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The article presents information on the finding of goodness-of-fit method in order to test the parametric specification for the function of Archimedean copulas. It informs that copulas are an important tool in multivariate statistics and they capture the pure dependence structure between random variables irrespective of their marginal distributions. Random number generation from Archimedean copulas is stated to be easy because of their simple structure. It is stated that if the dependence structure of the return vector can accurately be described by a parametric Archimedean copula, one could aim at developing credit portfolio models that capture the behavior of joint returns more realistically.

The multivariate tail dependence describes the amount of dependence in the upper-orthant tail or lower-orthant tail of a multivariate distribution and can be used in the study of dependence among extreme values. We derive an explicit expression of tail dependence of multivariate survival Marshall–Olkin copulas, and obtain a sufficient condition under which tail dependencies of two survival Marshall–Olkin copulas can be compared. Some examples are also presented to illustrate our results. [ABSTRACT FROM AUTHOR]

We generalize the extreme value analysis for Archimedean copulas (seeAlink,Löwe andWüthrich, 2003) to the non-Archimedean case: Assume we have d≥2 exchangeable and continuously distributed risks X1,..., X d. Under appropriate assumptions there is a constant q d such that, for all large u, we have . The constant q d describes the asymptotic dependence structure. Typically, q d will depend on more aspects of this dependence structure than the well-known tail dependence coefficient. [ABSTRACT FROM AUTHOR]

A bivariate distribution function H(x, y) with marginals F(x) and G(y) is said to be generated by an Archimedean copula if it can be expressed in the form H(x, y) = φ-1[φ{F(x)} + φ{G(y)}] for some convex, decreasing function φ defined on (0, 1] in such a way that φ(1) = 0. Many well-known systems of bivariate distributions belong to this class, including those of Gumbel, Ali-Mikhail-Haq-Thélot, Clayton, Frank, and Hougaard. Frailty models also fall under that general prescription. This article examines the problem of selecting an Archimedean copula providing a suitable representation of the dependence structure between two variates X and Y in the light of a random sample (X1, Y1), ,(Xn, Yn) The key to the estimation procedure is a one-dimensional empirical distribution function that can be constructed whether the uniform representation of X and Y is Archimedean or not, and independently of their marginals. This semiparametric estimator, based on a decomposition of Kendall's tau statistic, is seen to be √n-consistent, and an explicit formula for its asymptotic variance is provided. This leads to a strategy for selecting the parametric family of Archimedean copulas that provides the best possible fit to a given set of data. To illustrate these procedures, a uranium exploration data set is reanalyzed. Although the presentation is restricted to problems revolving a random sample from a bivariate distribution, extensions to situations involving multivariate or censored data could be envisaged. [ABSTRACT FROM AUTHOR]