Some benefits of standardisation for conditional extremes

A key aspect where extreme values methods differ from standard statistical models is through having asymptotic theory to provide a theoretical justification for the nature of the models used for extrapolation. In multivariate extremes, many different asymptotic theories have been proposed, partly as a consequence of the lack of ordering property with vector random variables. One class of multivariate models, based on conditional limit theory as one variable becomes extreme has received wide practical usage. The underpinning value of this approach has been supported by further theoretical characterisations of the limiting relationships. However, the paper “Conditional extreme value models: fallacies and pitfalls” by Holger Drees and Anja Janßen provides a number of counterexamples to these results. This paper studies these counterexamples in a conditional extremes framework which involves marginal standardisation to a common exponentially decaying tailed marginal distribution. Our calculations show that some of the issues identified can be addressed in this way.