Bootstrap approximation of tail dependence function. (English) Zbl 1284.62301

Summary: 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 and J. de Ronde, Extremes 1, No. 1, 7–46 (1998; Zbl 0921.62144)]). A simple, but effective estimator for the tail dependence function is the tail empirical distribution function, see [X. Huang, “Statistics of bivariate extreme values”, Tinbergen Institute Research Series, Ph.D. Thesis (1992); R. Schmidt and U. Stadtmüller, Scand. J. Stat. 33, No. 2, 307–335 (2006; Zbl 1124.62016)]. 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 and S.-H. Lo [Probab. Theory Relat. Fields 107, No. 2, 197–217 (1997; Zbl 0868.60033)], 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.


62G32 Statistics of extreme values; tail inference
62G09 Nonparametric statistical resampling methods
Full Text: DOI


[1] Chen, K.; Lo, S.H., On a mapping approach to investigating the bootstrap accuracy, Probab. theory related fields, 107, 197-217, (1997) · Zbl 0868.60033
[2] Danielsson, J.; de Haan, L.; Peng, L.; de Vries, C.G., Using a bootstrap method to choose the sample fraction in tail index estimation, J. multivariate anal., 76, 226-248, (2001) · Zbl 0976.62044
[3] Drees, H.; Huang, X., Best attainable rates of convergence for estimates of the stable tail dependence functions, J. multivariate anal., 64, 25-47, (1998) · Zbl 0953.62046
[4] Einmahl, J.H.J.; de Haan, L.; Li, D., Weighted approximations of tail copula processes with application to testing the multivariate extreme value distribution, Ann. statist., 34, 4, 1987-2014, (2006) · Zbl 1246.60051
[5] Einmahl, J.H.J.; Mason, D.M., Strong limit theorems for weighted quantile process, Ann. probab., 16, 1623-1643, (1988) · Zbl 0659.60052
[6] Einmahl, J.H.J.; Ruymgaart, F.H., The almost sure behavior of the oscillation modulus of the multivariate empirical process, Statist. probab. lett., 6, 87-96, (1987) · Zbl 0644.62058
[7] El-Nouty, C.; Guillou, A., On the bootstrap accuracy of the Pareto index, Statist. decisions, 18, 3, 275-289, (2000) · Zbl 1169.62319
[8] Geluk, J.; de Haan, L., On bootstrap sample size in extreme value theory, Publ. de l’inst. math. nouvelle serie, 71, 85, 21-25, (2002) · Zbl 1034.60043
[9] de Haan, L.; de Ronde, J., Sea and wind: multivariate extremes at work, Extremes, 1, 7-45, (1998) · Zbl 0921.62144
[10] de Haan, L.; Sinha, A.K., Estimating the probability of a rare event, Ann. statist., 27, 732-759, (1999) · Zbl 1105.62344
[11] Hall, P., Using the bootstrap to estimate Mean squared error and selecting parameter in nonparametric problem, J. multivariate anal., 32, 177-203, (1990) · Zbl 0722.62030
[12] X. Huang, Statistics of Bivariate Extreme Values, Ph.D. Thesis, Tinbergen Institute Research Series 1992
[13] Klüppelberg, C.; Kuhn, G.; Peng, L., Estimating the tail dependence of an elliptical distribution, Bernoulli, 13, 1, 229-251, (2007) · Zbl 1111.62048
[14] McNeil, A.J.; Frey, R.; Embrechts, P., Quantitative risk management, (2005), Princeton University Press · Zbl 1089.91037
[15] Peng, L.; Qi, Y., Partial derivatives and confidence intervals for bivariate tail dependence functions, J. statist. plann. inference, 137, 2089-2101, (2007) · Zbl 1120.62032
[16] Qi, Y., Almost sure convergence of the stable tail empirical dependence function in multivariate extreme statistics, Acta math. appl. sin. engl. ser., 13, 2, 167-175, (1997) · Zbl 0904.62061
[17] Schmidt, R.; Stadtmüller, U., Nonparametric estimation of tail dependence, Scand. J. statist., 33, 307-335, (2006) · Zbl 1124.62016
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.