# Hanna Ahmed & Clément Dombry

## 2018/10/08

At Delft University of Technology

## Hanna Ahmed (Tilburg University)

Title: Improved estimation of the extreme value index using a related variables

Abstract: Heavy tailed phenomena are naturally analyzed by extreme value statistics. A crucial step in such an analysis is the estimation of the extreme value index, which describes the tail heaviness of the underlying probability distribution. We consider the situation where we have next to the n observations of interest another n + m observations of one or more related variables, like, e.g., financial losses due to earthquakes and the related amounts of energy released, for a longer period than that of the losses. Based on such a data set, we present an adapted version of the Hill estimator that shows greatly improved behavior and we establish the asymptotic normality of this estimator. For this adaptation the tail dependence between the variable of interest and the related variable(s) plays an important role. A simulation study confirms the substantially improved performance of our adapted estimator relative to the Hill estimator. We also present an application to the aforementioned earthquake losses.

This is a joint work with John H.J. Einmahl.

## Clément Dombry (University of Franche-Comté)

Title: Analysis of the proportional tail model for extreme quantile regression via a coupling approach

Abstract: Extreme quantile regression is a fundamental problem in extreme value theory. Assume that we observe an $n$-sample $(x_1,y_1),\ldots,(x_n,y_n)$ of a random variable $Y\in\mathbb{R}$ together with covariates $X\in\mathbb{R}^p$. Our goal is to estimate the conditional quantile of order $1-p$ of $Y$ given $X=x$. When $p$ is small, there is not enough observations and extrapolation further in the tail distribution is needed. We face an extreme value problem.

The purpose of the talk is to present an ongoing joint work with B.Bobbia and D.Varron on the proportional tail model where we assume that the conditional tails are asymptotically proportional to the unconditional tail, that is $P(Y>y\mid X=x)\sim \sigma(x)P(Y>y)$ as $y\to y^*$, the upper endpoint of the distribution. This framework was introduced in the slightly different context of heteroscedastic extremes in Einmahl et al. (JRSSB 2016) and the function $\sigma$ was coined the skedasis function. Assuming an extreme value condition for $Y$ together with the proportional tail model, the extreme quantile regression is reduced to the estimation of the skedasis function and the extreme value index. We present our results for such an estimation. Interestingly, we introduce different techniques for the proof as Einmahl et al.: we introduce coupling arguments relying on total variation and Wasserstein distances, whereas the original proof relies mostly on empirical process theory.