| Title:
|
On extremal dependence of block vectors (English) |
| Author:
|
Ferreira, Helena |
| Author:
|
Ferreira, Marta |
| Language:
|
English |
| Journal:
|
Kybernetika |
| ISSN:
|
0023-5954 |
| Volume:
|
48 |
| Issue:
|
5 |
| Year:
|
2012 |
| Pages:
|
988-1006 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Due to globalization and relaxed market regulation, we have assisted to an increasing of extremal dependence in international markets. As a consequence, several measures of tail dependence have been stated in literature in recent years, based on multivariate extreme-value theory. In this paper we present a tail dependence function and an extremal coefficient of dependence between two random vectors that extend existing ones. We shall see that in weakening the usual required dependence allows to assess the amount of dependence in $d$-variate random vectors based on bidimensional techniques. Simple estimators will be stated and can be applied to the well-known stable tail dependence function. Asymptotic normality and strong consistency will be derived too. An application to financial markets will be presented at the end. (English) |
| Keyword:
|
multivariate extreme value theory |
| Keyword:
|
tail dependence |
| Keyword:
|
extremal coefficients |
| MSC:
|
60G70 |
| idMR:
|
MR3086865 |
| . |
| Date available:
|
2012-12-17T13:41:11Z |
| Last updated:
|
2013-09-24 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143095 |
| . |
| Reference:
|
[1] Beirlant, J., Goegebeur, Y., Segers, J., Teugels, J.: Statistics of Extremes: Theory and Application..John Wiley, Chichester 2004. MR 2108013 |
| Reference:
|
[2] Coles, S., Heffernan, J., Tawn, J.: Dependence measures for extreme value analysis..Extremes 2 (1999), 339-366. 10.1023/A:1009963131610 |
| Reference:
|
[3] Draisma, G., Drees, H., Ferreira, A., Haan, L. de: Bivariate tail estimation: dependence in asymptotic independence..Bernoulli 10 (2004), 251-280. Zbl 1058.62043, MR 2046774, 10.3150/bj/1082380219 |
| Reference:
|
[4] Drees, H., Müller, P.: Fitting and validation of a bivariate model for large claims..Insurance Math. Econom. 42 (2008), 638-650. Zbl 1152.91578, MR 2404319, 10.1016/j.insmatheco.2007.07.001 |
| Reference:
|
[5] Embrechts, P., Lindskog, F., McNeil, A.: Modelling dependence with copulas and applications to risk management..In: Handbook of Heavy Tailed Distibutions in Finance (S. Rachev, ed.), Elsevier, Amsterdam 2003, pp. 329-384. |
| Reference:
|
[6] Fermanian, J. D., Radulović, D., Wegkamp, M.: Weak convergence of empirical copula processes..Bernoulli 10 (2004), 5, 847-860. Zbl 1068.62059, MR 2093613, 10.3150/bj/1099579158 |
| Reference:
|
[7] Frahm, G.: On the extremal dependence coefficient of multivariate distributions..Statist. Probab. Lett. 76 (2006), 1470-1481. Zbl 1120.62035, MR 2245567, 10.1016/j.spl.2006.03.006 |
| Reference:
|
[8] Frahm, G., Junker, M., Schmidt, R.: Estimating the tail-dependence coefficient: properties and pitfalls..Insurance Math. Econom. 37 (2005), 1, 80-100. Zbl 1101.62012, MR 2156598, 10.1016/j.insmatheco.2005.05.008 |
| Reference:
|
[9] Gilat, D., Hill, T.: One-sided refinements of the strong law of large numbers and the Glivenko-Cantelli theorem..Ann. Probab. 20 (1992), 1213-1221. Zbl 0762.60025, MR 1175259, 10.1214/aop/1176989688 |
| Reference:
|
[10] Hua, L., Joe, H.: Tail order and intermediate tail dependence of multivariate copulas..J. Multivariate Anal. 102 (2011), 10, 1454-1471. Zbl 1221.62079, MR 2819962, 10.1016/j.jmva.2011.05.011 |
| Reference:
|
