Tail dependence in small samples: from theory to practice

Abstract

Tail dependence is a probability-based concept meant to address the challenge of detecting and modeling the extreme comovements that can be observed in many real-life situations. Huge financial losses for a bank, floods and epidemics are obvious instances of such extreme comovements. Like extreme value theory in the univariate case, tail dependence depends on asymptotic theory. Therefore, the statistical assessment of tail dependence faces exactly the same problem as extreme value theory: a scarcity of extreme event observations. In the field of dependence modeling, copulas have stood out as a tool of singular importance. They are widely used to account for the various dependence structures that can be encountered in real life. In 2009, Genest et al provided a series of tests to achieve copula selection but showed that these tests were not greatly powerful. This is all the more true when it comes to selecting a copula where tail dependence is crucial. In this paper, we suggest the use of tail indexes in order to detect the presence of tail dependence in a given data set and thus improve the process of selecting a copula. Because tail dependence often goes with data scarcity, we focus on this specific issue through an application to operational losses in the banking industry and propose a way to apply the benefits from theory in practice, while being conscious of the boundaries of such a notion.