Product Copula Research Articles

Stochastic dominance and statistical preference for random variables coupled by arbitrary copulas

Recently, results have been published showing that first order stochastic dominance implies statistical preference and diff-stochastic dominance, when the copula relating the compared variables is either Archimedean, the product copula, or one of the Fréchet-Hoeffding bounds.In the present paper, we rely on known results on multivariate stochastic orders to extend these results and simplify the proofs. The results are expanded in two directions: First, we show that it suffices for the copula to be symmetric. Second, we reveal that first stochastic dominance entails a wider range of stochastic preferences beyond statistical preference and diff-stochastic dominance.We further analyze whether first stochastic dominance implies statistical preference for the case of asymmetric copulas. We observe that, when at least one of the marginal cumulative distribution functions has no discontinuity jumps, the family of asymmetric copulas for which the implication holds is at least as large as the one for which it does not.

Characterization of pre-idempotent Copulas

Abstract Copulas C C for which ( C t C ) 2 = C t C {({C}^{t}C)}^{2}={C}^{t}C are called pre-idempotent copulas, of which well-studied examples are idempotent copulas and complete dependence copulas. As such, we shall work mainly with the topology induced by the modified Sobolev norm, with respect to which the class ℛ {\mathcal{ {\mathcal R} }} of pre-idempotent copulas is closed and the class of factorizable copulas is a dense subset of ℛ {\mathcal{ {\mathcal R} }} . Identifying copulas with Markov operators on L 2 {L}^{2} , the one-to-one correspondence between pre-idempotent copulas and partial isometries is one of our main tools. In the same spirit as Darsow and Olsen’s work on idempotent copulas, we obtain an explicit characterization of pre-idempotent copulas, which is split into cases according to the atomicity of its associated σ \sigma -algebras, where the nonatomic case gives all factorizable copulas and the totally atomic case yields conjugates of ordinal sums of copies of the product copula.

A Collection of Two-Dimensional Copulas Based on an Original Parametric Ratio Scheme

The creation of two-dimensional copulas is crucial for the proposal of novel families of two-dimensional distributions and the analysis of original dependence structures between two quantitative variables. Such copulas can be developed in a variety of ways. In this article, we provide theoretical contributions to this subject; we emphasize a new parametric ratio scheme to create copulas of the following form: C(x,y)=(b+1)xy/[b+ϕ(x,y)], where b is a constant and ϕ(x,y) is a two-dimensional function. As a notable feature, this form can operate an original trade-off between the product copula and more versatile copulas (not symmetric, with tail dependence, etc.). Instead of a global study, we examine seven concrete examples of such copulas, which have never been considered before. Most of them are extended versions of existing non-ratio copulas, such as the Celebioglu–Cuadras, Ali-Mikhail-Haq, and Gumbel–Barnett copulas. We discuss their attractive properties, including their symmetry, dominance, dependence, and correlation features. Some graphics and tables are given as complementary works. Our findings expand the horizons of new two-dimensional distributional or dependence modeling.

Modelling multivariate data using product copulas and minimum distance estimators: an exemplary application to ecological traits

Modelling and applying multivariate distributions is an important topic in ecology. In particular in plant ecology, the multidimensional nature of plant traits comes with challenges such as wide ranges in observations as well as correlations between several characteristics. In other disciplines (e.g., finances and economics), copulas have been proven as a valuable tool for modelling multivariate distributions. However, applications in ecology are still rarely used. Here, we present a copula-based methodology of fitting multivariate distributions to ecological data. We used product copula models to fit multidimensional plant traits, on example of observations from the global trait database TRY. The fitting procedure is split into two parts: fitting the marginal distributions and fitting the copula. We found that product copulas are well suited to model ecological data as they have the advantage of being asymmetric (similar to the observed data). Challenges in the fitting were mainly addressed to limited amount of data. In view of growing global databases, we conclude that copula modelling provides a great potential for ecological modelling.

Copula-Based Analysis of Power Allocation for Two-Sensor Composite Fading MAC With Dependent Channel Coefficients

In this article, the problem of power allocation in a wireless multiple access channel with two sensors is addressed, where squared channel coefficients, and, hence, signal-to-noise ratios are assumed to have Gaussian mixture distributions and the dependence between channel coefficients is described by using copula theory. For this model, outage probability is analyzed as the objective function for formulating the power allocation, and for the product copula function or in the independent channels case, a closed form of outage probability is obtained. Considering that the corresponding optimizations do not lead to convex optimization problems, MATLAB simulation is employed to solve the power allocation problem. Various Gaussian mixture distribution parameters are exploited for the links between the sensors and the fusion center with different dependencies and the results are compared.

Markov product invariance in classes of bivariate copulas characterized by univariate functions

We extend and sharpen some results in the literature concerning the notion of Markov product idempotence in some well-known classes of copulas. Focusing on families of copulas which are characterized by univariate functions we show that in the class of extreme-value copulas, in the class of diagonal copulas and in some special class of copulas represented by measure-preserving transformations only the usual suspects (if contained in the class) are idempotent, namely the product copula Π and minimum copula M. Additionally, we prove a conjecture going back to Albanese and Sempi in 2016 saying that the only idempotent Archimedean copula is the product copula Π.

