Estimating the limiting shape of bivariate scaled sample clouds for self-consistent inference of extremal dependence properties

An integral part of carrying out statistical analysis for bivariate extreme events is characterising the tail dependence relationship between the two variables. For instance, we may be interested in identifying the presence of asymptotic dependence and/or in determining whether an individual variable can be large while the other is of smaller order. In the extreme value theory literature, various techniques are available to assess or model different aspects of tail dependence; currently, inference must be carried out separately for each of these, with the possibility of contradictory conclusions. Recent developments by Nolde and Wadsworth (2022) have established theoretical links between different characterisations of extremal dependence, through studying the limiting shape of an appropriately-scaled sample cloud. We exploit these results for inferential purposes, by first developing an estimator for the sample limit set and then using this to deduce self-consistent estimates for the extremal dependence properties of interest. In simulations, the limit set estimates are shown to be successful across a range of distributions, and the estimates of dependence features are individually competitive with existing estimation techniques, and jointly provide a major improvement. We apply the approach to a data set of sea wave heights at pairs of locations, where the estimates successfully capture changes in the limiting shape of the sample cloud as the distance between the locations increases, including the weakening extremal dependence that is expected in environmental applications.