On the Extremal Behaviour of Generalised Periodic Sub-Sampled Moving Average Models with Regularly Varying Tails

Abstract

Let {X k } be a stationary moving average sequence of the form X k = Σj = −∞∞ β j *Zk − j where {Z k } is an iid sequence of random variables with regularly varying tails, and the operator * denotes multiplication if Z k is continuous and binomial thinning if Z k is a non-negative integer-valued. Let \(g{\left( k \right)}:\mathbb{N}_{{\text{0}}} \to \mathbb{N}_{{\text{0}}} \) be a strictly increasing sequence with a periodic pattern of the form g(k + I) = g(k) + M for some fixed integers I and M verifying 1 ≤ I ≤ M. Define Y k = Xg(k) as the generalised periodic sub-sampled moving average sequence. In this work we look at the extremal properties of {Y k }. In particular, we investigate the limiting distribution of the sample maxima and the corresponding extremal index. Motivation comes from the comparison of schemes for monitoring a variety of medical, finance, environmental, and social science data sets.