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Inference for Archimax copulas. (English) Zbl 1450.62046

Consider a \(d\)-dimensional Archimax copula \(C\) with the representation \[C_{\psi,\ell}\left(\mathbf{u} \right) = \psi\left[\ell\left\lbrace \phi\left(u_{1} \right),\dots,\phi\left(u_{d} \right) \right\rbrace \right], \] where \(\ell: \mathbb{R}_{+}^{d} \to \mathbb{R}_{+}\) is a \(d\)-variate stable tail dependence function and \(\psi : \left[ 0,\infty\right) \to \left[0,1 \right] \) is an Archimedean generator with inverse \(\phi\). The class of Archimax copulas includes both Archimedean and extreme-value copulas.
From authors’ summary: “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.”

MSC:

62H05 Characterization and structure theory for multivariate probability distributions; copulas
62H12 Estimation in multivariate analysis
62G05 Nonparametric estimation
62G20 Asymptotic properties of nonparametric inference
62G32 Statistics of extreme values; tail inference
60G70 Extreme value theory; extremal stochastic processes
62P12 Applications of statistics to environmental and related topics

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References:

[1] 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. · Zbl 0171.38503
[2] 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. · Zbl 1227.93120
[3] Bacigál, T. and Mesiar, R. (2012). 3-dimensional Archimax copulas and their fitting to real data. In COMPSTAT 2012. · Zbl 1227.93120
[4] Barbe, P., Genest, C., Ghoudi, K. and Rémillard, B. (1996). On Kendall’s process. J. Multivariate Anal. 58 197-229. · Zbl 0862.60020 · doi:10.1006/jmva.1996.0048
[5] Belzile, L. R. and Nešlehová, J. G. (2017). Extremal attractors of Liouville copulas. J. Multivariate Anal. 160 68-92. · Zbl 1372.60071 · doi:10.1016/j.jmva.2017.05.008
[6] 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. · Zbl 1191.62083 · doi:10.1002/cjs.10034
[7] Berghaus, B., Bücher, A. and Volgushev, S. (2017). Weak convergence of the empirical copula process with respect to weighted metrics. Bernoulli 23 743-772. · Zbl 1367.60026 · doi:10.3150/15-BEJ751
[8] 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
[9] Capéraà, P., Fougères, A.-L. and Genest, C. (2000). Bivariate distributions with given extreme value attractor. J. Multivariate Anal. 72 30-49. · Zbl 0978.62043
[10] Charpentier, A. and Segers, J. (2009). Tails of multivariate Archimedean copulas. J. Multivariate Anal. 100 1521-1537. · Zbl 1165.62038 · doi:10.1016/j.jmva.2008.12.015
[11] Charpentier, A., Fougères, A.-L., Genest, C. and Nešlehová, J. G. (2014). Multivariate Archimax copulas. J. Multivariate Anal. 126 118-136. · Zbl 1349.62173
[12] 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.
[13] Coles, S., Heffernan, J. and Tawn, J. (1999). Dependence measures for extreme value analyses. Extremes 2 339-365. · Zbl 0972.62030 · doi:10.1023/A:1009963131610
[14] de Haan, L. and Ferreira, A. (2006). Extreme Value Theory: An Introduction. Springer Series in Operations Research and Financial Engineering. Springer, New York. · Zbl 1101.62002
[15] 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. · Zbl 1402.62088 · doi:10.1007/s10687-017-0303-7
[16] Fougères, A.-L., de Haan, L. and Mercadier, C. (2015). Bias correction in multivariate extremes. Ann. Statist. 43 903-934. · Zbl 1312.62061 · doi:10.1214/14-AOS1305
[17] Fougères, A.-L., Mercadier, C. and Nolan, J. P. (2013). Dense classes of multivariate extreme value distributions. J. Multivariate Anal. 116 109-129. · Zbl 1277.62143 · doi:10.1016/j.jmva.2012.11.015
[18] Genest, C. and Ghoudi, K. (1994). Une famille de lois bidimensionnelles insolite. C. R. Acad. Sci., Sér. 1 Math. 318 351-354. · Zbl 0797.60017
[19] 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
