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Title: The proportional tail framework for extreme quantile regression Authors:  Clement Dombry - Universite de Franche Comte (France) [presenting]
Benjamin Bobbia - Universite de Franche Comte (France)
Davit Varron - University of Franche-Comte (France)
Abstract: Extreme quantile regression is a long-standing issue in extreme value theory. The goal is to predict the quantile of order $1-p$ of the response $Y\in\mathbb{R}$ given covariates $X\in\mathbb{R}^d$ when $p=p(n)$ goes to zero as the sample size $n$ goes to infinity. It has many applications, for instance in the context of risk management to assess the Value at Risk of the daily log-return of an asset given covariates that account for the market situation. The purpose is to present new results for extreme quantile regression in the proportional tail framework. This is closely related to the framework of heteroscedastic extremes, where the extremes of independent non-identically random variables are considered - the extremes depends on time through the so-called skedasis function $\sigma$. The main assumptions are that the response variable $Y$ is heavy tailed and that the conditional tail function $\bar F_{Y\mid X=x}$ of $Y$ given $X=x$ is asymptotically proportional to the unconditional one $\bar F_Y$. We present an analysis of the proportional tail framework based on coupling techniques and focus on properties of estimators of the extreme value index, the skedasis function and the extreme conditional quantiles.