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.

On the maximum term of MA and Max-AR models with margins in Anderson's class

“…Nevertheless, there has been some interest in this topic and several authors have already studied the extremal behavior of some integer-valued time series models. These in clude a paper by Serfozo [19] which considers queue lengths in M/G/l and GI/M/1 systems, two papers by McCormick and Park [14] and [15] which consider AR negative binomial processes and queue lengths of M/M/s sys tems, a paper by Hall [12] considering a max-AR process, and two other papers by Hall [10] and [8] considering sequences within a generalized class of integer-valued MA models when driven by independent and identically distributed (i.i.d.) heavy tailed or exponential type tailed innovations.…”

Extremes of Integer-Valued Moving Average Models with Exponential Type Tails

“…For dependent data, McCormick and Park (1992a) have extended Theorem 1 for this class of distributions as follows.…”

“…Like the continuous analogue, the INAR(1) may be written as an infinite moving average, now of form (2) with a particular dependence structure for the thinning operations (see Al-Osh and Alzaid, (1988)). The marginal distribution of this model must be discrete self-decomposable (in fact any discrete self-decomposable distribution may be a marginal distribution for {X n }, see for instance McCormick and Park, (1992a)). A discrete distribution of N 0 with probability generating functionP P h…”

Extremes of integer-valued moving average sequences

“…sequences. 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. 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.…”