The Annals of Statistics

Bias correction in multivariate extremes

Anne-Laure Fougères, Laurens de Haan, and Cécile Mercadier

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Abstract

The estimation of the extremal dependence structure is spoiled by the impact of the bias, which increases with the number of observations used for the estimation. Already known in the univariate setting, the bias correction procedure is studied in this paper under the multivariate framework. New families of estimators of the stable tail dependence function are obtained. They are asymptotically unbiased versions of the empirical estimator introduced by Huang [Statistics of bivariate extremes (1992) Erasmus Univ.]. Since the new estimators have a regular behavior with respect to the number of observations, it is possible to deduce aggregated versions so that the choice of the threshold is substantially simplified. An extensive simulation study is provided as well as an application on real data.

Article information

Source
Ann. Statist., Volume 43, Number 2 (2015), 903-934.

Dates
First available in Project Euclid: 23 March 2015

Permanent link to this document
https://projecteuclid.org/euclid.aos/1427115291

Digital Object Identifier
doi:10.1214/14-AOS1305

Mathematical Reviews number (MathSciNet)
MR3325714

Zentralblatt MATH identifier
1312.62061

Subjects
Primary: 62G32: Statistics of extreme values; tail inference 62G05: Estimation 62G20: Asymptotic properties
Secondary: 60F05: Central limit and other weak theorems 60G70: Extreme value theory; extremal processes

Keywords
Multivariate extreme value theory tail dependence bias correction threshold choice

Citation

Fougères, Anne-Laure; de Haan, Laurens; Mercadier, Cécile. Bias correction in multivariate extremes. Ann. Statist. 43 (2015), no. 2, 903--934. doi:10.1214/14-AOS1305. https://projecteuclid.org/euclid.aos/1427115291


