Tail Parameter Estimation using Non-Linear Regression on Tail Model Plots

The identification of an appropriate statistical model is a critical first step in many statistical analyses, including the estimation of extreme values. Probability and quantile plotting is often used to visualize the data and identify patterns that suggest an appropriate model. In extreme value estimation, some estimators of tail parameters can conveniently be interpreted as slope estimators in case of linear plots. One popular estimator for tail parameters is the Hill estimator, which can be viewed as an estimator of the slope in a log-log plot near the largest observations when a linear pattern appears in that area. This paper proposes a novel approach that generalizes the existing linear regression approach to non-linear regression, enabling the identification of appropriate tail models for basic distributions such as Pareto, log-normal, and Weibull models. Specifically, the proposed methodology uses the slope and non-linearity parameters to determine an appropriate tail model and support the estimation of important tail characteristics, such as extreme quantiles and return periods of extreme events.