At Delft University of Technology

## Yi He (University of Amsterdam)

**Title**: Rethinking Extreme Value Statistics

**Abstract**: I discuss three papers and invite rethinking of the fundamental designs in extreme value statistics. First, the tail dependence inference formulas for a diverging threshold is often redundant with that for an adaptive threshold, and the latter is more relevant in practice. Second, the univariate peaks-over-threshold method is no better than fitting a generalized Pareto model beyond an unknown finite threshold where naive bootstrapping is valid with a flexible choice of threshold. Finally, we show how nonrandom heterogeneous data generate heavy tails and propose novel modeling techniques for high dimensional datasets.

## Anne Sabourin (Télécom Paris, Institut polytechnique de Paris)

**Title**: Nonasymptotic analysis of the empirical angular measure for multivariate extremes, with applications to classification and minimum volume set estimation

**Abstract**: In multivariate extreme value theory, the angular measure characterizes the first order 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 provide a nonasymptotic bound for the uniform deviations of an 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 log(k)/√k up to a logarithmic factor. This nonasymptotic study is, to the best of our knowledge, the first of its kind in this domain. In addition we propose a modification of the classical empirical estimator based on the rank-transformed sample, based on intermediate data, i.e. 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. Our error bound for this modified estimator does not suffer from a logarithmic factor, but includes a multiplicative term depending on the truncation level. The relative merits of both versions of the empirical measure are illustrated by numerical experiments. As an application, we provide finite sample guarantees for classification in extreme regions and anomaly detection via minimum-volume sets estimation on the sphere. This is a joint work with Stéphan Clémençon, Hamid Jalalzai and Johan Segers