Extremes of integer-valued moving average sequences

Abstract

Extreme value theory, Binomial thinning, Extremal index, Integer-valued sequences, 60G70,

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

“…r.v's concentrated on the interval false(0;1false) with common distribution FA. The authors proved that if process has finite first and second moments then italicCorrfalse(Xt,Xt−kfalse)=false(αEfalse[Afalse])k,thinmathspacek∈ℕ. Moreover, extensions for the INMA model in 2.5 based on stochastic thinning sequences were considered by Hall et al. (2010), who assume that the βj’s form a sequence of independent r.v's independent of false(Ztfalse).…”

“…(2010), who assume that the βj’s form a sequence of independent r.v's independent of false(Ztfalse). Hall et al. (2010) analyzed the extremal properties (in particular the limiting distribution of the normalized maxima) of this class of INMA models.…”

A Periodic Bivariate Integer-Valued Autoregressive Model

“…Figure 7 shows the h-step-ahead joint predictive distribution P((x 1,T +h , x 2,T +h )|(x 1,T , x 2,T )), h = 1, 2, 3, 4. For the first season the mode is (10,11), for the second season this pair is (11,9), for the third season is (10,9) and in the last season is (11,9).…”

“…An application of the model proposed by [15] for the analysis of the number of hospital admissions per week caused by influenza can be found in [16]. In [11] was introduced a general class of periodic non-negative integer-valued moving average processes driven by a sequence of periodic integer-valued random variables with regularly varying tails. The authors analyzed some extremal properties related with this class of processes.…”

Subsampling weakly dependent time series and application to extremes

“…Of course this assumption can be relaxed when only considering the convergence of the normalized sample maximum: one may consider the example of an integer-valued moving average sequence given in Hall et al (2010), where it is shown that the limiting distribution of the normalized maxima is the same distribution as it would be obtained in the continuous case. This result has to be linked with the fact that the stationary distribution is heavy-tailed (see Andersson 1970 for further details).…”