Articles |
Modelling pairwise dependence of maxima in space
Laboratoire des Sciences du Climat et de l'Environnement, LSCE-IPSL-CNRS, 91191 Gif-sur-Yvette, France naveau{at}lsce.ipsl.fr
Institut de Recherche Mathématique Avancée, Université de Strasbourg, 67084 Strasbourg Cedex, France guillou{at}math.u-strasbg.fr
Department of Statistics, Colorado State University, Fort Collins, Colorado 80523-1877, U.S.A. cooleyd{at}stat.colostate.edu
Laboratoire Analyse et Mathématiques Appliquées, CNRS, Université de Marne-la Vallée, 77454 Marne-la Vallée, France e.diebolt{at}noos.fr
Received for publication 1 May 2006. Revision received 1 August 2008.
We model pairwise dependence of temporal maxima, such as annual maxima of precipitation, that have been recorded in space, either on a regular grid or at irregularly spaced locations. The construction of our estimators stems from the variogram concept. The asymptotic properties of our pairwise dependence estimators are established through properties of empirical processes. The performance of our approach is illustrated by simulations and by the treatment of a real dataset. In addition to bringing new results about the asymptotic behaviour of copula estimators, the latter being linked to first-order variograms, one main advantage of our approach is to propose a simple connection between extreme value theory and geostatistics.
Key Words: Copula • Extreme value theory • Spatial statistic • Variogram
References
-
Ancona-Navarrete M. A., Tawn J. A. Diagnostics for pairwise extremal dependence in spatial processes. Extremes (2002) 5:271–85.[CrossRef]
Baron S., Liflyand E., Stadtmüller U. Complementary spaces and multipliers of double Fourier series for functions of bounded variation. J. Math. Anal. Appl. (2000) 250:706–21.[CrossRef][Web of Science]
Beniston M., Stephenson D., Christensen O., Ferro C., Frei C., Goyette S., Halsnaes K., Holt T., Jylha K., Koffi B., Palutikof J., Scholl R., Semmler T., Woth K. Future extreme events in European climate: an exploration of regional climate model projections. Climatic Change (2007) 81:71–95.
Capéraà P., Fougères A.-L., Genest C. A nonparametric estimation procedure for bivariate extreme value copulas. Biometrika (1997) 84:567–77.
Casson E. A., Coles S. Spatial regression for extremes. Extremes (1999) 1:449–68.[CrossRef]
Coles S. G., Heffernan J. E., Tawn J. A. Dependence measures for extreme value analyses. Extremes (1999) 2:339–65.[CrossRef]
Cooley D., Naveau P., Poncet P. Variograms for Spatial Max-stable Random Fields (2006) 373–90. Lecture Notes in Statistics 187. New York: Springer.
Cressie N. A. C. Statistics for Spatial Data (1993) New York: John Wiley.
Davis R., Resnick S. Prediction of stationary max-stable processes. Ann. Appl. Prob. (1993) 3:497–525.[CrossRef]
de Haan L., Pereira T. Spatial extremes: the stationary case. Ann. Statist. (2006) 34:146–68.[CrossRef]
de Haan L., Resnick S. Limit theory for multivariate sample extremes. Z. Wahr. verw. Geb. (1977) 4:317–37.
Deheuvels P. La fonction de dépendance empirique et ses propriétés. Un test non paramétrique d'indépendance. Acad. R. Belg. Bull. Clin. Sci. (1979) 65:274–92.
Deheuvels P. On the limiting behaviour of the Pickands estimator for bivariate extreme-value distributions. Statist. Prob. Lett. (1991) 12:429–39.[CrossRef]
Embrechts P., Klüppelberg C., Mikosch T. Modelling Extremal Events for Insurance and Finance (1997) Berlin: Springer.
Fermanian J.-D., Radulovi
D., Wegkamp M. Weak convergence of empirical copula processes. Bernoulli (2004) 10:847–60.[CrossRef][Web of Science]
Fougères A.-L. Multivariate extremes. In: Extreme Values in Finance, Telecommunications and the Environment—Finkenstadt B., Rootzen H., eds. (2004) London: Chapman & Hall/CRC Press. 373–88.
Frei C., Scholl R., Fukutome S., Schmidli J., Vidale P. L. Future change of precipitation extremes in Europe: intercomparison of scenarios from regional climate models. J. Geophys. Res. (2006) 111:1–22.[CrossRef]
Gumbel E. J. Distributions des valeurs extrêmes en plusieurs dimensions. Pub. Inst. Statist. Univ. Paris (1960) 9:171–3.
Hall P., Tajvidi N. Distribution and dependence function estimation for bivariate extreme-value distributions. Bernoulli (2000) 6:835–44.[CrossRef][Web of Science]
Heffernan J. E., Tawn J. A. A conditional approach for multivariate extreme values (with Discussion). J. R. Statist. Soc. (2004) B 66:497–546.[CrossRef]
Hsing T., Kluppelberg C., Kuhn G. Dependence estimation and visualization in multivariate extremes with applications to financial data. Extremes (2004) 7:99–121.[CrossRef]
Joe H. Parametric family of multivariate distributions with given margins. J. Mult. Anal. (1993) 46:262–82.[CrossRef]
Ledford A., Tawn J. Modelling dependence within joint tail regions. J. R. Statist. Soc. (1997) B 59:475–99.[CrossRef]
Matheron G. Suffit-il, pour une covariance, d'être de type positif? Sci. Terre. Info. Geol. (1987) 26:51–66.
Mikosch T. Copulas: tales and facts. Extremes (2006) 9:3–20.[CrossRef]
Neuhaus G. On the weak convergence of stochastic processes with multidimensional time parameter. Ann. Math. Statist. (1971) 42:1285–95.[CrossRef]
Pickands J. II. Multivariate extreme value distributions. Bull. Int. Statist. Inst. (1981) 49:859–78.
R Development Core Tea. R: a Language and Environment for Statistical Computing (2006) Vienna: Springer-/R Foundation for Statistical Computing.
Resnick S. Extreme Values, Regular Variation, and Point Processes (1987) New York: Springer.
Schlather M. Models for stationary max-stable random fields. Extremes (2002) 5:33–44.[CrossRef]
Schlather M., Tawn J. A dependence measure for multivariate and spatial extreme values: properties and inference. Biometrika (2003) 90:139–56.
Smith R. Statistics of extremes, with applications in environment, insurance and finance. Extreme Values in Finance, Telecommunications and the Environment—Finkenstadt B., Rootzén H., eds. (2004) pp. 1–78. London: Chapman & Hall/CRC Press.
Tawn J. A. Bivariate extreme value theory: models and estimation. Biometrika (1988) 75:397–415.
Tawn J. A. Modelling multivariate extreme value distributions. Biometrika (1990) 77:245–53.
van der Vaart A., Wellner J. A. Weak Convergence and Empirical Processes (1996) New York: Springer.
Wackernagel H. Multivariate Geostatistics. An, 3rd ed. Heidelberg (2003) Introduction with Applications: Springer.
CiteULike 
Connotea 
Del.icio.us What's this?
| ||||||||||||||||||||||||||||||||||||||||||||||||||