Assessing and Modeling Asymmetry in Bivariate Continuous Data

A bivariate copula is the cumulative distribution function of a pair (U, V ) of uniform random variables. This copula is said to be symmetric if and only if (V, U) and (U, V ) have the same distribution. Many standard bivariate parametric families of copulas have this property; Archimedean and meta-elliptical copulas are prime examples. In practice, however, dependence is often asymmetric. This paper revisits key aspects of this issue from a modeling perspective. Measures of asymmetry and rank-based estimators thereof are discussed, along with recently proposed tests of symmetry. Several techniques for the construction of asymmetric dependence structures are critically reviewed. A hydrological data set is used for illustration purposes.

[1]  Christian Genest,et al.  Tests of symmetry for bivariate copulas , 2011, Annals of the Institute of Statistical Mathematics.

[2]  Fabrizio Durante,et al.  Invariant dependence structure under univariate truncation , 2012 .

[3]  C. Genest,et al.  Bivariate Distributions with Given Extreme Value Attractor , 2000 .

[4]  M. Wegkamp,et al.  Weak Convergence of Empirical Copula Processes , 2004 .

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

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

[7]  Karl Friedrich Siburg,et al.  Symmetry of functions and exchangeability of random variables , 2011 .

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

[9]  B. V. Mendes.  Asymmetric extreme interdependence in emerging equity markets , 2005 .

[10]  Christian Genest,et al.  “Understanding Relationships Using Copulas,” by Edward Frees and Emiliano Valdez, January 1998 , 1998 .

[11]  Radko Mesiar,et al.  Non-exchangeable random variables, Archimax copulas and their fitting to real data , 2011, Kybernetika.

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

[13]  José Juan Quesada-Molina,et al.  On the construction of copulas and quasi-copulas with given diagonal sections , 2008 .

[14]  H. Joe.  Multivariate models and dependence concepts , 1998 .

[15]  Christian Genest,et al.  A goodness-of-fit test for bivariate extreme-value copulas , 2011, 1102.2078.

[16]  Jun Yan,et al.  Nonparametric rank-based tests of bivariate extreme-value dependence , 2010, J. Multivar. Anal..

[17]  Radko Mesiar,et al.  How non-symmetric can a copula be? , 2006 .

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

[19]  E. Klement,et al.  Measures of non-exchangeability for bivariate random vectors , 2010 .

[20]  Roger B. Nelsen,et al.  Extremes of nonexchangeability , 2007 .

[21]  J. Segers.  Asymptotics of empirical copula processes under non-restrictive smoothness assumptions , 2010, 1012.2133.

[22]  Radko Mesiar,et al.  Univariate conditioning of copulas , 2008, Kybernetika.

[23]  P. Embrechts,et al.  Quantitative Risk Management: Concepts, Techniques, and Tools , 2005 .

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

[25]  Alexander J. McNeil,et al.  Multivariate Archimedean copulas, $d$-monotone functions and $\ell_1$-norm symmetric distributions , 2009, 0908.3750.

[26]  José M. González-Barrios,et al.  On the Construction of Families of Absolutely Continuous Copulas with Given Restrictions , 2006 .

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

[28]  Fabrizio Durante,et al.  Componentwise Concave Copulas and Their Asymmetry , 2009, Kybernetika.

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

[30]  Radko Mesiar,et al.  L ∞ -measure of non-exchangeability for bivariate extreme value and Archimax copulas , 2010 .

[31]  Andrew J. Patton.  (IAM Series No 001) On the Out-Of-Sample Importance of Skewness and Asymetric Dependence for Asset Allocation , 2002 .

[32]  Holger Dette,et al.  A note on bootstrap approximations for the empirical copula process , 2010 .

[33]  Radko Mesiar,et al.  Asymmetric semilinear copulas , 2007, Kybernetika.

[34]  Christian Genest,et al.  Copula Modeling for Extremes , 2013 .

[35]  Radko Mesiar,et al.  Copulas with Given Diagonal Sections: Novel Constructions and Applications , 2007, Int. J. Uncertain. Fuzziness Knowl. Based Syst..

[36]  R. Mesiar,et al.  Invariant dependence structures and Archimedean copulas , 2011 .

[37]  Alexander J. McNeil,et al.  From Archimedean to Liouville copulas , 2010, J. Multivar. Anal..

[38]  F. Durante,et al.  Non-exchangeability of negatively dependent random variables , 2010 .

[39]  Aurélien Alfonsi,et al.  New Families of Copulas Based on Periodic Functions , 2005 .

[40]  H. Tsukahara.  Semiparametric estimation in copula models , 2005 .

[41]  Eckhard Liebscher.  Erratum to "Construction of asymmetric multivariate copulas" [J. Multivariate Anal. 99(2008) 2234-2250] , 2011, J. Multivar. Anal..

[42]  F. Durante.  Construction of non-exchangeable bivariate distribution functions , 2009 .

[43]  A. V. D. Vaart,et al.  Asymptotic Statistics: Frontmatter , 1998 .

[44]  Alexander J. McNeil,et al.  Likelihood inference for Archimedean copulas in high dimensions under known margins , 2012, J. Multivar. Anal..