## Short course by Prof. John Einmahl and Prof. Laurens de Haan

University of Aveiro, Portugal

Department of Mathematics, Room Sousa Pinto

Duration: 12 hours

Department of Mathematics, Room Sousa Pinto

Duration: 12 hours

## Course description

Many results in extreme value statistics have been derived with the aid of empirical process theory. It is the purpose of this short course to show that (and how) empirical processes, and in the univariate case also the related quantile processes, are an indispensable tool for extreme value statistics. First we will present and derive results for empirical processes, in particular for (weighted) tail empirical processes. The results include probability inequalities and various weak convergence (asymptotic normality) results. Also results for tail quantile processes will be discussed. Then it will be shown how this theory can be used in univariate and multivariate extreme value statistics: we will derive the asymptotic normality of estimators of the extreme value index and of estimators of the tail dependence structure. The thus obtained results will be used in the estimation of high quantiles and of probabilities of extreme events. We will also use tail empirical process theory to handle heteroscedastic extremes.

## Lecturers

**John H.J. Einmahl**is professor of Statistics at the Department of Econometrics and research fellow at CentER, both at Tilburg University. He obtained his Ph.D. from Nijmegen University and is an elected fellow of the Institute of Mathematical Statistics.

John H.J. Einmahl has published in the leading journals in Statistics and Probability Theory. His main research area is nonparametric statistics and its ramifications, including empirical processes, statistics of extremes, empirical likelihood and generalized quantiles. He is or was an Associate Editor of several journals including The Annals of Statistics, The Annals of Probabilty, and Extremes. He was a co-director of the Stochastics of Extremes and Risk Analysis program at Eurandom (Eindhoven) from 2001-2004. In 1998 he visited Florida State University as a Senior Fulbright Scholar.

**Laurens de Haan**received his MA in Mathematics from the University of Amsterdam in 1966, and his Ph.D. in Mathematics in 1970 for the thesis "On regular variation and its application to the weak convergence of sample extremes".

Laurens de Haan started his academic career in 1966 as researcher probability and statistics at the Mathematisch Centrum, Amsterdam. In the years 1971-72 he was Visiting Assistant Professor at Stanford University. In 1977 he was appointed Professor of Probability and Mathematical Statistics at the Erasmus Universiteit, where he stayed until his retirement in 1998. From 1990 to 1992 he was Associate Dean of the School of Economics. From 2008 to 2011 he was part time Professor of Statistics at the University of Tilburg. In 1977 he was elected Fellow I.M.S, and he was Guest Professor at the Peking University in 1994. He was awarded a Doctor

*honoris causa*from the Universidade de Lisboa in 2000 and the Medallion lecture at the I.M.S. annual meeting in Gothenburg in 2000 [wikipedia].

Short course organised by DEXTE - Development of Extremes in Time and Space, project EXPL/MAT-STA/0622/2013