On the tail dependence in bivariate hydrological frequency analysis

• Autorzy: Alexandre Lekina , Fateh Chebana , Taha B. M. J. Ouarda
• Języki publikacji: EN

Abstrakty

EN

In Bivariate Frequency Analysis (BFA) of hydrological events, the study and quantification of the dependence between several variables of interest is commonly carried out through Pearson’s correlation (r), Kendall’s tau (τ) or Spearman’s rho (ρ). These measures provide an overall evaluation of the dependence. However, in BFA, the focus is on the extreme events which occur on the tail of the distribution. Therefore, these measures are not appropriate to quantify the dependence in the tail distribution. To quantify such a risk, in Extreme Value Analysis (EVA), a number of concepts and methods are available but are not appropriately employed in hydrological BFA. In the present paper, we study the tail dependence measures with their nonparametric estimations. In order to cover a wide range of possible cases, an application dealing with bivariate flood characteristics (peak flow, flood volume and event duration) is carried out on three gauging sites in Canada. Results show that r, τ and ρ are inadequate to quantify the extreme risk and to reflect the dependence characteristics in the tail. In addition, the upper tail dependence measure, commonly employed in hydrology, is shown not to be always appropriate especially when considered alone: it can lead to an overestimation or underestimation of the risk. Therefore, for an effective risk assessment, it is recommended to consider more than one tail dependence measure.

Słowa kluczowe

EN

Asymptotic; Extreme; Copula; Bivariate distribution; Non-parametric estimation

• Wydawca: De Gruyter Open
• Czasopismo: Dependence Modeling
• Rocznik: 2015
• Tom: 3
• Numer: 1

Daty

otrzymano

2015-07-07

zaakceptowano

2015-10-22

online

2015-11-19

Twórcy

autor

Alexandre Lekina

• ITE - VEDECOM, Institut de la Transition Énergétique du Véhicule Décarboné et Communicant et de sa Mobilité. Fondation Partenariale de l’Université de Versailles Saint-Quentin-en-Yvelines, 77 rue des Chantiers, 78000- Versailles,France
• Institut National de la Recherche Scientifique, INRS-ETE, 490 rue de la Couronne, Québec (Québec) G1K 9A9,
  Canada

autor

Fateh Chebana

• Institut National de la Recherche Scientifique, INRS-ETE, 490 rue de la Couronne, Québec (Québec) G1K 9A9,
  Canada

autor

Taha B. M. J. Ouarda

• Institut National de la Recherche Scientifique, INRS-ETE, 490 rue de la Couronne, Québec (Québec) G1K 9A9,
  Canada
• Masdar Institute of Science and technology, PO Box 54224, Abu Dhabi, UAE

