zbMATH — the first resource for mathematics

Weighted estimation of the dependence function for an extreme-value distribution. (English) Zbl 06168761
Summary: Bivariate extreme-value distributions have been used in modeling extremes in environmental sciences and risk management. An important issue is estimating the dependence function, such as the Pickands dependence function. Some estimators for the Pickands dependence function have been studied by assuming that the marginals are known. Recently, Genest and Segers [Ann. Statist. 37 (2009) 2990–3022] derived the asymptotic distributions of those proposed estimators with marginal distributions replaced by the empirical distributions. In this article, we propose a class of weighted estimators including those of Genest and Segers (2009) as special cases. We propose a jackknife empirical likelihood method for constructing confidence intervals for the Pickands dependence function, which avoids estimating the complicated asymptotic variance. A simulation study demonstrates the effectiveness of our proposed jackknife empirical likelihood method.

62G05 Nonparametric estimation
62G09 Nonparametric statistical resampling methods
62G32 Statistics of extreme values; tail inference
62H20 Measures of association (correlation, canonical correlation, etc.)
Full Text: DOI Euclid arXiv
[1] Bücher, A. and Dette, H. (2010). A note on bootstrap approximations for the empirical copula process. Statist. Probab. Lett. 80 1925-1932. · Zbl 1202.62055 · doi:10.1016/j.spl.2010.08.021
[2] Bücher, A., Dette, H. and Volgushev, S. (2011). New estimators of the Pickands dependence function and a test for extreme-value dependence. Ann. Statist. 39 1963-2006. · Zbl 1306.62087 · doi:10.1214/11-AOS890
[3] Capéraà, P., Fougères, A.L. and Genest, C. (1997). A nonparametric estimation procedure for bivariate extreme value copulas. Biometrika 84 567-577. · Zbl 1058.62516 · doi:10.1093/biomet/84.3.567
[4] Chen, J., Peng, L. and Zhao, Y. (2009). Empirical likelihood based confidence intervals for copulas. J. Multivariate Anal. 100 137-151. · Zbl 1151.62037 · doi:10.1016/j.jmva.2008.04.005
[5] Deheuvels, P. (1991). On the limiting behavior of the Pickands estimator for bivariate extreme-value distributions. Statist. Probab. Lett. 12 429-439. · Zbl 0749.62033 · doi:10.1016/0167-7152(91)90032-M
[6] Falk, M. and Reiss, R.D. (2005). On Pickands coordinates in arbitrary dimensions. J. Multivariate Anal. 92 426-453. · Zbl 1068.60025 · doi:10.1016/j.jmva.2003.10.006
[7] Fermanian, J.D., Radulović, D. and Wegkamp, M. (2004). Weak convergence of empirical copula processes. Bernoulli 10 847-860. · Zbl 1068.62059 · doi:10.3150/bj/1099579158
[8] Genest, C. and Segers, J. (2009). Rank-based inference for bivariate extreme-value copulas. Ann. Statist. 37 2990-3022. · Zbl 1173.62013 · doi:10.1214/08-AOS672
[9] Gong, Y., Peng, L. and Qi, Y. (2010). Smoothed jackknife empirical likelihood method for ROC curve. J. Multivariate Anal. 101 1520-1531. · Zbl 1186.62053 · doi:10.1016/j.jmva.2010.01.012
[10] Hall, P. and Tajvidi, N. (2000). Distribution and dependence-function estimation for bivariate extreme-value distributions. Bernoulli 6 835-844. · Zbl 1067.62540 · doi:10.2307/3318758
[11] Jing, B.Y., Yuan, J. and Zhou, W. (2009). Jackknife empirical likelihood. J. Amer. Statist. Assoc. 104 1224-1232. · Zbl 1388.62136 · doi:10.1198/jasa.2009.tm08260
[12] Kojadinovic, I. and Yan, J. (2010). Nonparametric rank-based tests of bivariate extreme-value dependence. J. Multivariate Anal. 101 2234-2249. · Zbl 1201.62056 · doi:10.1016/j.jmva.2010.05.004
[13] Owen, A.B. (1988). Empirical likelihood ratio confidence intervals for a single functional. Biometrika 75 237-249. · Zbl 0641.62032 · doi:10.1093/biomet/75.2.237
[14] Owen, A. (1990). Empirical likelihood ratio confidence regions. Ann. Statist. 18 90-120. · Zbl 0712.62040 · doi:10.1214/aos/1176347494
[15] Owen, A. (2001). Empirical Likelihood . New York: Chapman & Hall/CRC. · Zbl 0989.62019
[16] Pickands, J. III (1981). Multivariate extreme value distributions. Bull. Inst. Internat. Statist. 49 859-878. · Zbl 0518.62045
[17] Qin, J. and Lawless, J. (1994). Empirical likelihood and general estimating equations. Ann. Statist. 22 300-325. · Zbl 0799.62049 · doi:10.1214/aos/1176325370
[18] Segers, J. (2012). Asymptotics of empirical copula processes under nonrestrictive smoothness assumptions. Bernoulli 18 764-782. · Zbl 1243.62066 · doi:10.3150/11-BEJ387
[19] Zhou, M. emplik: Empirical likelihood ratio for censored/truncated data. R package version 0.9-3-1. Available at . · www.ms.uky.edu
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.