Bootstrap approximation of tail dependence function

For estimating a rare event via the multivariate extreme value theory, the so-called tail dependence function has to be investigated (see L. de Haan, J. de Ronde, Sea and wind: Multivariate extremes at work, Extremes 1 (1998) 7-45]). A simple, but effective estimator for the tail dependence function is the tail empirical distribution function, see X. Huang, Statistics of Bivariate Extreme Values, Ph.D. Thesis, Tinbergen Institute Research Series, 1992] or R. Schmidt, U. Stadtmüller, Nonparametric estimation of tail dependence, Scand. J. Stat. 33 (2006) 307-335]. In this paper, we first derive a bootstrap approximation for a tail dependence function with an approximation rate via the construction approach developed by K. Chen, S.H. Lo, On a mapping approach to investigating the bootstrap accuracy, Probab. Theory Relat. Fields 107 (1997) 197-217], and then apply it to construct a confidence band for the tail dependence function. A simulation study is conducted to assess the accuracy of the bootstrap approach.