A goodness-of-fit test for bivariate extreme-value copulas

Resume: It is often reasonable to assume that the dependence structure of a bivariate continuous distribution belongs to the class of extreme-value copulas. The latter are characterized by their Pickands dependence function. The talk is concerned with a procedure for testing whether this function belongs to a given parametric family. The test is based on a Cramer-von Mises statistic measuring the distance between an estimate of the parametric Pickands dependence function and either one of two nonparametric estimators thereof studied by Genest and Segers (2009). As the limiting distribution of the test statistic depends on unknown parameters, it must be estimated via a parametric bootstrap procedure, whose validity is established. Monte Carlo simulations are used to assess the power of the test, and an extension to dependence structures that are left-tail decreasing in both variables is considered.

[1]  Bill Ravens,et al.  An Introduction to Copulas , 2000, Technometrics.

[2]  Michel Denuit,et al.  Actuarial Theory for Dependent Risks: Measures, Orders and Models , 2005 .

[3]  Jun Yan,et al.  Comparison of three semiparametric methods for estimating dependence parameters in copula models , 2010 .

[4]  P. Hall,et al.  Distribution and dependence-function estimation for bivariate extreme-value distributions , 2000 .

[5]  B. Rémillard,et al.  Validity of the parametric bootstrap for goodness-of-fit testing in semiparametric models , 2005 .

[6]  Gunky Kim,et al.  Comparison of semiparametric and parametric methods for estimating copulas , 2007, Comput. Stat. Data Anal..

[7]  C. Genest,et al.  A semiparametric estimation procedure of dependence parameters in multivariate families of distributions , 1995 .

[8]  Nonparametric Estimation of the Dependence Function in Bivariate Extreme Value Distributions , 2001 .

[9]  H. Joe.  Multivariate extreme value distributions , 1997 .

[10]  Collin Carbno,et al.  Actuarial Theory for Dependent Risks: Measures, Orders, and Models , 2007, Technometrics.

[11]  Yanqin Fan,et al.  Pseudo‐likelihood ratio tests for semiparametric multivariate copula model selection , 2005 .

[12]  Carlo De Michele,et al.  Extremes in Nature : an approach using Copulas , 2007 .

[13]  Ludger Rüschendorf,et al.  Asymptotic Distributions of Multivariate Rank Order Statistics , 1976 .

[14]  Bruce L. Jones,et al.  Multivariate Extreme Value Theory And Its Usefulness In Understanding Risk , 2006 .

[15]  Jun Yan,et al.  Modeling Multivariate Distributions with Continuous Margins Using the copula R Package , 2010 .

[16]  M. Sklar.  Fonctions de repartition a n dimensions et leurs marges , 1959 .

[17]  Jun Yan,et al.  Package R copula : "Multivariate dependence with copulas", version 0.9-7 , 2011 .

[18]  Johan Segers,et al.  Nonparametric estimation of an extreme-value copula in arbitrary dimensions , 2009, J. Multivar. Anal..

[19]  Ana Isabel Garralda Guillem.  Structure de dépendance des lois de valeurs extrêmes bivariées , 2000 .

[20]  Christian Genest,et al.  On the Ghoudi, Khoudraji, and Rivest test for extreme‐value dependence , 2009 .

[21]  George E. P. Box,et al.  Empirical Model‐Building and Response Surfaces , 1988 .

[22]  Emiliano A. Valdez,et al.  Understanding Relationships Using Copulas , 1998 .

[23]  B. Rémillard,et al.  Goodness-of-fit tests for copulas: A review and a power study , 2006 .

[24]  J. Segers,et al.  RANK-BASED INFERENCE FOR BIVARIATE EXTREME-VALUE COPULAS , 2007, 0707.4098.

[25]  Christian Genest,et al.  A nonparametric estimation procedure for bivariate extreme value copulas , 1997 .

[26]  Jon A. Wellner,et al.  Weak Convergence and Empirical Processes: With Applications to Statistics , 1996 .

[27]  Christine M. Anderson-Cook,et al.  Book review: quantitative risk management: concepts, techniques and tools, revised edition, by A.F. McNeil, R. Frey and P. Embrechts. Princeton University Press, 2015, ISBN 978-0-691-16627-8, xix + 700 pp. , 2017, Extremes.

[28]  Eckhard Liebscher,et al.  Construction of asymmetric multivariate copulas , 2008 .

[29]  P. Gaenssler,et al.  Seminar on Empirical Processes , 1987 .

[30]  Liang Peng,et al.  Nonparametric estimation of the dependence function for a multivariate extreme value distribution , 2008 .

[31]  I. Keilegom,et al.  Bivariate Archimedean copula models for censored data in non-life insurance , 2006 .

[32]  T. Louis,et al.  Inferences on the association parameter in copula models for bivariate survival data. , 1995, Biometrics.

[33]  Kilani Ghoudi,et al.  Propriétés statistiques des copules de valeurs extrêmes bidimensionnelles , 1998 .

[34]  E. Luciano,et al.  Copula methods in finance , 2004 .

[35]  Stuart A. Klugman,et al.  Fitting bivariate loss distributions with copulas , 1999 .

[36]  Bruno Rémillard,et al.  Goodness‐of‐fit Procedures for Copula Models Based on the Probability Integral Transformation , 2006 .

[37]  Daniel Berg.  Copula goodness-of-fit testing: an overview and power comparison , 2009 .

[38]  Rolf-Dieter Reiss,et al.  On Pickands coordinates in arbitrary dimensions , 2005 .