A recent development in extreme value modeling uses the geometry of the dataset to perform inference on the multivariate tail. A key quantity in this inference is the gauge function, whose values define this geometry. Methodology proposed to date for capturing the gauge function either lacks flexibility due to parametric specifications, or relies on complex neural network specifications in dimensions greater than three. We propose a semiparametric gauge function that is piecewise-linear, making it simple to interpret and provides a good approximation for the true underlying gauge function. This linearity also makes optimization tasks computationally inexpensive. The piecewise-linear gauge function can be used to define both a radial and an angular model, allowing for the joint fitting of extremal pseudo-polar coordinates, a key aspect of this geometric framework. We further expand the toolkit for geometric extremal modeling through the estimation of high radial quantiles at given angular values via kernel density estimation. We apply the new methodology to air pollution data, which exhibits a complex extremal dependence structure.

翻译：近期极值建模领域的一项进展利用数据集的几何结构进行多元尾部推断。该推断中的一个关键量是规范函数，其数值定义了这种几何结构。迄今为止提出的捕捉规范函数的方法，要么因参数化设定而缺乏灵活性，要么在维度大于三时依赖于复杂的神经网络设定。我们提出了一种分段线性的半参数规范函数，其结构简单易于解释，并能对真实的基础规范函数提供良好近似。这种线性特性也使得优化任务在计算上成本低廉。分段线性规范函数可用于定义径向模型和角模型，从而实现对极值伪极坐标的联合拟合——这是该几何框架的一个关键方面。我们进一步通过核密度估计在给定角度值下估计高径向分位数，从而扩展了几何极值建模的工具集。我们将新方法应用于具有复杂极值依赖结构的空气污染数据。