Conditional Extreme Value Models: Fallacies and Pitfalls

TL;DR

This paper critically examines conditional extreme value models, highlighting their differences from classical multivariate extreme value theory, and discusses potential pitfalls and nuances in their application and interpretation.

Contribution

It clarifies the relationship between conditional and classical multivariate extreme value models, illustrating common misconceptions with examples and discussing the implications of marginal standardization.

Findings

Conditional models can behave counterintuitively compared to classical theory.

Marginal standardization may obscure true asymptotic behavior.

Relaxing model conditions can yield more comprehensive asymptotic characterizations.

Abstract

Conditional extreme value models have been introduced by Heffernan and Resnick (2007) to describe the asymptotic behavior of a random vector as one specific component becomes extreme. Obviously, this class of models is related to classical multivariate extreme value theory which describes the behavior of a random vector as its norm (and therefore at least one of its components) becomes extreme. However, it turns out that this relationship is rather subtle and sometimes contrary to intuition. We clarify the differences between the two approaches with the help of several illuminative (counter)examples. Furthermore, we discuss marginal standardization, which is a useful tool in classical multivariate extreme value theory but, as we point out, much less straightforward and sometimes even obscuring in conditional extreme value models. Finally, we indicate how, in some situations, a more comprehensive characterization of the asymptotic behavior can be obtained if the conditions of conditional extreme value models are relaxed so that the limit is no longer unique.