Moving-maximum models for extrema of time series

Abstract

We discuss moving-maximum models, based on weighted maxima of independent random variables, for extreme values from a time series. The models encompass a range of stochastic processes that are of interest in the context of extreme-value data. We show that a stationary stochastic process whose finite-dimensional distributions are extreme-value distributions may be approximated arbitrarily closely by a moving-maximum process with extreme-value marginals. It is demonstrated that bootstrap techniques, applied to moving-maximum models, may be used to construct confidence and prediction intervals from dependent extrema. Moreover, it is shown that bootstrapped moving-maximum models may be used to capture the dominant features of a range of processes that are not themselves moving maxima. Connections of moving-maximum models to more conventional, moving-average processes are addressed. In particular, it is proved that a moving-maximum process with extreme-value distributed marginals may be approximated by powers of moving-average processes with stably distributed marginals.