Semi-parametric approximation of Kendall’s distribution function and multivariate Return Periods
[Approximation semi-paramétrique de la fonction de distribution de Kendall et périodes de retour multivariées]
Journal de la société française de statistique, Numéro spécial sur les copules, Tome 154 (2013) no. 1, pp. 151-173.

Dans le cas bivarié, une approximation de la fonction de distribution de Kendall ainsi que de la période de retour de Kendall sont proposées. Le contexte théorique est celui de la théorie des copules. L’approche semi-paramétrique suggérée consiste à approcher la fonction de distribution de Kendall empirique par une fonction linéaire par morceaux sur l’intervalle unité. La robustesse de l’approche proposée est étudiée empiriquement par le biais de simulations.

In this work we outline a constructive approach for the approximation of Kendall’s distribution function and Kendall’s Return Period in the bivariate case. First, we introduce a suitable theoretical framework, based on the Theory of Copulas, where to embed the issue. Then, we outline an original construction procedure to approximate the empirical Kendall distribution function estimated using the available data. The whole approach is semi-parametric: the empirical Kendall distribution function is approximated via a (suitable) continuous piece-wise linear function on the unit interval. A sensitivity analysis is carried out via a simulation procedure, in order to investigate the robustness of the approach proposed against several relevant factors.

Keywords: Kendall’s distribution function, multivariate return periods, copulas, risk assessment
Mot clés : fonction de distribution de Kendall, périodes de retour multivariés, copules, évaluation de risque 
@article{JSFS_2013__154_1_151_0,
     author = {Salvadori, Gianfausto and Durante, Fabrizio and Perrone, Elisa},
     title = {Semi-parametric approximation of {Kendall{\textquoteright}s} distribution function and multivariate {Return} {Periods}},
     journal = {Journal de la soci\'et\'e fran\c{c}aise de statistique},
     pages = {151--173},
     publisher = {Soci\'et\'e fran\c{c}aise de statistique},
     volume = {154},
     number = {1},
     year = {2013},
     zbl = {1316.62072},
     language = {en},
     url = {http://archive.numdam.org/item/JSFS_2013__154_1_151_0/}
}
TY  - JOUR
AU  - Salvadori, Gianfausto
AU  - Durante, Fabrizio
AU  - Perrone, Elisa
TI  - Semi-parametric approximation of Kendall’s distribution function and multivariate Return Periods
JO  - Journal de la société française de statistique
PY  - 2013
SP  - 151
EP  - 173
VL  - 154
IS  - 1
PB  - Société française de statistique
UR  - http://archive.numdam.org/item/JSFS_2013__154_1_151_0/
LA  - en
ID  - JSFS_2013__154_1_151_0
ER  - 
%0 Journal Article
%A Salvadori, Gianfausto
%A Durante, Fabrizio
%A Perrone, Elisa
%T Semi-parametric approximation of Kendall’s distribution function and multivariate Return Periods
%J Journal de la société française de statistique
%D 2013
%P 151-173
%V 154
%N 1
%I Société française de statistique
%U http://archive.numdam.org/item/JSFS_2013__154_1_151_0/
%G en
%F JSFS_2013__154_1_151_0
Salvadori, Gianfausto; Durante, Fabrizio; Perrone, Elisa. Semi-parametric approximation of Kendall’s distribution function and multivariate Return Periods. Journal de la société française de statistique, Numéro spécial sur les copules, Tome 154 (2013) no. 1, pp. 151-173. http://archive.numdam.org/item/JSFS_2013__154_1_151_0/

[1] Extremes in a Changing Climate (AghaKouchak, A.; Easterling, D.; Hsu, K.; Schubert, S.; Sorooshian, S., eds.), Water Science and Technology Library, 65, Springer Science & Business Media, Dordrecht, 2013 (ISBN 978-94-007-4479-0)

[2] Belzunce, F.; Castaño, A.; Olvera-Cervantes, A.; Suárez-Llorens, A. Quantile curves and dependence structure for bivariate distributions, Computational Statistics & Data Analysis, Volume 51 (2007) no. 10, p. 5112--5129 | Zbl

