Annals of Statistics

Inference for Archimax copulas

Simon Chatelain, Anne-Laure Fougères, and Johanna G. Nešlehová

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Abstract

Archimax copula models can account for any type of asymptotic dependence between extremes and at the same time capture joint risks at medium levels. An Archimax copula is characterized by two functional parameters: the stable tail dependence function $\ell $, and the Archimedean generator $\psi $ which distorts the extreme-value dependence structure. This article develops semiparametric inference for Archimax copulas: a nonparametric estimator of $\ell $ and a moment-based estimator of $\psi $ assuming the latter belongs to a parametric family. Conditions under which $\psi $ and $\ell $ are identifiable are derived. The asymptotic behavior of the estimators is then established under broad regularity conditions; performance in small samples is assessed through a comprehensive simulation study. The Archimax copula model with the Clayton generator is then used to analyze monthly rainfall maxima at three stations in French Brittany. The model is seen to fit the data very well, both in the lower and in the upper tail. The nonparametric estimator of $\ell $ reveals asymmetric extremal dependence between the stations, which reflects heavy precipitation patterns in the area. Technical proofs, simulation results and $\mathsf{R}$ code are provided in the Online Supplement.

Article information

Source
Ann. Statist., Volume 48, Number 2 (2020), 1025-1051.

Dates
Received: June 2018
Revised: February 2019
First available in Project Euclid: 26 May 2020

Permanent link to this document
https://projecteuclid.org/euclid.aos/1590480044

Digital Object Identifier
doi:10.1214/19-AOS1836

Mathematical Reviews number (MathSciNet)
MR4102686

Subjects
Primary: 62H12: Estimation 62G05: Estimation 62G20: Asymptotic properties 62G32: Statistics of extreme values; tail inference
Secondary: 60G70: Extreme value theory; extremal processes

Keywords
Copulas empirical processes multivariate extremes subasymptotic modeling

Citation

Chatelain, Simon; Fougères, Anne-Laure; Nešlehová, Johanna G. Inference for Archimax copulas. Ann. Statist. 48 (2020), no. 2, 1025--1051. doi:10.1214/19-AOS1836. https://projecteuclid.org/euclid.aos/1590480044


