Asymptotic analysis of extremes from autoregressive negative binomial processes

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.

Thinning-based models in the analysis of integer-valued time series: a review

“…Anderson (1970) gave a remarkable contribution by defining a particular class of discrete distributions for which the maximum term (under an i.i.d. setting) possesses an almost stable behaviour; extensions for certain stationary sequences were proposed by McCormick and Park (1992) and Hall (1996). The analysis of the extremal behaviour INMA-type models driven by innovations with distribution belonging either to Anderson's class and to the class of heavy-tailed distribution became a topic of lively research; details can be found in Turkman et al (2014).…”

Scaling laws for reliable data dissemination in shared loss multicast trees

“…Based on Andeson's work, several stationary integer-valued models have been studied. McCormick et al [20] analysed the limit distribution of a stationary sequence satisfying Leadbetter's conditions D(u n ) and D (u n ), and applied it to first order Auto Regressive (AR(1)) NB and geometric processes, using an operation called binomial thinning [52]. Hall [21] extended his result to a family of stationary processes, satisfying conditions D(u n ) and D k (u n ), and applied it to Max AR(1) processes.…”

On the maximum of periodic integer-valued sequences with exponential type tails via max-semistable laws

“…sequences and, as an example of application, the author analyzed the behavior of the maximum queue length for M/M/1 queues. McCormick and Park (1992) were the first to study the extremal properties of some models obtained as discrete analogues of continuous models, replacing scalar multiplication by random thinning. More recently, Hall (2001) provided results regarding the limiting distribution of the maximum of sequences within a generalized class of integer-valued moving averages driven by i.i.d.…”