Dependence Properties of Meta-Elliptical Distributions (original) (raw)

General Multivariate Dependence Using Associated Copulas

SSRN Electronic Journal, 2011

This paper studies the general multivariate dependence of a random vector using associated copulas. We extend definitions and results of positive dependence to the general dependence case. This includes associated tail dependence functions and associated tail dependence coefficients. We derive the relationships among associated copulas and study the associated copulas of the perfect dependence cases and elliptically contoured distributions. We present the expression for the associated tail dependence function of the multivariate Student-t copula, which accounts for all types of tail dependence. Date: April 20th, 2012. This paper is based on results from the first chapters of my doctorate thesis supervised by Dr. Wing Lon Ng. I would like to thank Mexico's CONACYT, for the funding during my studies. I would also like to thank my examiners Dr. Aristidis Nikoloulopoulos and Dr. Nick Constantinou for the corrections, suggestions and comments that have made this work possible.

Some results on weak and strong tail dependence coefficients for means of copulas

2007

Copulas represent the dependence structure of multivariate distributions in a natural way. In order to generate new copulas from given ones, several proposals found its way into statistical literature. One simple approach is to consider convex-combinations (i.e. weighted arithmetic means) of two or more copulas. Similarly, one might consider weighted geometric means. Consider, for instance, the Spearman copula, defined as the geometric mean of the maximum and the independence copula. In general, it is not known whether weighted geometric means of copulas produce copulas, again. However, applying a recent result of Liebscher (2006), we show that every weighted geometric mean of extreme-value copulas produces again an extreme-value copula. The second contribution of this paper is to calculate extremal dependence measures (e.g. weak and strong tail dependence coe±cients) for (weighted) geometric and arithmetic means of two copulas. --

Further Results for a General Family of Bivariate copulas

Communications in Statistics Theory and Methods, 2013

A bivariate family of copulas has been initiated by Cuadras-Augé (1981) and Marshall (1996). Recently, Durante (2007) considered this family as a general family of symmetric bivariate copulas indexed by a generator function and studied some of its dependence properties. In this article, we obtain and describe further aspects of dependence for this family. For example, we have proved that the family has positive likelihood ratio dependence structure if and only if the family reduces to some well-known copulas. We also derive several proper forms for the generator function of this family. Considering a multivariate extension of the bivariate family of copulas provided by Durante et al. (2007), some dependence properties are studied. Finally, some positive dependence stochastic orderings for two random vectors having a copula from the proposed families, are discussed.

Symmetry and dependence properties within a semiparametric family of bivariate copulas

Journal of Nonparametric Statistics, 2002

In this paper, we study a semiparametric family of bivariate copulas. The family is generated by an univariate function, determining the symmetry (radial symmetry, joint symmetry) and dependence property (quadrant dependence, total positivity, ...) of the copulas. We provide bounds on different measures of association (such as Kendall's Tau, Spearman's Rho) for this family and several choices of generating functions allowing to reach these bounds.

The Meta-elliptical Distributions with Given Marginals

Journal of Multivariate Analysis, 2002

Based on an analysis of copulas of elliptically contoured distributions, joint densities of continuous variables with given strictly increasing marginal distributions are constructed. A method utilized for this procedure is to embed the spherical distribution quantile transformation of each variable into an elliptically contoured distribution. The new class of distributions is then called meta-elliptical distributions. The corresponding analytic forms of the density, conditional distribution functions, and dependence properties are derived. This new class of distributions has the same Kendall's rank correlation coefficient as meta-Gaussian distributions. As an extension of elliptically contoured distributions, some new classes of distributions are also obtained.

Modelling the Dependence of Parametric Bivariate Extreme Value Copulas

Asian Journal of Mathematics & Statistics, 2009

In this study, we consider the situation where contraints are made on the domains of two random variables whose joint copula is an extreme value model. We introduce a new measure which characterize these conditional dependence. We proved that every bivariate extreme value copulas is totally characterized by a conditional dependence function. Every twodimensional distribution is also shown to be max-infinite divisible under a restriction on the new measure. The average and median values of the measure have been computed for the main bivariate families of parametric extreme value copulas.

On the relationships between copulas of order statistics and marginal distributions

Statistics Probability Letters, 2010

In this paper we study the relationships between copulas of order statistics from heterogeneous samples and the marginal distributions of the parent random variables. Specifically, we study the copula of the order statistics obtained from a general random vector X = (X 1 , X 2 , . . . , X n ). We show that the copula of the order statistics from X only depends on the copula of X and on the marginal distributions of X 1 , X 2 , . . . , X n through an exchangeable copula and the average of the marginal distribution functions. We study in detail some relevant cases.

On tests for symmetry and radial symmetry of bivariate copulas towards testing for ellipticity

Computational Statistics

Very simple non-parametric tests are proposed to detect symmetry and radial symmetry in the dependence structure of bivariate copula data. The performance of the proposed tests is illustrated in an intensive simulation study and compared to the one of similar more advanced tests, which do not require known margins. Further, a powerful non-parametric testing procedure to decide whether the dependence structure of the underlying bivariate copula data may be captured by an elliptical copula is provided. The testing procedure makes use of intrinsic properties of bivariate elliptical copulas such as symmetry, radial symmetry, and equality of Kendall's tau and Blomqvist's beta. The proposed tests as well as the testing procedure are very simple to use in applications. For an illustration of the testing procedure for ellipticity, financial and insurance data is analyzed.

Copulas: A Review and Recent Developments

Stochastic Models, 2006

In this review paper we outline some recent contributions to copula theory. Several new author's investigations are presented brie°y, namely: order statistics copula, copulas with given multivariate marginals, copula representation via a local dependence measure and applications of extreme value copulas.