[3] Bücher, A.; Dette, H.; Volgushev, S. A test for Archimedeanity in bivariate copula models, J. Multivariate Anal., Volume 110 (2012), pp. 121-132 | Zbl

[4] Barbe, P.; Genest, C.; Ghoudi, K.; Rémillard, B. On Kendall’s process, J. Multivariate Anal., Volume 58 (1996) no. 2, pp. 197-229 | MR | Zbl

[5] Brechmann, E. C. Hierarchical Kendall copulas: Properties and inference, arXiv:1202.1998v2 (2012)

[6] Chaouch, M.; Goga, C. Design-Based estimation for geometric quantiles with application to outlier detection, Computational Statistics & Data Analysis, Volume 54 (2010) no. 10, pp. 2214-2229 | Zbl

[7] Chebana, F.; Ouarda, T. B. M. J. Index flood-based multivariate regional frequency analysis, Water Resour. Res., Volume 45 (2009) | DOI

[8] Chebana, F.; Ouarda, T. B. M. J. Multivariate quantiles in hydrological frequency analysis, Environmetrics, Volume 22 (2011) no. 1, pp. 63-78 | DOI

[9] Charpentier, A.; Segers, J. Convergence of Archimedean copulas, Statist. Probab. Lett., Volume 78 (2008) no. 412–419 | Zbl

[10] Durante, F.; Fernández-Sánchez, J.; Sempi, C. Sklar’s Theorem obtained via regularization techniques, Nonlinear Anal., Volume 75 (2012) no. 2, pp. 769-774 | Zbl

[11] Durante, F.; Sempi, C. Copula theory: an introduction, Copula Theory and its Applications (Jaworki, P.; Durante, F.; Härdle, W.; Rychlik, T., eds.) (Lecture Notes in Statistics - Proceedings), Volume 198, Springer, Berlin Heidelberg, 2010, pp. 3-31

[12] Durante, F.; Salvadori, G. On the construction of Multivariate Extreme Value models via copulas, Environmetrics, Volume 21 (2010), pp. 143-161

[13] De Michele, C.; Salvadori, G.; Passoni, G.; Vezzoli, R. A multivariate model of sea storms using copulas, Coastal Engineering, Volume 54 (2007), pp. 734-751

[14] Ghoudi, K.; Khoudraji, A.; Rivest, L.-P. Propriétés statistiques des copules de valeurs extrêmes bidimensionnelles, Canad. J. Statist., Volume 26 (1998), pp. 187-197 | Zbl

[15] Genest, C.; Nes̆lehová, J.; Ben Ghorbal, N. Estimators based on Kendall’s tau in multivariate copula models, Australian and New Zealand Journal of Statistics, Volume 53 (2011), pp. 157-177 | Zbl

[16] Genest, C.; Nešlehová, J.; Quessy, J.-F. Tests of symmetry for bivariate copulas, Ann. Inst. Statist. Math., Volume 4 (2012), pp. 811-834 | Zbl

[17] Genest, C.; Nes̆lehová, J.; Ziegel, J. Inference in multivariate Archimedean copula models (with discussion), Test, Volume 20 (2011), pp. 223-292 | Zbl

[18] Genest, C.; Rivest, L.-P. On the multivariate probability integral transformation, Statist. Probab. Lett., Volume 53 (2001) no. 4, pp. 391-399 | MR | Zbl

[19] Genest, C.; Rivest, L.-P. Statistical inference procedures for bivariate Archimedean copulas, J. Amer. Statist. Assoc., Volume 88 (1993) no. 423, pp. 1034-1043 | MR | Zbl

[20] Gräler, B.; van den Berg, M. J.; Vandenberghe, S.; Petroselli, A.; Grimaldi, S.; De Baets, B.; Verhoest, N. E. C. Multivariate return periods in hydrology: a critical and practical review focusing on synthetic design hydrograph estimation, Hydrol. Earth Syst. Sci., Volume 17 (2013), pp. 1281-1296 | DOI

[21] Hofert, M.; Mächler, M.; McNeil, A. J. Likelihood inference for Archimedean copulas in high dimensions under known margins, J. Multivariate Anal., Volume 110 (2012), pp. 133-150 | Zbl