Bibliografia

• [1] Ané, T. and C. Kharoubi (2003). Dependence structure and risk measure. J. Business 76, 411–438. [Crossref]
• [2] Bacro, J. N. (2005). Dépendance des extrêmes. Tech. rep., Institut de Mathématiques et de Modélisation de Montpellier.
• [3] Beirlant, J., Y. Goegebeur, J. Teugels, and J. Segers (2004). Statistics of Extremes: Theory and Applications. John Wiley and Sons, Chichester.
• [4] Ben Aissia, M.-A., F. Chebana, T. B. M. J. Ouarda, L. Roy, G. Desrochers, I. Chartier, and E. Robichaud (2012). Multivariate analysis of flood characteristics in a climate change context of thewatershed of the Baskatong reservoir, Province of Québec, Canada. Hydrol. Process. 26(1), 130–142. [Crossref]
• [5] Bingham, N. H., C. M. Goldie, and J. L. Teugels (1987). Regular Variation. Cambridge University Press.
• [6] Bobée, B. and F. Ashkar (1991). The Gamma Family and Derived Distributions Applied in Hydrology. Water Resources Publications, Colorado.
• [7] Bouchouicha, R. (2010). Dépendance entre risques extrêmes: application aux hedge funds. Tech. Rep. 1013, Groupe d’Analyse et de Théorie Economique (GATE), Centre national de la recherche scientifique (CNRS), Université Lyon 2, Ecole Normale Supérieure.
• [8] Capéraà, P., A.-L. Fougères, and C. Genest (1997). A nonparametric estimation procedure for bivariate extreme value copulas. Biometrika 84(3), 567–577. [Crossref]
• [9] Chebana, F. (2013). Multivariate analysis of hydrological variables. In Encyclopedia of Environmetrics, Second Edition, A. H. El-Shaarawi and W. W. Piegorsch eds., 1676–1681. John Wiley and Sons, Chichester.
• [10] Chebana, F. and T. Ouarda (2011). Depth-based multivariate descriptive statistics with hydrological applications. J. Geophys. Res. 116(D10), 1–19. [Crossref]
• [11] Chebana, F. and T. Ouarda (2011). Multivariate quantiles in hydrological frequency analysis. Environmetrics 22(1), 63–78. [Crossref]
• [12] Coles, S., J. Heffernan, and J. Tawn (1999). Dependence measures for extremes value analyses. Extremes 2(4), 339–365. [Crossref]
• [13] Cont, R. (2009). La statistique face aux événements rares. Pour la science 385, 116–123.
• [14] Deheuvels, P. (1979). La fonction de dépendance empirique et ses propriétés. Un test non paramétrique d’indépendance. Acad. Roy. Belg. Bull. Cl. Sci. 5(65), 274–292.
• [15] Dobric, J. and F. Schmid (2005). Nonparametric estimation of the lower tail dependence λL in bivariate copulas. J. Appl. Statist. 32, 387–407. [Crossref]
• [16] Draisma, G., H. Drees, A. Ferreira, and L. de Haan (2004). Bivariate tail estimation: dependence in asymptotic independence. Bernoulli 10(2), 251–280. [Crossref]
• [17] Drees, H. and X. Huang (1998). Best attainable rates of convergence for estimators of the stable tail dependence function. J. Multivariate Anal. 64(1), 25–47. [Crossref]
• [18] Dupuis, D. J. (2007). Using copulas in hydrology: benefits, cautions, and issues. J. Hydrol. Eng. 12(4), 381–393. [Crossref]
• [19] Durante, F., J. Fernández-Sánchez, and R. Pappadà (2015). Copulas, diagonals, and tail dependence. Fuzzy Set. Syst. 264, 22–41.
• [20] Einmahl, J. H. J., A. Krajina, and J. Segers (2008). A method of moments estimator of tail dependence. Bernoulli 14, 1003– 1026. [Crossref]
• [21] Embrechts, P., A. McNeil, and D. Straumann (1999). Correlation: Pitfalls and alternatives. Risk Magazine May, 69–71.
• [22] Embrechts, P., A. Mcneil, and D. Straumann (2002). Correlation and dependence in riskmanagement: Properties and pitfalls. In Risk Management: Value at Risk and Beyond, M.A.H. Dempster ed., Cambridge University Press, 176-223.
• [23] Fischer, M. and M. Dörflinger (2006). A note on a non-parametric tail dependence estimator. Discussion Papers 76/2006, Friedrich-Alexander-University Erlangen-Nuremberg.
