## Extreme quantile regression : a coupling approach and Wasserstein distance.

par - publié le

Vendredi 27 novembre 2020
Benjamin Bobbia
(Université de Bourgogne Franche-Comté)

In this work, we develop two coupling approaches for extreme quantile regression. We consider i.i.d copies of $Y \in \mathbb{R}$ and $X \in \mathbb{R}^d$and we want an estimation of the conditional quantile of $Y$ given $X = x$ of order $1-\alpha$ for a very small $\alpha > 0$.
We introduce the proportional tail model, strongly inspired by the heteroscedastic
extremes developped by Einmahl, de Haan and Zhou, where $Y$ has an heavy tail $\bar{F}$ with extreme value index $\gamma > 0$ and the conditional tails $\bar{F}_{x}$ are asymptotically equivalent to $\sigma(x) \bar{ F}$. We propose and study estimators of both model parameters and conditional quantile with are studied by coupling methods.
The first method is based on coupling of empirical processes while the second is related with optimal transport. Even if we establish the asymptotic normality of parameters estimators with both methods, the first is focused on the proper quantile estimation whereas the second is more focused on the estimation of in presence of bias and the elaboration of a validation procedure for our model.
Moreover, we also develop the optimal coupling approach in the general case of uni-variate extreme value theory.