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

摘要

Let {X_k} be a stationary moving average sequence of the form X_k = ∑ from j=-∞ to ∞ of β_j ~* Z_(k-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(k) : N_0 → N_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 = X_(g(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.