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Name: Michaël Lalancette

Talk Title: Concentration and asymptotic normality of the empirical variogram, with application to structure learning

Abstract: Multivariate peaks-over-threshold methods often seek to fit a multivariate Pareto distribution to a subset of the data where at least one measured variable is large. An important property of multivariate Pareto distributions is a certain variogram matrix. For instance, if the data is attracted to a Hüsler-Reiss distribution, its extremal dependence structure is fully characterized by this matrix. In this talk we introduce novel theory for the empirical variogram, a natural nonparametric estimator of the variogram, both in the finite sample and asymptotic regimes.

We first present concentration inequalities that guarantee uniform consistency of the empirical variogram, while allowing the dimension of the matrix to grow superpolynomially in the sample size. We further showcase how the bounds may be used to justify different model selection techniques for graphical extreme value theory in high dimension. Examples include a minimum weight spanning tree and a graphical LASSO procedure.

In fixed dimension, we also characterize the asymptotic distribution of the empirical variogram. For this purpose, we prove weighted uniform weak convergence of a certain tail empirical process on infinite L-shaped sets, a result of its own interest.

This talk is an invited talk at EVA 2021.