At Erasmus University of Rotterdam

## Pasquale Cirillo (Delft University of Technology)

**Title**: Risk concentration and the inequality of tail

**Abstract**: Refurbishing the well-known Gini index as a measure of tail risk, I discuss a brand new set of inequality-based tools for the study of fat tails. As a side result, I will also deal with the estimation of the Gini index in the case of infinite variance, showing why its commonly used nonparametric estimator should be avoided under extremely fat tails. I will naturally explain the main theoretical aspects and properties, but I will also discuss heuristics and applications on interesting actual data, not so commonly found in the literature, related to war casualties, terrorism, op losses and bit coins.

## Axel Bücher (Ruhr-University Bochum)

**Title**: On a pseudo-maximum likelihood estimator for the extremal index

**Abstract**: The extremes of a stationary time series typically occur in clusters. A primary measure for this phenomenon is the extremal index, representing the reciprocal of the expected cluster size. Both a disjoint and a sliding blocks estimator for the extremal index, essentially due to Northrop (2015) [An efficient semiparametric maxima estimator of the extremal index. Extremes 18, 585–603], are analyzed in detail. In contrast to many competitors, the estimators only depend on the choice of one parameter sequence. We derive an asymptotic expansion, prove asymptotic normality and show consistency of an estimator for the asymptotic variance. Explicit calculations in certain models and a finite-sample Monte Carlo simulation study reveal that the sliding blocks estimator outperforms other blocks estimators, and that it is competitive to runs- and inter-exceedance estimators in various models. The methods are applied to a variety of financial time series.