Mixed effects models for extreme value index regression

Abstract: Extreme value theory (EVT) provides an elegant mathematical tool for the statistical analysis of rare events. When data are collected from multiple population subgroups, because some subgroups may have less data available for extreme value analysis, a scientific interest of many researchers would be to improve the estimates obtained directly from each subgroup. 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 producing reliable estimates for subgroups with small sample sizes, i.e., ``small areas.'' The key idea of MEM is to incorporate information from all subgroups into a single model and to borrow strength from all subgroups to improve estimates for each subgroup. Using this property, in extreme value analysis, the MEM may contribute to reducing the bias and variance of the direct estimates from each subgroup. This prompts us to evaluate the effectiveness of the MEM for EVT through theoretical studies and numerical experiments, including its application to the risk assessment of a number of stocks in the cryptocurrency market.