Rectangular Patchwork for Bivariate Copulas and Tail Dependence

We present a method for constructing bivariate copulas by changing the values that a given copula assumes on some subrectangles of the unit square. Some applications of this method are discussed, especially in relation to the construction of copulas with different tail dependencies.     

Further Results for a General Family of Bivariate Copulas

A bivariate family of copulas has been initiated by Cuadras-Augé (1981 Cuadras, C.M., Augé, J. (1981). A continuous general multivariate distribution and its properties. Commun. Statist. (A) Theor. Meth. 10:339–353.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) and Marshall (1996 Marshall, A.W. (1996). Copulas, marginals, and joint distributions. In: Distributions with fixed marginals and related topics. IMS Lecture Notes Monogr. Ser. 28:213–222.[Crossref] , [Google Scholar]). Recently, Durante (2007 Durante, F. (2007). A new family of symmetric bivariate copulas. C. R. Math. Acad. Sci. Paris 344:195–198.[Crossref], [Web of Science ®] , [Google Scholar]) 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 Durante, F., Quesada-Molina, J.J., Flores, M. (2007). On a family of multivariate copulas for aggregation processes. Inform. Sci. 177(24):5715–5724.[Crossref], [Web of Science ®] , [Google Scholar]), some dependence properties are studied. Finally, some positive dependence stochastic orderings for two random vectors having a copula from the proposed families, are discussed.     

Developing a Wide Easy-to-Generate Class of Bivariate Copulas

As of late, copulas have drawn great attention in stochastic simulation, financial engineering, and risk management. Their power lies under their ability of modeling dependent random variables. Using a known theorem in probability which proves that the fractional part of the sum of a uniform and an arbitrary independent continuous random variable follows a uniform distribution, we construct a wide class of bivariate copulas in which bivariate random vector generation can be performed easily. Some important members of this new class and their properties together with two invariant correlation measures and some insights in their application are presented.     

Absolutely Continuous Copulas with Given Diagonal Sections

Necessary and sufficient conditions are given in order to ensure that a function δ:[0, 1] → [0, 1] is the diagonal section of an absolutely continuous copula. Explicit constructions are provided.     

A Goodness-of-fit Test for Copulas

We propose a new rank-based goodness-of-fit test for copulas. It uses the information matrix equality and so relates to the White (1982 White , H. ( 1982 ). Maximum likelihood estimation of misspecified models . Econometrica 50 : 1 – 26 .[Crossref], [Web of Science ®] , [Google Scholar]) specification test. The test avoids parametric specification of marginal distributions, it does not involve kernel weighting, bandwidth selection, or any other strategic choices, it is asymptotically pivotal with a standard distribution, and it is simple to compute compared to available alternatives. The finite-sample size of this type of tests is known to deviate from their nominal size based on asymptotic critical values, and bootstrapping critical values could be a preferred alternative. A power study shows that, in a bivariate setting, the test has reasonable properties compared to its competitors. We conclude with an application in which we apply the test to two stock indices.     

Constructing Copulas with Given Diagonal and Opposite Diagonal Sections

A method for constructing copulas with given diagonal and opposite diagonal sections is presented. It makes use of a recently developed method for constructing cross-copulas with given horizontal and vertical sections. Conditions guaranteeing the existence of a cross-copula with the given diagonal and opposite diagonal sections are derived. It is shown how the new method facilitates the construction of families of copulas that simultaneously model tail dependences of upper-upper, upper-lower, lower-lower and lower-upper type.     

Distorted Copulas: Constructions and Tail Dependence

Given a copula C, we examine under which conditions on an order isomorphism ψ of [0, 1] the distortion C _ψ: [0, 1]² → [0, 1], C _ψ(x, y) = ψ{C[ψ^?1(x), ψ^?1(y)]} is again a copula. In particular, when the copula C is totally positive of order 2, we give a sufficient condition on ψ that ensures that any distortion of C by means of ψ is again a copula. The presented results allow us to introduce in a more flexible way families of copulas exhibiting different behavior in the tails.     

A Non‐parametric Test of Exchangeability for Extreme‐Value and Left‐Tail Decreasing Bivariate Copulas

Abstract. A non‐parametric rank‐based test of exchangeability for bivariate extreme‐value copulas is first proposed. The two key ingredients of the suggested approach are the non‐parametric rank‐based estimators of the Pickands dependence function recently studied by Genest and Segers, and a multiplier technique for obtaining approximate p‐values for the derived statistics. The proposed approach is then extended to left‐tail decreasing dependence structures that are not necessarily extreme‐value copulas. Large‐scale Monte Carlo experiments are used to investigate the level and power of the various versions of the test and show that the proposed procedure can be substantially more powerful than tests of exchangeability derived directly from the empirical copula. The approach is illustrated on well‐known financial data.     

Bivariate Contours of Copula

Quantile functions associated with bivariate copulas are considered. Some of their structural properties are studied. Quantile functions allow one to express the cdf of the random variable C(X, Y), where (X, Y) is distributed as C(x, y) and where C is a copula. Quantile functions provide also a simple algorithm for simulating random observations.     