[22] Jaworski, P. Testing Archimedeanity, Combining Soft Computing and Statistical Methods in Data Analysis (Borgelt, C.; González-Rodríguez, G.; Trutschnig, W.; Lubiano, M.; Gil, M.; Grzegorzewski, P.; Hryniewicz, O., eds.) (Advances in Intelligent and Soft Computing), Volume 77, Springer, Berlin, 2010, pp. 353-360

[23] Copula Theory and its Applications (Jaworski, P.; Durante, F.; Haerdle, W.; Rychlik, T., eds.), Lecture Notes in Statistics - Proceedings, 198, Springer, Berlin Heidelberg, 2010 | Zbl

[24] Joe, H. Multivariate models and dependence concepts, Chapman & Hall, London, 1997 | Zbl

[25] Kojadinovic, I.; Yan, J. A nonparametric test of exchangeability for extreme-value and left-tail decreasing bivariate copulas, The Scandinavian Journal of Statistics, Volume 39 (2012) no. 3, pp. 480-496 | Zbl

[26] McNeil, A. J.; Nešlehová, J. Multivariate Archimedean copulas, d -monotone functions and ℓ 1 -norm symmetric distributions, The Annals of Statistics, Volume 37 (2009), pp. 3059-3097 | Zbl

[27] Nelsen, R.B. An introduction to copulas, Springer-Verlag, New York, 2006 | Zbl

[28] Nelsen, R. B.; Quesada-Molina, J. J.; Rodríguez-Lallena, J. A.; Úbeda-Flores, M. Distribution functions of copulas: a class of bivariate probability integral transforms, Statist. Probab. Lett., Volume 54 (2001) no. 3, pp. 277-282 | Zbl

[29] Nelsen, R. B.; Quesada-Molina, J. J.; Rodríguez-Lallena, J. A.; Úbeda-Flores, M. Kendall distribution functions, Statist. Probab. Lett., Volume 65 (2003) no. 3, pp. 263-268 | MR | Zbl

[30] Nelsen, R. B.; Quesada-Molina, J. J.; Rodríguez-Lallena, J. A.; Úbeda-Flores, M. Kendall distribution functions and associative copulas, Fuzzy Sets and Systems, Volume 160 (2009) no. 1, pp. 52-57 | Zbl

[31] Nappo, G.; Spizzichino, F. Kendall distributions and level sets in bivariate exchangeable survival models, Information Sciences, Volume 179 (2009) no. 17, pp. 2878-2890 | Zbl

[32] Salvadori, G. Bivariate return periods via 2-copulas, Statist. Methodol., Volume 1 (2004), pp. 129-144 | Zbl

[33] Salvadori, G.; De Michele, C. Frequency analysis via Copulas: theoretical aspects and applications to hydrological events, Water Resour. Res., Volume 40 (2004) | DOI

[34] Salvadori, G.; De Michele, C. Multivariate multiparameter extreme value models and return periods: A copula approach, Water Resour. Res., Volume 46 (2010) | DOI

[35] Salvadori, G.; De Michele, C.; Durante, F. On the return period and design in a multivariate framework, Hydrol. Earth Syst. Sci., Volume 15 (2011), pp. 3293-3305 | DOI

[36] Salvadori, G.; De Michele, C.; Kottegoda, N.T.; Rosso, R. Extremes in nature. An approach using copulas, Water Science and Technology Library, 56, Springer, Dordrecht, 2007

[37] Serfling, R. Quantile functions for multivariate analysis: approaches and applications, Statist. Neerlandica, Volume 56 (2002) no. 2, pp. 214-232 | Zbl

[38] Singh, V.P.; Jain, S.K.; Tyagi, A.K. Risk and Reliability Analysis, ASCE Press, Reston, Virginia, 2007

[39] Sklar, A. Fonctions de répartition à n dimensions et leurs marges, Publ. Inst. Statist. Univ. Paris, Volume 8 (1959), pp. 229-231 | MR | Zbl

[40] Vandenberghe, S.; Verhoest, N. E. C.; Buyse, E.; Baets, B. De A stochastic design rainfall generator based on copulas and mass curves, Hydrol. Earth Syst. Sci., Volume 14 (2010), pp. 2429-2442