Extreme Value Theory: A Projection Estimator for the Angular Dependence Function

Extreme value theory provides a principal framework for modeling rare and extreme events, with applications in fields such as environmental science, finance, and engineering. Classical multivariate extreme value theory describes the limiting behaviour of normalised random vectors, particularly when the components are asymptotically dependent. However, many practical applications require a more flexible description of extremal dependence that can accommodate both asymptotic dependence and asymptotic independence. The Angular Dependence Function (ADF), introduced by Wadsworth and Tawn (2013), provides a flexible and interpretable way to characterize extremal dependence by describing how the rate of joint tail decay varies with the relative contribution of each component. Accurate estimation of the ADF is critical for understanding and modeling joint extremes.

While several estimators have been proposed for the ADF, they face significant limitations in finite samples, including high variability, irregular behavior across the domain, and violations of key theoretical constraints. The violations of the theoretical constraints are problematic and current approaches to addressing these issues are typically ad hoc, involving post hoc adjustments.

To resolve the issue of the violations, this work introduces a projection-based estimator for the ADF. Inspired by projection methods for the Pickands dependence function introduced in Fils-Villetard et al. (2008), the proposed method projects an initial non-parametric estimate onto a closed, convex set of admissible functions in L²([0, 1]). By construction, this estimator strictly enforces the theoretical upper and lower bounds. A simulation study across various Gaussian dependence levels demonstrates its ability to preserve validity and eliminate violations. This proposed method focuses on convex ADFs under positive quadrant dependence, reflecting the dependence structures most commonly observed in practice and other literature.

To join this seminar virtually, please request Zoom connection details from hr.ops@stat.ubc.ca.