Max-stable Processes and Spatial Extremes

Max-stable processes arise from an infinite-dimensional generalisation of extreme value theory. They form a natural class of processes when sample maxima are observed at each site of a spatial process, a problem of particular interest in connection with regional estimation methods in hydrology. A general representation of max-stable processes due to de Haan and Vatan is discussed, and examples are given to show how it may be used to generate explicit examples of max-stable process. As a side-product, it is possible to generate a number of known multivariate extreme value families in this way, and in one case this suggests an extension of the family. The main contribution of the paper, however, is to define two new max-stable stochastic processes, related to the multivariate normal and multivariate t distributions. Statistical estimation and model checking are discussed, and the concepts illustrated by being applied to rainfall data.