and stability. Ann. Prob. 7, 893–899.CrossRefGoogle Scholar