Mixed effects models for large sized clustered extremes

Extreme value theory (EVT) provides an elegant mathematical tool for statistical analysis of rare events. Typically, when data are collected from multiple clusters, analysts want to preserve cluster information, such as region, period, and group. To consider large-sized cluster information in extreme value analysis, we incorporate the mixed effects model (MEM) into the regression technique in EVT. In the field of small area estimation, it is well known that the MEM is an important tool for providing reliable estimates of large-sized clusters with small sample sizes. In the context of EVT for rare event analysis, the sample size of extreme value data for each cluster is often small. Therefore, the MEM may contribute to improving the predictive accuracy of extreme value analysis. This motivates us to verify the effectiveness of the MEM in EVT through theoretical studies and numerical experiments, including its application to the risk assessment of heavy rainfall in Japan.