On Multivariate Extremes

Multivariate extreme value theory has traditionally been developed by studying the componentwise maxima of the observations. Recently, alternative approaches based on measuring the outlyingness of observations, using for example the Mahalanobis distance from the center of the distribution, have been proposed. This thesis features efforts to advance this point of view by, both, developing new results for the previously published methods and proposing new methods. The existing methods, the separating multivariate Hill estimator in particular, are developed further by settling important open questions concerning their asymptotic properties. As a salient example: We show that, under natural conditions, the separating Hill estimator is both consistent and asymptotically normal not only when the parameters of the underlying distribution are known, but in the practical case when the observations follow an unknown elliptical distribution. As a somewhat tangential aspect, we study a new way, the Delaunay outlyingness, to measure the outlyingness of multivariate observations that is based on the geometry of the sample. We study Delaunay outlyingness in the case of a compact convex region and a finite number of outliers and show, that at least in this situation, it has favorable asymptotic properties. Delaunay outlyingness is also studied through simulations which suggest that it's applicable beyond compact convex regions.