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References

  • Abramowitz, M. and Stegun, I. A. (1964). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards Applied Mathematics Series 55. For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C.
  • Bacigál, T., Jágr, V. and Mesiar, R. (2011). Non-exchangeable random variables, Archimax copulas and their fitting to real data. Kybernetika (Prague) 47 519–531.
  • Bacigál, T. and Mesiar, R. (2012). 3-dimensional Archimax copulas and their fitting to real data. In COMPSTAT 2012.
  • Barbe, P., Genest, C., Ghoudi, K. and Rémillard, B. (1996). On Kendall’s process. J. Multivariate Anal. 58 197–229.
  • Belzile, L. R. and Nešlehová, J. G. (2017). Extremal attractors of Liouville copulas. J. Multivariate Anal. 160 68–92.
  • Ben Ghorbal, N., Genest, C. and Nešlehová, J. (2009). On the Ghoudi, Khoudraji, and Rivest test for extreme-value dependence. Canad. J. Statist. 37 534–552.
  • Berghaus, B., Bücher, A. and Volgushev, S. (2017). Weak convergence of the empirical copula process with respect to weighted metrics. Bernoulli 23 743–772.
  • Capéraà, P., Fougères, A.-L. and Genest, C. (1997). A nonparametric estimation procedure for bivariate extreme value copulas. Biometrika 84 567–577.
  • Capéraà, P., Fougères, A.-L. and Genest, C. (2000). Bivariate distributions with given extreme value attractor. J. Multivariate Anal. 72 30–49.
  • Charpentier, A. and Segers, J. (2009). Tails of multivariate Archimedean copulas. J. Multivariate Anal. 100 1521–1537.
  • Charpentier, A., Fougères, A.-L., Genest, C. and Nešlehová, J. G. (2014). Multivariate Archimax copulas. J. Multivariate Anal. 126 118–136.
  • Chatelain, S., Fougères, A.-L. and Nešlehová, J. G. (2020). Supplement to “Inference for Archimax copulas.” https://doi.org/10.1214/19-AOS1836SUPPA, https://doi.org/10.1214/19-AOS1836SUPPB.
  • Coles, S., Heffernan, J. and Tawn, J. (1999). Dependence measures for extreme value analyses. Extremes 2 339–365.
  • de Haan, L. and Ferreira, A. (2006). Extreme Value Theory: An Introduction. Springer Series in Operations Research and Financial Engineering. Springer, New York.
  • Einmahl, J. H. J., Kiriliouk, A. and Segers, J. (2018). A continuous updating weighted least squares estimator of tail dependence in high dimensions. Extremes 21 205–233.
  • Fougères, A.-L., de Haan, L. and Mercadier, C. (2015). Bias correction in multivariate extremes. Ann. Statist. 43 903–934.
  • Fougères, A.-L., Mercadier, C. and Nolan, J. P. (2013). Dense classes of multivariate extreme value distributions. J. Multivariate Anal. 116 109–129.
  • Genest, C. and Ghoudi, K. (1994). Une famille de lois bidimensionnelles insolite. C. R. Acad. Sci., Sér. 1 Math. 318 351–354.
  • Genest, C. and Segers, J. (2009). Rank-based inference for bivariate extreme-value copulas. Ann. Statist. 37 2990–3022.
  • Ghoudi, K., Khoudraji, A. and Rivest, L.-P. (1998). Propriétés statistiques des copules de valeurs extrêmes bidimensionnelles. Canad. J. Statist. 26 187–197.
  • Gudendorf, G. and Segers, J. (2011). Nonparametric estimation of an extreme-value copula in arbitrary dimensions. J. Multivariate Anal. 102 37–47.
  • Gudendorf, G. and Segers, J. (2012). Nonparametric estimation of multivariate extreme-value copulas. J. Statist. Plann. Inference 142 3073–3085.
  • Hall, P. and Tajvidi, N. (2000). Distribution and dependence-function estimation for bivariate extreme-value distributions. Bernoulli 6 835–844.
  • Hofert, M. (2008). Sampling Archimedean copulas. Comput. Statist. Data Anal. 52 5163–5174.
  • Hofert, M. and Maechler, M. (2016). Parallel and other simulations in $\mathsf{R}$ made easy: An end-to-end study. J. Stat. Softw. 69 1–44.
  • Huang, X. (1992). Statistics of Bivariate Extreme Values. Tinbergen Institute Research Series 22. Thesis (Ph.D.)–Erasmus Univ. Rotterdam.
  • Huser, R., Opitz, T. and Thibaud, E. (2017). Bridging asymptotic independence and dependence in spatial extremes using Gaussian scale mixtures. Spat. Stat. 21 166–186.
  • Joe, H. (2015). Dependence Modeling with Copulas. Monographs on Statistics and Applied Probability 134. CRC Press, Boca Raton, FL.
  • Kendall, M. G. and Smith, B. B. (1940). On the method of paired comparisons. Biometrika 31 324–345.
  • Kojadinovic, I., Segers, J. and Yan, J. (2011). Large-sample tests of extreme-value dependence for multivariate copulas. Canad. J. Statist. 39 703–720.
  • Larsson, M. and Nešlehová, J. (2011). Extremal behavior of Archimedean copulas. Adv. in Appl. Probab. 43 195–216.
  • Ledford, A. W. and Tawn, J. A. (1996). Statistics for near independence in multivariate extreme values. Biometrika 83 169–187.
  • Malov, S. V. (2001). On finite-dimensional Archimedean copulas. In Asymptotic Methods in Probability and Statistics with Applications (N. Balakrishnan, I. Ibragimov and V. Nevzorov, eds.) 19–35. Birkhäuser, Basel.
  • McNeil, A. J. and Nešlehová, J. (2009). Multivariate Archimedean copulas, $d$-monotone functions and $l_{1}$-norm symmetric distributions. Ann. Statist. 37 3059–3097.
  • Mesiar, R. and Jágr, V. (2013). $d$-dimensional dependence functions and Archimax copulas. Fuzzy Sets and Systems 228 78–87.
  • Morillas, P. M. (2005). A characterization of absolutely monotonic $(\Delta)$ functions of a fixed order. Publ. Inst. Math. (Beograd) (N.S.) 78 93–105.
  • Naveau, P., Huser, R., Ribereau, P. and Hannart, A. (2016). Modeling jointly low, moderate, and heavy rainfall intensities without a threshold selection. Water Resour. Res. 52 1–17.
  • Nelsen, R. B. (2006). An Introduction to Copulas, 2nd ed. Springer Series in Statistics. Springer, New York.
  • Pickands, J. III (1981). Multivariate extreme value distributions. In Proceedings of the 43rd Session of the International Statistical Institute, Vol. 2 (Buenos Aires, 1981) 49 859–878, 894–902.
  • Resnick, S. I. (1987). Extreme Values, Regular Variation, and Point Processes. Applied Probability. A Series of the Applied Probability Trust 4. Springer, New York.
  • Ressel, P. (2013). Homogeneous distributions—and a spectral representation of classical mean values and stable tail dependence functions. J. Multivariate Anal. 117 246–256.
  • Segers, J. (2012). Asymptotics of empirical copula processes under non-restrictive smoothness assumptions. Bernoulli 18 764–782.
  • Sklar, M. (1959). Fonctions de répartition à $n$ dimensions et leurs marges. Publ. Inst. Stat. Univ. Paris 8 229–231.
  • van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes. Springer Series in Statistics. Springer, New York.
  • Wadsworth, J. L., Tawn, J. A., Davison, A. C. and Elton, D. M. (2017). Modelling across extremal dependence classes. J. R. Stat. Soc. Ser. B. Stat. Methodol. 79 149–175.
  • Zhang, D., Wells, M. T. and Peng, L. (2008). Nonparametric estimation of the dependence function for a multivariate extreme value distribution. J. Multivariate Anal. 99 577–588.
  • Zhao, Z. and Zhang, Z. (2018). Semiparametric dynamic max-copula model for multivariate time series. J. R. Stat. Soc. Ser. B. Stat. Methodol. 80 409–432.

Supplemental materials

  • Supplement to “Inference for Archimax copulas”. This file contains the proofs of Sections 2, 4 and 6. It also contains detailed results of the simulation study from Section 5.
  • R code for “Inference for Archimax copulas”. The functions necessary for fitting the Clayton-Archimax model as was done in Section 8 are provided.