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

Abstract

In this paper we study the limiting distribution of the maximum term of non-negative integervalued moving average sequences of the form X n ¼ P 1 i ¼ À1 i Z n À i where {Z n } is an iid sequence of nonnegative integer-valued random variables with exponential type tails of the formand ) denotes binomial thinning. Several models are considered allowing different dependence structures of the thinning operations. For these models results are established which present similarities with those obtained for the classic linear Fmoving average: {X n } behaves as if it was iid regarding the limiting distribution of the maximum term. The paper concludes with some examples which apply the results to a particular model, the INAR(1), and with a simulation study to illustrate the results.

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

“…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.…”

“…Extensions for exponential type-tails innovations have been studied by Hall (2003). Also of interest for the present work is the paper by Hall (1996) who studies the extremes of a maxautoregressive model for integer-valued data.…”

“…Most of the work on extremes of integer-valued models consider stationary models and as far as we know only one recent work considers the extremes of non-stationary sequences: Hall and Scotto (2006) who consider the extremes of periodic integer-valued models with exponential type tails. Hall and Scotto (2006) consider periodic integer-valued sequences with exponential type tails, but using a different approach.…”

“…Hall and Scotto (2006) consider periodic integer-valued sequences with exponential type tails, but using a different approach. In all cases considered, the marginal distribution of the process and the distribution of the innovations do not belong to the domain of attraction of any extreme value distribution but are included in a class of distributions considered by Anderson (1970) …”

“…The second class is a first order max-autoregressive periodic sequence for which clusters of exceedances are formed and therefore the extremal index is less than unity. Both these classes have been considered by Hall and Scotto (2006) although in this work we use a different approach. Whereas Hall and Scotto only obtain bounds for the limiting distribution of the maximum we manage to obtain a non-degenerate limiting distribution for the maximum under the MSS setup.…”