ESTIMATING THE EXTREMAL INDEX by RICHARD L . SMITH

The extremal index is an important parameter measuring the degree of clustering of extremes in a stationary process. If we consider the point process of exceedance times over a high threshold, then this can be shown to converge asymptotically to a clustered Poisson process. The extremal index, a parameter between 0 and 1, is the reciprocal of the mean cluster size. Apart from being of interest in its own right, it is a crucial parameter for determining the limiting distribution of extreme values from the process. In this paper we review current work on statistical estimation of the extremal index, and consider an optimality criterion based on a bias-variance trade-off. Theoretical results are presented for some simple stochastic processes, and the practical implications are examined through simulations and some real data analysis.