• [24] Frahm, G., M. Junker, and R. Schmidt (2005). Estimating the tail-dependence coefficient: properties and pitfalls. Insurance Math. Econ. 37(1), 80–100.
• [25] Galambos, J. (1978). The Asymptotic Theory of Extreme Order Statistics. Wiley, New York.
• [26] Geffroy, J., 1958: Contribution à la théorie des valeurs extrêmes. Public. Inst. Stat. Paris 7 7, 37–121.
• [27] Geffroy, J. (1959). Contribution à la théorie des valeurs extrêmes. Public. Inst. Stat. Paris 7 8, 123–184.
• [28] Genest, C. and F. Chebana (2015). Copula modeling in hydrological frequency analysis. To appear in Chow’s Handbook of Applied Hydrology, Second Edition, McGraw-Hill.
• [29] Genest, C. and A.-C. Favre (2007). Everything you always wanted to know about copula modeling but were afraid to ask. J. Hydrol. Eng. 12(4), 347–368. [Crossref]
• [30] Genest, C., I. Kojadinovic, J. Nešlehová, and J. Yan (2010). A goodness-of-fit test for bivariate extreme-value copulas. Bernoulli 17(1), 253–275. [Crossref]
• [31] Genest, C. and J. Segers (2009). Rank-based inference for bivariate extreme-value copulas. Ann. Stat. 37, 2990–3022. [Crossref]
• [32] Ghosh, S. (2010). Modelling bivariate rainfall distribution and generating bivariate correlated rainfall data in neighbouring meteorological subdivisions using copula. Hydrol. Process. 24(24), 3558–3567.
• [33] Ghoudi, K., A. Khoudraji, and L.-P. Rivest (1998). Propriétés statistiques des copules de valeurs extrêmes bidimensionnelles. Canad. J.Stat. 26(1), 187–197.
• [34] Heffernan, J. (2000). A directory of coefficients of tail dependence. Extremes 3(3), 279–290. [Crossref]
• [35] Hill, B. (1975). A simple general approach to inference about the tail of a distribution. Ann. Stat. 3(5), 1163–1174. [Crossref]
• [36] Huang, X. (1992). Statistics of Bivariate Extreme Values. Ph.D. thesis, Erasmus University Rotterdam.
• [37] Joe, H. (1997). Multivariate Models and Dependence Concepts. Chapman and Hall, London.
• [38] Joe, H., R. L. Smith, and I. Weissman (1992). Bivariate threshold methods for extremes. J. R. Stat. Soc. Ser. B 54(1), 171–183.
• [39] Juri, A. and M. Wuthrich (2002). Copula convergence theorems for tail events. Insurance Math. Econ. 30(3), 405–420.
• [40] Kendall, M. G. (1938). A new measure of rank correlation. Biometrika 30(1/2), 81–93. [Crossref]
• [41] Kojadinovic, I. and J. Yan (2010). Nonparametric rank-based tests of bivariate extreme-value dependence. J. Multivariate Anal. 101, 2234–2249. [Crossref]
• [42] Kratz, M. and S. Resnick (1996). The QQ-estimator and heavy tails. Stochastic Models 12(4), 699–724.
• [43] Ledford, A. W. and J. A. Tawn (1996). Statistics for near independence in multivariate extreme values. Biometrika 83(1), 169–187. [Crossref]
• [44] Ledford, A. W. and J. A. Tawn (1997). Modelling dependence within joint tail regions. J. R. Stat. Soc. Ser. B 59(2), 475–499. [Crossref]
• [45] Ledford, A. W. and J. A. Tawn (1998). Concomitants of extremes. Adv. Appl. Probab. 30(1), 197–215. [Crossref]
• [46] Lee, T., R. Modarres, and T. Ouarda (2013). Data based analysis of bivariate copula tail dependence for drought duration and severity. Hydrol. Process. 27(10), 1454–1463. [Crossref]
• [47] Lekina, A., (2010). Estimation Non-paramétrique des Quantiles Extrêmes Conditionnels. Ph.D. thesis, Université de Grenoble.
• [48] Lekina, A., F. Chebana, and T. Ouarda (2014). Weighted estimate of extreme quantile: an application to the estimation of high flood return periods. Stoch. Env. Res. Risk. A. 28(2), 147–165. [Crossref]
• [49] Longin, F. M. and B. Solnik (2001). Extreme correlations of international equity markets during extremely volatile periods. J. Finance 56, 649–676. [Crossref]
• [50] Malevergne, Y. and D. Sornette (2004). Investigating extreme dependences. Review Financ. Studies, in press.