基于Copula方法的国债市场相依风险度量

本文讨论了如何利用Copula连接函数对多元金融数据的相依结构进行统计建模,首先对几种常用的Copula连接函数进行了介绍,分析了不同边际分布和不同Copula函数的选取对联合分布产生的影响,然后讨论了Copula函数的选取和其参数的估计问题,最后利用我国国债数据进行实证分析,得到了不同组合的风险值。     

The Joy of Copulas: Bivariate Distributions with Uniform Marginals

We describe a class of bivariate distributions whose marginals are uniform on the unit interval. Such distributions are often called “copulas.” The particular copulas we present are especially well suited for use in undergraduate mathematical statistics courses, as many of their basic properties can be derived using elementary calculus. In particular, we show how these copulas can be used to illustrate the existence of distributions with singular components and to give a geometric interpretation to Kendall's tau.     

New Families of Copulas Based on Periodic Functions

Abstract

Although there exists a large variety of copula functions, only a few are practically manageable, and often the choice in dependence modeling falls on the Gaussian copula. Furthermore most copulas are exchangeable, thus implying symmetric dependence. We introduce a way to construct copulas based on periodic functions. We study the two-dimensional case based on one dependence parameter and then provide a way to extend the construction to the n-dimensional framework. We can thus construct families of copulas in dimension n and parameterized by n ? 1 parameters, implying possibly asymmetric relations. Such “periodic” copulas can be simulated easily.     

A Note on the Relations Between Some Concepts of Positive Dependence

In this article, we provide some explicit examples showing that weak association in sequence is strictly weaker than weak association and strictly stronger than positive supermodular dependence. Furthermore, we show that strongly positive orthant dependence is strictly weaker than weakly association and strictly stronger than positive orthant dependence. Finally, we also show that positive supermodular dependence is not stronger than strongly positive orthant dependence.     

基于Copula函数的金融时间序列模型述评

结合当前Copula函数及其应用的热点问题,着重评述了基于Copula函数的金融时间序列模型的应用。鉴于利用Copula可以将边际分布和变量间的相依结构分开来研究这一优良性质,在设定和估计模型时便显得极为方便和灵活。从模型的构造、Copula函数的选择、模型的估计以及拟合优度检验等几方面展开阐述和评价,介绍了Copula模型在金融领域中的几类应用,并对Copula理论和应用的新视角进行了展望。     

Semiparametric Estimation in Copulas with the Same Marginals

We consider semiparametric multivariate data models based on copula representation of the common distribution function. A copula is characterized by a parameter of association and marginal distribution functions. This parameter and the marginal distributions are unknown. In this article, we study the estimator of the parameter of association in copulas with the marginal distribution functions assumed as nuisance parameters restricted by the assumption that the components are identically distributed. Results of this work could be used to construct special kinds of tests of homogeneity for random vectors having dependent components.     

Mixtures of Farlie-Gumbel-Morgenstern Copulas

Some modifications of bivariate Farlie-Gumbel-Morgenstern (FGM) copulas are going to be explained in this article. These modifications are generated by using mixtures of bivariate FGM copula functions. The main goal of this study is to determine both the ranges of association parameter and the rate of correlation, and also observe the changes in local dependence function. An application, which is related with simulated data, is conducted and results are illustrated.     

On Censored Bivariate Random Variables: Copula,Characterization, and Estimation

Let (X, Y) be a bivariate random vector with joint distribution function F_{X, Y}(x, y) = C(F(x), G(y)), where C is a copula and F and G are marginal distributions of X and Y, respectively. Suppose that (X_i, Y_i), i = 1, 2, …, n is a random sample from (X, Y) but we are able to observe only the data consisting of those pairs (X_i, Y_i) for which X_i ? Y_i. We denote such pairs as (X*_i, Y_i*), i = 1, 2, …, ν, where ν is a random variable. The main problem of interest is to express the distribution function F_{X, Y}(x, y) and marginal distributions F and G with the distribution function of observed random variables X* and Y*. It is shown that if X and Y are exchangeable with marginal distribution function F, then F can be uniquely determined by the distributions of X* and Y*. It is also shown that if X and Y are independent and absolutely continuous, then F and G can be expressed through the distribution functions of X* and Y* and the stress–strength reliability P{X ? Y}. This allows also to estimate P{X ? Y} with the truncated observations (X*_i, Y_i*). The copula of bivariate random vector (X*, Y*) is also derived.     

Bivariate censored regression relying on a new estimator of the joint distribution function

In this paper we study a class of M  -estimators in a regression model under bivariate random censoring and provide a set of sufficient conditions that ensure asymptotic n^1/2-convergence$n^{1/2}\mathrm{-}\mathrm{convergence}$. The cornerstone of our approach is a new estimator of the joint distribution function of the censored lifetimes. A copula approach is used to modelize the dependence structure between the bivariate censoring times. The resulting estimators present the advantage of being easily computable. A simulation study enlighten the finite sample behavior of this technique.     

Using Convex Combination of Copulas and EM Algorithm for Credibility

In this article, using longitudinal data, we develop the theory of credibility by copula model. The convex combination of copulas is used to describe the dependencies among claims. Finally, for comparing with the results of a single copula, using EM algorithm, some simulations of Massachusetts automobile claims are presented.