Excess Functions and Estimation of the Extreme-Value Index

Abstract

A general class of estimators of the extreme-value index is generated using estimates of mean, medium and trimmed excess functions. Special cases yield earlier proposals in the literature, such as Pickands' (1975) estimator. A particular restatement of the mean excess function yields an estimator which can be derived from the slope at the right upper tail from a generalized quantile plot. From this viewpoint algorithms can be constructed to search for the number of extremes needed to minimize the mean square error of the estimator. Basic asymptotic properties of this estimator are derived. The method is applied in case studies of size distributions for alluvial diamonds and of wind speeds.

Extreme versions of Wang risk measures and their estimation for heavy-tailed distributions

“…The first motivating problem for extreme value theory is arguably to determine 1 how high the dykes surrounding the areas below sea level in the Netherlands should be so as to protect these zones from flood risk in case of extreme storms affecting Northern Europe, see de Haan and Ferreira (2006). Further climate-related examples are the estimation of extreme rainfall at a given location (Koutsoyiannis, 2004), the estimation of extreme daily wind speeds (Beirlant et al, 1996) or the modeling of large forest fires (Alvarado et al, 1998). Another stimulating topic comes from the fact that extreme phenomena may have strong adverse effects on financial institutions or insurance companies, and the investigation of those effects on financial returns makes up a large part of the recent extreme value literature.…”

Semi-parametric estimation for heavy tailed distributions

“…This estimator is the well-known Hill estimator (1975) when the parameter α and weights are one. The class of the so-called kernel estimators given by Csörgő et al (1985), as the one studied in Beirlant et al (1996), corresponds to a specific form of the weighted function with moreover α equal to one. Note also that Martins (1999, 2001) and Segers (2001) considered such type of estimators when the weighted function g is identically equal to one and α is some positive real.…”

Business Groups as Hierarchies of Firms: Determinants of Vertical Integration and Performance

“…Given their extremely long right tails, we resort to mean excess plots of Figure C.1, where graphs are the results of the following so called excess mean function (see Beirlant, Vynckier and Teugels, 1996;Coles, 2001):…”