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Characterization and Estimation of the Multivariate Extremal Index

The multivariate extremal index has been introduced by Nandagopalan as a measure of the clustering among the extreme values of a multivariate stationary process. In this paper, we derive some additional properties and use those to construct a statistical estimation scheme. Central to the discussion is a class of processes we call M 4 processes, which are characterized by means of a multivariate generalization of a characterization due to Deheuvels for univariate max-stable processes. The multivariate extremal index for M 4 processes is derived, and certain properties established. We then discuss estimation of the multivariate extremal index.