[20] Ghoudi, K., Khoudraji, A. and Rivest, L.-P. (1998). Propriétés statistiques des copules de valeurs extrêmes bidimensionnelles. Canad. J. Statist. 26 187-197. · Zbl 0899.62071 · doi:10.2307/3315683
[21] Gudendorf, G. and Segers, J. (2011). Nonparametric estimation of an extreme-value copula in arbitrary dimensions. J. Multivariate Anal. 102 37-47. · Zbl 1352.62048 · doi:10.1016/j.jmva.2010.07.011
[22] Gudendorf, G. and Segers, J. (2012). Nonparametric estimation of multivariate extreme-value copulas. J. Statist. Plann. Inference 142 3073-3085. · Zbl 1349.62207 · doi:10.1016/j.jspi.2012.05.007
[23] 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
[24] Hofert, M. (2008). Sampling Archimedean copulas. Comput. Statist. Data Anal. 52 5163-5174. · Zbl 1452.62070 · doi:10.1016/j.csda.2008.05.019
[25] 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.
[26] Huang, X. (1992). Statistics of Bivariate Extreme Values. Tinbergen Institute Research Series 22. Thesis (Ph.D.)-Erasmus Univ. Rotterdam.
[27] 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.
[28] Joe, H. (2015). Dependence Modeling with Copulas. Monographs on Statistics and Applied Probability 134. CRC Press, Boca Raton, FL.
[29] Kendall, M. G. and Smith, B. B. (1940). On the method of paired comparisons. Biometrika 31 324-345. · Zbl 0023.14803 · doi:10.1093/biomet/31.3-4.324
[30] Kojadinovic, I., Segers, J. and Yan, J. (2011). Large-sample tests of extreme-value dependence for multivariate copulas. Canad. J. Statist. 39 703-720. · Zbl 1284.62333 · doi:10.1002/cjs.10110
[31] Larsson, M. and Nešlehová, J. (2011). Extremal behavior of Archimedean copulas. Adv. in Appl. Probab. 43 195-216. · Zbl 1213.62084 · doi:10.1239/aap/1300198519
[32] Ledford, A. W. and Tawn, J. A. (1996). Statistics for near independence in multivariate extreme values. Biometrika 83 169-187. · Zbl 0865.62040 · doi:10.1093/biomet/83.1.169
[33] 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. · Zbl 1065.62099
[34] McNeil, A. J. and Nešlehová, J. (2009). Multivariate Archimedean copulas, \(d\)-monotone functions and \(l_1\)-norm symmetric distributions. Ann. Statist. 37 3059-3097. · Zbl 1173.62044 · doi:10.1214/07-AOS556
[35] Mesiar, R. and Jágr, V. (2013). \(d\)-dimensional dependence functions and Archimax copulas. Fuzzy Sets and Systems 228 78-87. · Zbl 1284.62345
[36] Morillas, P. M. (2005). A characterization of absolutely monotonic \((\Delta)\) functions of a fixed order. Publ. Inst. Math. (Beograd) (N.S.) 78 93-105. · Zbl 1119.26013
[37] 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.
[38] Nelsen, R. B. (2006). An Introduction to Copulas, 2nd ed. Springer Series in Statistics. Springer, New York. · Zbl 1152.62030
[39] 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. · Zbl 0518.62045
[40] Resnick, S. I. (1987). Extreme Values, Regular Variation, and Point Processes. Applied Probability. A Series of the Applied Probability Trust 4. Springer, New York. · Zbl 0633.60001
[41] Ressel, P. (2013). Homogeneous distributions—and a spectral representation of classical mean values and stable tail dependence functions. J. Multivariate Anal. 117 246-256. · Zbl 1283.60021 · doi:10.1016/j.jmva.2013.02.013
[42] Segers, J. (2012). Asymptotics of empirical copula processes under non-restrictive smoothness assumptions. Bernoulli 18 764-782. · Zbl 1243.62066 · doi:10.3150/11-BEJ387
[43] Sklar, M. (1959). Fonctions de répartition à \(n\) dimensions et leurs marges. Publ. Inst. Stat. Univ. Paris 8 229-231. · Zbl 0100.14202
[44] van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes. Springer Series in Statistics. Springer, New York. · Zbl 0862.60002
[45] 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. · Zbl 1414.62165 · doi:10.1111/rssb.12157
[46] 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. · Zbl 1333.62140 · doi:10.1016/j.jmva.2006.09.011
[47] Zhao, Z. · Zbl 06849261 · doi:10.1111/rssb.12256
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