Threshold Selection for Peak Over Threshold Models Using Logistic Regression

Abstract

Threshold selection remains a critical challenge in the application of Extreme Value Theory (EVT), particularly in actuarial science, where accurate modeling of extreme insurance claims is vital for solvency, capital adequacy, and reinsurance pricing. Traditional graphical tools, such as the mean residual life (MRL) plot, are highly dependent on visual interpretation and expert judgment, which limits reproducibility and consistency between practitioners. This paper proposes a machine learning approach using logistic regression to assign extremeness probabilities to insurance claims. The model is trained on labels generated from an initial quantile-based rule, classifying claims above the 90th percentile as extreme. The optimal threshold is determined as the smallest amount of claim with predicted probability exceeding a predefined cut-off (e.g., $90\%$). This probability-based rule provides an objective alternative to visual diagnostics and aligns well with classical EVT tools such as the MRL plot. After identifying exceedances above the selected threshold, a Generalized Pareto Distribution (GPD) is fitted using maximum likelihood estimation. Tail-based risk measures, including Value at Risk (VaR) and Expected Shortfall (ES), are then computed at the $99\%$ confidence interval to quantify the severity of potential extreme losses. The proposed framework is interpretable, reproducible, and readily applicable in actuarial workflows, offering a more consistent and automated solution for tail risk modeling.