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.

翻译：我们研究了监督降维问题，特别关注用协变量向量$X \in \mathbb{R}^p$解释目标变量$Y\in\mathbb{R}$的极端值。其总体目标是定义并估计一个协变量空间中的低维子空间投影，该子空间足以用于预测目标变量超过高阈值的极端事件。我们提出了匹配这一目标的“尾部条件独立性”原创定义。受切片逆回归（Sliced Inverse Regression，SIR）方法的启发，我们发展了一个新框架（TIREX，Tail Inverse Regression for EXtreme response），用于估计极端充分降维（SDR）空间，其潜在维度低于经典SDR空间。我们证明了估计过程中涉及的尾部经验过程的弱收敛性，并通过模拟数据和真实世界数据验证了所提方法的相关性。