This work focuses on dimension-reduction techniques for modelling conditional extreme values. Specifically, we investigate the idea that extreme values of a response variable can be explained by nonlinear functions derived from linear projections of an input random vector. In this context, the estimation of projection directions is examined, as approached by the Extreme Partial Least Squares (EPLS) method--an adaptation of the original Partial Least Squares (PLS) method tailored to the extreme-value framework. Further, a novel interpretation of EPLS directions as maximum likelihood estimators is introduced, utilizing the von Mises-Fisher distribution applied to hyperballs. The dimension reduction process is enhanced through the Bayesian paradigm, enabling the incorporation of prior information into the projection direction estimation. The maximum a posteriori estimator is derived in two specific cases, elucidating it as a regularization or shrinkage of the EPLS estimator. We also establish its asymptotic behavior as the sample size approaches infinity. A simulation data study is conducted in order to assess the practical utility of our proposed method. This clearly demonstrates its effectiveness even in moderate data problems within high-dimensional settings. Furthermore, we provide an illustrative example of the method's applicability using French farm income data, highlighting its efficacy in real-world scenarios.

翻译：本研究聚焦于条件极值建模中的降维技术。具体而言，我们探讨响应变量的极值可通过输入随机向量线性投影所导出的非线性函数进行解释这一思想。在此框架下，我们研究了投影方向的估计问题，该方法由极端偏最小二乘法（EPLS）实现——这是对原始偏最小二乘法（PLS）的改进，专门适用于极值分析框架。进一步地，我们提出了一种将EPLS方向解释为最大似然估计量的新视角，该方法利用了应用于超球的von Mises-Fisher分布。通过引入贝叶斯范式，我们增强了降维过程，使得先验信息能够融入投影方向的估计中。我们在两种特定情形下推导了最大后验估计量，阐明其本质是对EPLS估计量的正则化或收缩处理。同时，我们建立了当样本量趋于无穷时的渐近性质。通过模拟数据研究评估了所提方法的实际效用，结果明确表明即使在高维情境下的中等规模数据问题中，该方法仍具有显著有效性。此外，我们以法国农场收入数据为例展示了该方法在实际场景中的应用，突显了其在现实问题中的卓越性能。