On tail dependence parameters for non-continuous and autocorrelated margins
Victory Idowu
TL;DR
This work addresses the limitation that classical tail dependence measures $\lambda_U$ and $\lambda_L$ assume continuous margins, restricting their applicability to non-continuous or mixed margins. It introduces a volume-based generalized tail dependence defined through the copula-volume $V_C$, with $\tilde{\lambda}_U$ and $\tilde{\lambda}_L$, and extends the framework to generalized autocorrelation tail dependence using $V_{\tilde{C}}$. The authors prove that the generalized measures reduce to the standard ones under continuity and establish their well-definedness within a non-standard bivariate regular variation setting. The proposed geometric, volume-based approach provides a robust tool for assessing tail dependence across discrete and autocorrelated margins, with potential implications for risk modeling in finance and insurance.
Abstract
Tail dependence plays an essential role in the characterization of joint extreme events in multivariate data. However, most standard tail dependence parameters assume continuous margins. This note presents a form of tail dependence suitable for non-continuous and discrete margins. We derive a representation of tail dependence based on the volume of a copula and prove its properties. We utilize a bivariate regular variation to show that our new metric is consistent with the standard tail dependence parameters on continuous margins. We further define tail dependence on autocorrelated margins where the tail dependence parameter examine lagged correlation on the sample.