Extreme value theory (EVT) provides an elegant mathematical tool for statistically analyzing rare events. When data are collected from multiple population subgroups, the scientific interest of researchers would generally be to improve the estimates obtained directly from each subgroup because some subgroups may have less data available for extreme value analysis. To achieve this, we incorporate the mixed effects model (MEM) into the regression technique in EVT. In small area estimation, the MEM has attracted considerable attention as a primary tool for providing reliable estimates for subgroups with small sample sizes, that is, ``small area.'' The key idea of the MEM is to incorporate information from all subgroups into a single model and to borrow strength from other subgroups to improve estimates by subgroup. Using this property, the MEM may contribute to reducing the bias and variance of the direct estimates for each subgroup, which result from the asymptotic specification of EVT. This prompts us to evaluate MEM's effectiveness in EVT through theoretical studies and numerical experiments, including its application to assessing the risk of heavy rainfall in Japan.

翻译：极值理论（EVT）为罕见事件的统计分析提供了优雅的数学工具。当数据来自多个总体子组时，研究人员通常希望改善直接从各子组获得的估计结果，因为某些子组用于极值分析的数据可能较少。为实现这一目标，我们将混合效应模型（MEM）融入EVT的回归技术中。在小区域估计中，MEM作为为样本量较小的子组（即"小区域"）提供可靠估计的主要工具，已引起广泛关注。MEM的核心思想是将所有子组的信息整合到单一模型中，并借用其他子组的强度来改善各子组的估计。利用这一特性，MEM有助于减少各子组直接估计的偏差和方差，这些偏差和方差源于EVT的渐近设定。这促使我们通过理论研究和数值实验评估MEM在EVT中的有效性，包括将其应用于评估日本强降雨风险。