[11] Huang, X.: Statistics of Bivariate Extreme Values..Ph.D. Thesis, Tinbergen Institute Research Series 22, Erasmus University Rotterdam 1992. |
| Reference:
|
[12] Joe, H.: Multivariate Models and Dependence Concepts..Chapman and Hall, London 1997. Zbl 0990.62517, MR 1462613 |
| Reference:
|
[13] Krajina, A.: An M-Estimator of Multivariate Tail Dependence..Tilburg University Press 2010. |
| Reference:
|
[14] Ledford, A., Tawn, J.: Statistics for near independence in multivariate extreme values..Biometrika 83 (1996), 169-187. Zbl 0865.62040, MR 1399163, 10.1093/biomet/83.1.169 |
| Reference:
|
[15] Ledford, A., Tawn, J.: Modelling Dependence within joint tail regions..J. R. Statist. Soc. Ser. B Stat. Methodol. 59 (1997), 475-499. Zbl 0886.62063, MR 1440592, 10.1111/1467-9868.00080 |
| Reference:
|
[16] Li, H.: Tail Dependence of Multivariate Pareto Distributions..WSU Mathematics Technical Report 2006-6, Washington 2006. |
| Reference:
|
[17] Li, H.: Tail dependence comparison of survival Marshall-Olkin copulas..Methodol. Comput. Appl. Probab. 10 (2008), 1, 39-54. Zbl 1142.62035, MR 2394034, 10.1007/s11009-007-9037-3 |
| Reference:
|
[18] Li, H.: Orthant tail dependence of multivariate extreme value distributions..J. Multivariate Anal. 100 (2009), 1, 243-256. Zbl 1151.62041, MR 2460490, 10.1016/j.jmva.2008.04.007 |
| Reference:
|
[19] Li, H., Sun, Y.: Tail dependence for heavy-tailed scale mixtures of multivariate distributions..J. Appl. Probab. 46 (2009), 4, 925-937. Zbl 1179.62076, MR 2582698, 10.1239/jap/1261670680 |
| Reference:
|
[20] Marshall, A. W., Olkin, I.: A multivariate exponential distribution..J. Amer. Statist. Assoc. 62 (1967), 30-44. Zbl 0147.38106, MR 0215400, 10.1080/01621459.1967.10482885 |
| Reference:
|
[21] Nelsen, R. B.: Nonparametric measures of multivariate association..In: Distribution with fixed marginals and related topics, IMS Lecture Notes - Monograph Series, Vol. 28 (L. Rüschendorf et al., eds.) Hayward, Institute of Mathematical Statistics 1996, pp. 223-232. MR 1485534 |
| Reference:
|
[22] Nelsen, R. B.: An Introduction to Copulas. Second edition..Springer, New York 2006. MR 2197664 |
| Reference:
|
[23] Neuhaus, G.: On the weak convergence of stochastic processes with multidimensional time parameter..Ann. Math. Statist. 42 (1971), 1285-1295. MR 0293706, 10.1214/aoms/1177693241 |
| Reference:
|
[24] Schmid, F., Schmidt, R.: Multivariate conditional versions of Spearman's rho and related measures of tail dependence..J. Multivariate Anal. 98 (2007), 1123-1140. Zbl 1116.62061, MR 2326243, 10.1016/j.jmva.2006.05.005 |
| Reference:
|
[25] Schmidt, R., Stadtmüller, U.: Nonparametric estimation of tail dependence..Scand. J. Statist. 33 (2006), 307-335. Zbl 1124.62016, MR 2279645, 10.1111/j.1467-9469.2005.00483.x |
| Reference:
|
[26] Smith, R. L.: Max-stable processes and spatial extremes..Preprint, Univ. North Carolina, USA 1990. |
| Reference:
|
[27] Smith, R. L., Weissman, I.: Characterization and estimation of the multivariate extremal index..Manuscript, UNC 1996. |
| Reference:
|
[28] Sibuya, M.: Bivariate extreme statistics..Ann. Inst. Statist. Math. 11 (1960), 195-210. Zbl 0095.33703, MR 0115241, 10.1007/BF01682329 |
| Reference:
|
[29] Oliveira, J. Tiago de: Structure theory of bivariate extremes: extensions..Est. Mat. Estat. e Econ. 7 (1962/63), 165-195. |
| Reference:
|
[30] Wolff, E. F.: N-dimensional measures of dependence..Stochastica 4 (1980), 3, 175-188. Zbl 0482.62048, MR 0611502 |
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