Copulas Constructed from Horizontal Sections

In analogy with the study of copulas whose diagonal sections have been fixed, we study the set _h of copulas for which a horizontal section h has been given. We first show that this set is not empty, by explicitly writing one such copula, which we call horizontal copula. Then we find the copulas that bound both below and above the set _h. Finally, we determine the expressions for Kendall's tau and Spearman's rho for the horizontal and the bounding copulas.     

Semi-polynomial copulas

In this paper, we characterise a family of bivariate copulas whose sections between the main diagonal and the border of the unit square are polynomial, generalising several families of copulas, including those with quadratic and cubic sections. We also study a measure of association and the tail dependence for this class, illustrating our results with several examples.     

Geometry of a corelation coefficient under a copula

Spearman's rank correlation coefficient is shown to be a measure of distance on the unit square, which characterizes the concentration of the probability density under a copula. A distance function offers insight into structuring copulas with a desired degree of association.     

Copules archimédiennes et families de lois bidimensionnelles dont les marges sont données

Every bivariate distribution function with continuous marginals can be represented in terms of a unique copula, that is, in terms of a distribution function on the unit square with uniform marginals. This paper is concerned with a special class of copulas called Archimedean, which includes the uniform representation of many standard bivariate distributions. Conditions are given under which these copulas are stochastically ordered and pointwise limits of sequences of Archimedean copulas are examined. We also provide two new one-parameter families of bivariate distributions which include as limiting cases the Frechet bounds and the independence distribution.     

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.     

Sharp Bounds on a Class of Copulas with known Values at Several Points

We derive best-possible bounds on the class of copulas with known values at several points, under the assumption that the points are either in “increasing order” or in “decreasing order”. These bounds may be used to establish best-possible bounds on Kendall's τ and Spearman's ρ, for such copulas. An important special case is when the values of a copula are known at several diagonal points. We also use our results to establish best-possible bounds on the distribution function of the sum of two random variables with known marginal distributions when the values of the joint distribution function are known at several points.     

Various measures of dependence of a new asymmetric generalized Farlie–Gumbel–Morgenstern copulas

ABSTRACT

In this paper, we discuss an asymmetric extension of Farlie–Gumbel–Morgenstern copulas studied by several authors and obtain the range of the parameter. We derive an expression for regression function and the properties of these copulas are studied in detail. Also, explicit expressions for various measures of association are obtained. These measures are numerically compared for some particular parametric values of the copulas.     

Idempotent and multivariate copulas with fractal support

Using special iterated function systems (IFS) Fredricks et al. (2005) constructed two-dimensional copulas with fractal supports and showed that for every s∈(1,2)$s\in (\mathrm{1,2})$ there exists a copula A whose support has Hausdorff dimension s. In the current paper we present a stronger version and prove that the same result holds for the subclass of idempotent copulas. Additionally we show that every doubly stochastic idempotent matrix N (having neither minimum nor maximum rank) induces a family of idempotent copulas such that, firstly, the corresponding Markov kernels transform according to N   and, secondly, the set of Hausdorff dimensions of the supports of elements of the family covers (1,2). Furthermore we generalize the IFS approach to arbitrary dimensions d≥2$d\ge 2$ and show that for every s∈(1,d)$s\in (1,d)$ we can find a d-dimensional copula whose support has Hausdorff dimension s.     

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.     

Bivariate copulas on the Hotelling's T2 control chart

In this paper, we propose five types of copulas on the Hotelling's T² control chart when observations are from exponential distribution and use the Monte Carlo simulation to compare the performance of the control chart, which is based on the Average Run Length (ARL) for each copula. Five types of copulas function for specifying dependence between random variables are used and measured by Kendall's tau. The results show that the copula approach can be fitted the observation and we can use copula as an option for application on Hotelling's T² control chart.     

Extremes of nonexchangeability

Summary For identically distributed random variables X and Y with joint distribution function H, we show that the supremum of |H(x,y)-H(y,x)| is 1/3. Using copulas, we define a measure of nonexchangeability, and study maximally nonexchangeable random variables and copulas. In particular, we show that maximally nonexchangeable random variables are negatively correlated in the sense of Spearman's rho. An erratum to this article is available at .     

Some Observations on Copula Regression Functions

ABSTRACT

The main objective of this article is to introduce an alternative way of looking at regression analysis by using copulas. To achieve our objective we work on copula regression function, study its properties, and discuss advantages that will come out from our approach.     

Robust estimation of the correlation matrix of longitudinal data

We propose a double-robust procedure for modeling the correlation matrix of a longitudinal dataset. It is based on an alternative Cholesky decomposition of the form Σ=DLL ^? D where D is a diagonal matrix proportional to the square roots of the diagonal entries of Σ and L is a unit lower-triangular matrix determining solely the correlation matrix. The first robustness is with respect to model misspecification for the innovation variances in D, and the second is robustness to outliers in the data. The latter is handled using heavy-tailed multivariate t-distributions with unknown degrees of freedom. We develop a Fisher scoring algorithm for computing the maximum likelihood estimator of the parameters when the nonredundant and unconstrained entries of (L,D) are modeled parsimoniously using covariates. We compare our results with those based on the modified Cholesky decomposition of the form LD ² L ^? using simulations and a real dataset.     

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.     

Gluing Copulas

We present a new way of constructing n-copulas, by scaling and gluing finitely many n-copulas. Gluing for bivariate copulas produces a copula that coincides with the independence copula on some grid of horizontal and vertical sections. Examples illustrate how gluing can be applied to build complicated copulas from simple ones. Finally, we investigate the analytical as well as statistical properties of the copulas obtained by gluing, in particular, the behavior of Spearman's ρ and Kendall's τ.     

Copula-Based Models for the Power of Independence Tests

ABSTRACT

We consider independence tests and the methods to evaluate their efficiency. First, we observe that many of the most used independence tests are functions of the empirical copula, which is a sufficient statistic. Hence, the power of these tests, such as the tests based on Spearman's ρ, on Kendall's τ, and on Gini's γ, depend solely on the theoretical copula, and not on the marginal distributions. Then, we consider monotone dependence tests and we propose a parametric model to define the power function. Such a model is based on a path of copulas, from the copula of discordance to the copula of concordance, and can be characterized by the copula of the underlying joint distribution. Moreover, we introduce a consistent estimator of the path of copulas. Finally, we provide some examples of applications, and in particular, a bootstrap-plug-in estimator of the power curve, all useful for power comparison.     

Vine copula constructions of higher-dimensional dependent reliability systems

Copulas have proved to be very successful tools for the flexible modeling of dependence. Bivariate copulas have been deeply researched in recent years, while building higher-dimensional copulas is still recognized to be a difficult task. In this paper, we study the higher-dimensional dependent reliability systems using a type of decomposition called “vine,” by which a multivariate distribution can be decomposed into a cascade of bivariate copulas. Some equations of system reliability for parallel, series, and k-out-of-n systems are obtained and then decomposed based on C-vine and D-vine copulas. Finally, a shutdown system is considered to illustrate the results obtained in the paper.     

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.     

Copulas checker-type approximations: Application to quantiles estimation of sums of dependent random variables

Abstract

Several approximations of copulas have been proposed in the literature. By using empirical versions of checker-type copulas approximations, we propose non parametric estimators of the copula. Under some conditions, the proposed estimators are copulas and their main advantage is that they can be sampled from easily. One possible application is the estimation of quantiles of sums of dependent random variables from a small sample of the multivariate law and a full knowledge of the marginal laws. We show that estimations may be improved by including in an easy way in the approximated copula some additional information on the law of a sub-vector for example. Our approach is illustrated by numerical examples.