Title: Nonasymptotic analysis of the angular measure for extremes, application to classification
Authors: Anne Sabourin - Telecom Paris, Institut Polytechnique de Paris (France) [presenting]
Hamid Jalalzai - Telecom ParisTech (France)
Stephan Clemencon - Telecom ParisTech (France)
Abstract: In multivariate extreme value theory, the angular measure characterizes the dependence structure of multivariate heavy-tailed variables. In the case where the components have different tail indices, standardization using the rank-transformation (empirical distribution function) is a common practice. We propose a modification of the classical empirical estimator based on the rank-transformed sample, based on intermediate data, ie. upon data which norm rank among the largest of the observed sample, but not among the very largest. In other word we discard the very largest data. We provide a nonasymptotic bound for the uniform deviations of the empirical angular measure evaluated on rectangles of the unit sphere. Our bound scales as the squared root of the number of observations used for inference. This nonasymptotic study is, to the best of our knowledge, the first of its kind in this domain. As an application, we provide finite sample guarantees for classification in extreme regions and anomaly detection via minimum-volume sets estimation on the sphere.