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Threshold selection for multivariate heavy-tailed data



Regular variation is often used as the starting point for modeling multivariate heavy-tailed data. A random vector is regularly varying if and only if its radial part R is regularly varying and is asymptotically independent of the angular part Θ as R goes to infinity. The conditional limiting distribution of Θ given R is large characterizes the tail dependence of the random vector and hence its estimation is the primary goal of applications. A typical strategy is to look at the angular components of the data for which the radial parts exceed some threshold. While a large class of methods has been proposed to model the angular distribution from these exceedances, the choice of threshold has been scarcely discussed in the literature. In this paper, we describe a procedure for choosing the threshold by formally testing the independence of R and Θ using a measure of dependence called distance covariance. We generalize the limit theorem for distance covariance to our unique setting and propose an algorithm which selects the threshold for R. This algorithm incorporates a subsampling scheme that is also applicable to weakly dependent data. Moreover, it avoids the heavy computation in the calculation of the distance covariance, a typical limitation for this measure. The performance of our method is illustrated on both simulated and real data.


Heavy-tailed data Multivariate regular variation Threshold selection Distance covariance 

AMS 2000 Subject Classifications

62G32 60G70 


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The foreign exchange rate data are obtained from OANDA from through R-package ‘qrmtools’. We would like to thank Bodhisattva Sen for helpful discussions. We would also like to thank the editor and referees for their many constructive and insightful comments.


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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of StatisticsColumbia UniversityNew YorkUSA

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