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

Abstract

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.

Dependent conditional tail expectation for extreme levels

“…Similar investigations in the context of bivariate risk measures are worthwhile to pursue. In the framework of Theorem 4.1, i.e., in case k{pnpq Ñ 8, the limiting distribution is that of the tail index estimator used, and thus a bias-correction can be obtained by using a bias-corrected estimator for γ X , e.g., the ones proposed in Beirlant et al (1999) or Beirlant et al (2009). At the theoretical level this will require that we obtain the limiting distribution of such estimators for γ X in terms of the W R process.…”

Robust extreme quantile estimation for Pareto-type tails through an exponential regression model

“…, n − 1}. Note that in Beirlant et al (1999), the parameters of (2.7) are estimated via maximum likelihood principle and a similar quantile estimator of the form (2.10) can be obtained. This quantile estimator is used as a baseline for comparing the proposed estimator in the next section.…”

“…In this section, we compare the performance of the proposed quantile estimator with the equivalent minimum density power divergence quantile estimators of the Pareto-type tail index in the literature. Specifically, the proposed exponential regression model estimator based on log-spacings of order statistics, ERM Qp, the Dierckx et al (2013) estimator obtained from fitting an extended Pareto distribution to relative excesses, EPD Qp, and the simple non-robust maximum likelihood estimator of the extreme quantile estimator using exponential regression, MLE Qp, (Beirlant et al, 1999) i.e. the case where α = 0.…”

Transforming to Normality in Regression Analysis with Exponentially Residuals

“…ER finds extensive application in diverse fields, such as biology, economics, finance, epidemiology, and environmental sciences. This widely-used technique offers a potent tool for analyzing and comprehending exponential relationships, equipping researchers, analysts, and decision-makers with the means to make informed predictions and sound decisions based on the observed data [1].…”