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：本工作聚焦于用于建模条件极值的降维技术。具体而言，我们研究了响应变量的极值可通过输入随机向量的线性投影推导出的非线性函数来解释这一思想。在此背景下，我们考察了由极端偏最小二乘法（EPLS）——一种针对极值框架量身定制的原始偏最小二乘法（PLS）的变体——所采用的投影方向估计方法。进一步地，我们引入了一种对EPLS方向的新颖解释，将其视为基于超球面上冯·米塞斯-费希尔分布的最大似然估计量。通过贝叶斯范式增强了降维过程，使得先验信息能够融入投影方向估计中。在两种特定情形下推导了最大后验估计量，将其阐述为EPLS估计量的正则化或收缩方法。我们还建立了其随着样本量趋于无穷大时的渐近行为。为了评估所提方法的实际效用，我们开展了模拟数据研究。即使在中等数据规模的高维问题中，该方法也明确展示了其有效性。此外，我们使用法国农场收入数据提供了该方法适用性的示例，突显了其在实际场景中的效能。