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. of extremal dependence. For a given joint distribution, their results allow for the calculation of various extremal dependence measures, by considering the limiting shape of the sample cloud, under suitable scaling. This approach was motivated by results from Nolde (2014), who ﬁrst showed how to calculate the coeﬃcient of asymptotic independence of Ledford and Tawn (1996) in a similar way. We develop a ﬁrst approach to estimate the asymptotic, scaled sample cloud boundary. We then aim to exploit this, along with the theoretical results of Nolde (2014) and Nolde and Wadsworth (2022), to obtain estimates of several bivariate extremal dependence measures. This is the ﬁrst work to explore statistical inference based on these probabilistic limit results. Our approach ensures self-consistent inference across the measures we consider, for the ﬁrst time. In addition to this desirable feature, for individual extremal dependence measures, we show that our estimators are competitive with existing inferential methods. Abstract The aim of this Supplementary Material is to provide additional details for the paper ‘Estimating the limiting shape of bivariate scaled sample clouds for self-consistent inference of extremal dependence properties’. In Section A, we discuss the selection of tuning parameters for our limit set estimates ˆ G L and ˆ G , while Section B provides an illustration of our scaling/truncation approach and how we obtain estimates of several bivariate extremal dependence features from ˆ G . The remainder of the Supplementary Material provides additional simulation results for our approach: in Section C, we present estimates of ˆ G to test sensitivity to the extrapolation quantile q ; in Section D, we give results on estimating τ 1 ( δ ) for additional copula dependence parameters; Section E covers estimation of λ ( ω ); Section F provides results on the ( α i , β i ) parameters of the conditional extremes approach; and Section G provides additional simulation results for the asymmetric logistic copula discussed in Section 5.2.