  We consider the problem of supervised dimension reduction with a particular
focus on extreme values of the target $Y\in\mathbb{R}$ to be explained by a
covariate vector $X \in \mathbb{R}^p$. The general purpose is to define and
estimate a projection on a lower dimensional subspace of the covariate space
which is sufficient for predicting exceedances of the target above high
thresholds. We propose an original definition of Tail Conditional Independence
which matches this purpose. Inspired by Sliced Inverse Regression (SIR)
methods, we develop a novel framework (TIREX, Tail Inverse Regression for
EXtreme response) in order to estimate an extreme sufficient dimension
reduction (SDR) space of potentially smaller dimension than that of a classical
SDR space. We prove the weak convergence of tail empirical processes involved
in the estimation procedure and we illustrate the relevance of the proposed
approach on simulated and real world data.