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References

  • Abdous, B. and Ghoudi, K. (2005). Non-parametric estimators of multivariate extreme dependence functions. J. Nonparametr. Stat. 17 915–935.
    Mathematical Reviews (MathSciNet): MR2192166
    Digital Object Identifier: doi:10.1080/10485250500336379
  • Beirlant, J., Dierckx, G. and Guillou, A. (2011). Bias-reduced estimators for bivariate tail modelling. Insurance Math. Econom. 49 18–26.
    Mathematical Reviews (MathSciNet): MR2811890
  • Beirlant, J., Goegebeur, Y., Teugels, J. and Segers, J. (2004). Statistics of Extremes: Theory and Applications. Wiley, Chichester.
    Mathematical Reviews (MathSciNet): MR2108013
  • Bruun, J. T. and Tawn, J. A. (1998). Comparison of approaches for estimating the probability of coastal flooding. Appl. Statist. 47 405–423.
  • 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.
    Mathematical Reviews (MathSciNet): MR2893858
    Digital Object Identifier: doi:10.1214/11-AOS890
    Project Euclid: euclid.aos/1314190620
  • Caeiro, F., Gomes, M. I. and Rodrigues, L. H. (2009). Reduced-bias tail index estimators under a third-order framework. Comm. Statist. Theory Methods 38 1019–1040.
    Mathematical Reviews (MathSciNet): MR2522545
    Digital Object Identifier: doi:10.1080/03610920802361415
  • Cai, J.-J., de Haan, L. and Zhou, C. (2013). Bias correction in extreme value statistics with index around zero. Extremes 16 173–201.
    Mathematical Reviews (MathSciNet): MR3057195
    Digital Object Identifier: doi:10.1007/s10687-012-0158-x
  • Capéraà, P. and Fougères, A.-L. (2000). Estimation of a bivariate extreme value distribution. Extremes 3 311–329.
    Mathematical Reviews (MathSciNet): MR1870461
    Digital Object Identifier: doi:10.1023/A:1012241624430
  • Capéraà, P., Fougères, A.-L. and Genest, C. (2000). Bivariate distributions with given extreme value attractor. J. Multivariate Anal. 72 30–49.
    Mathematical Reviews (MathSciNet): MR1747422
    Digital Object Identifier: doi:10.1006/jmva.1999.1845
  • Ciuperca, G. and Mercadier, C. (2010). Semi-parametric estimation for heavy tailed distributions. Extremes 13 55–87.
    Mathematical Reviews (MathSciNet): MR2593951
    Digital Object Identifier: doi:10.1007/s10687-009-0086-6
  • Coles, S. G. and Tawn, J. A. (1991). Modelling extreme multivariate events. J. Roy. Statist. Soc. Ser. B 53 377–392.
    Mathematical Reviews (MathSciNet): MR1108334
  • de Haan, L. and Ferreira, A. (2006). Extreme Value Theory: An Introduction. Springer, New York.
    Mathematical Reviews (MathSciNet): MR2234156
  • de Haan, L., Mercadier, C. and Zhou, C. (2014). Adapting extreme value statistics to financial time series: Dealing with bias and serial dependence. Submitted.
  • de Haan, L. and Resnick, S. I. (1993). Estimating the limit distribution of multivariate extremes. Comm. Statist. Stochastic Models 9 275–309.
    Mathematical Reviews (MathSciNet): MR1213072
    Digital Object Identifier: doi:10.1080/15326349308807267
  • de Haan, L. and Sinha, A. K. (1999). Estimating the probability of a rare event. Ann. Statist. 27 732–759.
    Mathematical Reviews (MathSciNet): MR1714710
    Digital Object Identifier: doi:10.1214/aos/1018031214
    Project Euclid: euclid.aos/1018031214
  • Demarta, S. and McNeil, A. J. (2005). The t copula and related copulas. Int. Stat. Rev. 73 111–129.
  • Einmahl, J. H. J., de Haan, L. and Sinha, A. K. (1997). Estimating the spectral measure of an extreme value distribution. Stochastic Process. Appl. 70 143–171.
    Mathematical Reviews (MathSciNet): MR1475660
    Digital Object Identifier: doi:10.1016/S0304-4149(97)00065-3
  • Einmahl, J. H. J., Krajina, A. and Segers, J. (2008). A method of moments estimator of tail dependence. Bernoulli 14 1003–1026.
    Mathematical Reviews (MathSciNet): MR2543584
    Digital Object Identifier: doi:10.3150/08-BEJ130
    Project Euclid: euclid.bj/1225980569
  • Einmahl, J. H. J., Krajina, A. and Segers, J. (2012). An $M$-estimator for tail dependence in arbitrary dimensions. Ann. Statist. 40 1764–1793.
    Mathematical Reviews (MathSciNet): MR3015043
    Digital Object Identifier: doi:10.1214/12-AOS1023
    Project Euclid: euclid.aos/1349196391
  • Fils-Villetard, A., Guillou, A. and Segers, J. (2008). Projection estimators of Pickands dependence functions. Canad. J. Statist. 36 369–382.
    Mathematical Reviews (MathSciNet): MR2456011
    Digital Object Identifier: doi:10.1002/cjs.5550360303
  • Fraga Alves, M. I., de Haan, L. and Lin, T. (2003). Estimation of the parameter controlling the speed of convergence in extreme value theory. Math. Methods Statist. 12 155–176.
    Mathematical Reviews (MathSciNet): MR2025356
  • Goegebeur, Y. and Guillou, A. (2013). Asymptotically unbiased estimation of the coefficient of tail dependence. Scand. J. Stat. 40 174–189.
    Mathematical Reviews (MathSciNet): MR3024038
    Digital Object Identifier: doi:10.1111/j.1467-9469.2012.00800.x
  • Gomes, M. I., de Haan, L. and Peng, L. (2002). Semi-parametric estimation of the second order parameter in statistics of extremes. Extremes 5 387–414.
    Mathematical Reviews (MathSciNet): MR2002125
    Digital Object Identifier: doi:10.1023/A:1025128326588
  • Gomes, M. I., de Haan, L. and Rodrigues, L. H. (2008). Tail index estimation for heavy-tailed models: Accommodation of bias in weighted log-excesses. J. R. Stat. Soc. Ser. B. Stat. Methodol. 70 31–52.
    Mathematical Reviews (MathSciNet): MR2412630
  • Guillotte, S., Perron, F. and Segers, J. (2011). Non-parametric Bayesian inference on bivariate extremes. J. R. Stat. Soc. Ser. B. Stat. Methodol. 73 377–406.
    Mathematical Reviews (MathSciNet): MR2815781
    Digital Object Identifier: doi:10.1111/j.1467-9868.2010.00770.x
  • Hall, P. and Welsh, A. H. (1985). Adaptive estimates of parameters of regular variation. Ann. Statist. 13 331–341.
    Mathematical Reviews (MathSciNet): MR773171
    Digital Object Identifier: doi:10.1214/aos/1176346596
    Project Euclid: euclid.aos/1176346596
  • Huang, X. (1992). Statistics of bivariate extremes. Ph.D. thesis, Erasmus Univ. Rotterdam, Tinbergen Institute Research series No. 22.
  • Joe, H. (1990). Families of min-stable multivariate exponential and multivariate extreme value distributions. Statist. Probab. Lett. 9 75–81.
    Mathematical Reviews (MathSciNet): MR1035994
  • Joe, H., Smith, R. L. and Weissman, I. (1992). Bivariate threshold methods for extremes. J. Roy. Statist. Soc. Ser. B 54 171–183.
    Mathematical Reviews (MathSciNet): MR1157718
  • Kotz, S., Balakrishnan, N. and Johnson, N. L. (2000). Continuous Multivariate Distributions: Models and Applications, 2nd ed. Wiley Series in Probability and Statistics: Applied Probability and Statistics 1. Wiley, New York.
    Mathematical Reviews (MathSciNet): MR1788152
  • Krajina, A. (2012). A method of moments estimator of tail dependence in meta-elliptical models. J. Statist. Plann. Inference 142 1811–1823.
    Mathematical Reviews (MathSciNet): MR2903392
    Digital Object Identifier: doi:10.1016/j.jspi.2012.01.020
  • Ledford, A. W. and Tawn, J. A. (1996). Statistics for near independence in multivariate extreme values. Biometrika 83 169–187.
    Mathematical Reviews (MathSciNet): MR1399163
    Digital Object Identifier: doi:10.1093/biomet/83.1.169
  • Ledford, A. W. and Tawn, J. A. (1997). Modelling dependence within joint tail regions. J. Roy. Statist. Soc. Ser. B 59 475–499.
    Mathematical Reviews (MathSciNet): MR1440592
    Digital Object Identifier: doi:10.1111/1467-9868.00080
  • Peng, L. (1998). Asymptotically unbiased estimators for the extreme-value index. Statist. Probab. Lett. 38 107–115.
    Mathematical Reviews (MathSciNet): MR1627906
  • Peng, L. (2010). A practical way for estimating tail dependence functions. Statist. Sinica 20 365–378.
    Mathematical Reviews (MathSciNet): MR2640699
  • Resnick, S. I. (1986). Point processes, regular variation and weak convergence. Adv. in Appl. Probab. 18 66–138.
    Mathematical Reviews (MathSciNet): MR827332
    Digital Object Identifier: doi:10.2307/1427239
  • Tawn, J. A. (1988). Bivariate extreme value theory: Models and estimation. Biometrika 75 397–415.
    Mathematical Reviews (MathSciNet): MR967580
    Digital Object Identifier: doi:10.1093/biomet/75.3.397

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