• [51] Nelsen, R. B. (2006). An Introduction to Copulas, Second edition, Springer-Verlag New York.
• [52] Ouarda, T. B. M. J., M. Haché, P. Bruneau, and B. Bobée (2000). Regional flood peak and volume estimation in northern canadian basin. J. Cold Reg. Eng. 14(4), 176–191.
• [53] Patton, A. (2006). Estimation of multivariate models for time series of possibly different lengths. J. Appl. Econom. 21(2), 147–173. [Crossref]
• [54] Peng, L. (1999). Estimation of the coefficient of tail dependence in bivariate extremes. Stat. Probabil. Lett. 43(4), 399–409. [Crossref]
• [55] Pickands, J. (1975). Statistical inference using extreme order statistics. Ann. Stat. 3(1), 119–131. [Crossref]
• [56] Pickands, J. (1981). Multivariate extreme value distributions. Bull. Inst. Internat. Statist. 49(2), 859–878.
• [57] Poon, S.-H., M. Rockinger, and J. Tawn (2004). Extreme value dependence in financial markets: diagnostics, models, and financial implications. Review Financ. Studies 17(2), 581–610. [Crossref]
• [58] Poulin, A., D. Huard, A.-C. Favre, and S. Pugin (2007). Importance of tail dependence in bivariate frequency analysis. J. Hydrol. Eng. 12(4), 394–403. [Crossref]
• [59] Raynal-Villasenor, J. and J. Salas (1987). Multivariate extreme value distributions in hydrological analyses. In Water for the Future: Hydrology in Perspective, Rodda, J. C. and N.C. Matalas eds., 111–119.
• [60] Resnick, S. (1987). Extreme Values, Regular Variation, and Point Process. Springer Verlag, New-York.
• [61] Rohatgi, V. (1976). An Introduction to Probability Theory and Mathematical Statistics. John Wiley & Sons, New York.
• [62] Salvadori, G. and C. De Michele (2011). Estimating strategies for multiparameter multivariate extreme value copulas. Hydrol. Earth Syst. Sc. 15(1), 141–150. [Crossref]
• [63] Salvadori, G., C. De Michele, N. T. Kottegoda, and R. Rosso (2007). Extremes in Nature: an Approach Using Copulas. Springer, Dordrecht.
• [64] Schmidt, R. (2002). Tail dependence for elliptically contoured distributions. Math. Methods Oper. Res. 55, 301–327. [Crossref]
• [65] Schmidt, R. and U. Stadtmuller (2006). Non-parametric estimation of tail dependence. Scand. J. Statist. 33, 307–335. [Crossref]
• [66] Schultze, J. and J. Steinebach (1996). On least squares estimates of an exponential tail coefficient. Stat. Decisions 14(3), 353–372.
• [67] Segers, J. (2007). Non-parametric inference for bivariate extreme-value copulas. In Topics in Extreme Values, M. Ahsanullah and S. N. U. A. Kirmani eds., Nova Science Publishers Inc, New York, 181–203.
• [68] Serinaldi, F. (2008). Analysis of inter-gauge dependence by Kendall’s fiK, upper tail dependence coefficient, and 2-copulas with application to rainfall fields. Stoch. Env. Res. Risk. A. 22(6), 671–688.
• [69] Shiau, J. T. (2003). Return period of bivariate distributed extreme hydrological events. Stoch. Env. Res. Risk. A. 17, 42–57. [Crossref]
• [70] Shiau, J.-T., R. Modarres, and S. Nadarajah (2012). Assessing multi-site drought connections in Iran using empirical copula. Environ. Model. Assess. 17(5), 469–482. [Crossref]
• [71] Sklar, A. (1959). Fonstions de répartition à n dimensions et leurs marges. Publ. Inst. Statist. Univ. Paris 8, 229–231.
• [72] Venter, G. G., 2002: Tails of copulas. Proc. Casualty Actuarial Soc. 89, 68–113.
• [73] Wasserman, L. (2003). All of Statistics: a Concise Course in Statistical Inference. Springer, New York.
• [74] Yue, S., T. Ouarda, B. Bobée, P. Legendre, and P. Bruneau (1999). The Gumbel mixed model for flood frequency analysis. J. Hydrol. 226(1-2), 88–100. [Crossref]
• [75] Zhang, L. and V. P. Singh (2006). Bivariate flood frequency analysis using the copula method. J. Hydrol. Eng. 11(2), 150–164. [Crossref]

Identyfikatory

DOI

10.1515/demo-2015-0015