Some properties of double shuffles of bivariate copulas and (extreme) copulas invariant with respect to Lüroth double shuffles

Considering the well-known shuffling operation in x- and in y-direction yields so-called double shuffles of bivariate copulas. We study continuity properties of the double shuffle operator ST induced by pairs T=(T1×T2) of measure preserving transformations on ([0,1],B([0,1]),λ) on the family C of all bivariate copulas, analyze its interrelation with the star/Markov product, and show that for each left- and for each right-invertible copula A the set of all possible double shuffles of A is dense in C with respect to the uniform metric d∞. After deriving some general properties of the set ΩT of all ST-invariant copulas we focus on the situation where T1,T2 are strongly mixing and show that in this case the product copula Π is an extreme point of ΩT. Moreover, motivated by a recent paper by Horanská and Sarkoci (Fuzzy Sets and Systems 378, 2018) we then study double shuffles induced by pairs of so-called Lüroth maps and derive various additional properties of ΩT, including the surprising fact that ΩT contains uncountably many extreme points which (interpreted as doubly stochastic measures) are pairwise mutually singular with respect to each other and which allow for an explicit construction.

New results on perturbation-based copulas

AbstractA prominent example of a perturbation of the bivariate product copula (which characterizes stochastic independence) is the parametric family of Eyraud-Farlie-Gumbel-Morgenstern copulas which allows small dependencies to be modeled. We introduce and discuss several perturbations, some of them perturbing the product copula, while others perturb general copulas. A particularly interesting case is the perturbation of the product based on two functions in one variable where we highlight several special phenomena, e.g., extremal perturbed copulas. The constructions of the perturbations in this paper include three different types of ordinal sums as well as flippings and the survival copula. Some particular relationships to the Markov product and several dependence parameters for the perturbed copulas considered here are also given.

Copula theory and probabilistic sensitivity analysis: Is there a connection?

Copula theory is concerned with defining dependence structures given appropriate marginal distributions. Probabilistic sensitivity analysis is concerned with quantifying the strength of the dependence among the output of a simulator and the uncertain simulator inputs. In this work, we investigate the connection between these two families of methods. We define four classes of sensitivity measures based on the distance between the empirical copula and the product copula. We discuss the new classes in the light of transformation invariance and Rényi’s postulate D of dependence measures. The connection is constructive: the new classes extend the current definition of sensitivity measures and one gains an of understanding which sensitivity measures in use are, in fact, copula-based. Also a set of new visualization tools can be obtained. These tools ease the communication of results to the modeler and provide insights not only on statistical dependence but also on the partial behavior of the output as a function of the inputs. Application to the benchmark simulator for sensitivity analysis concludes the work.

A sharp inequality for Kendall’s τ and Spearman’s ρ of Extreme-Value Copulas

AbstractWe derive a new (lower) inequality between Kendall’s τ and Spearman’s ρ for two-dimensional Extreme-Value Copulas, show that this inequality is sharp in each point and conclude that the comonotonic and the product copula are the only Extreme-Value Copulas for which the well-known lower Hutchinson-Lai inequality is sharp.

Solution to two open problems on perturbations of the product copula

In this note we provide an answer to some open problems concerning perturbations of the product copula in terms of convex strong negations. We also approach the case in higher dimensions.

Extraction dependence structure of distorted copulas via a measure of dependence

Copulas are one of the most powerful tools in modeling dependence structure of multivariate variables. In Tran et al. (Integrated uncertainty in knowledge modelling and decision making. Springer, Berlin, pp 126–137, 2015), we have constructed a new measure of dependence, $$\lambda (C),$$ based on Sobolev norm for copula C which can be used to characterize comonotonicity, countermonotonicity and independence of random vectors. This paper aims to use the measure $$ \lambda (C) $$ to study how dependence structure of a distorted copula after being transformed by a distortion function is changed. Firstly, we propose two methods to estimate the measure $$\lambda (C)$$ , one for known copula C using conditional copula-based Monte Carlo simulation and the latter for unknown copula dealing with empirical data. Thereafter, PH-transform $$g_{ PH }$$ of extreme value copulas and Wang’s transform $$ g_\gamma $$ of normal and product copula are studied, and we observe their dependence behaviors changing through variability of the measure $$ \lambda (C) $$ . Our results show that dependence structure of distorted copulas is subject to comonotonicity as increasing the parametric $$ \gamma $$ .

On (a,b)-transformations of conjunctive functions

We introduce and study transformations that assign to each conjunctive real-valued function F:[0,1]2→R and any pair of parameters (a,b)∈[0,1]2 a function Fa,b:[0,1]2→R that is shown to be conjunctive, too. In more details, we study the proposed transformations in the classes of all binary copulas, quasi-copulas, conjunctive k-Lipschitz functions and the class of all conjunctive Lipschitz functions. We discuss the invariance of these classes with respect to all possible transformations. An interesting result is, that although the class of all copulas is invariant with respect to all considered transformations, the only copula that is not changed by any of them is the product copula. We also show that (a,b)-transformations of copulas yield the same results as constructions of copulas based on special measure-preserving transformations; and using measure-preserving transformations we extend the studied binary constructions to n-dimensional copulas. Finally, we focus our attention on consecutive realization of the proposed transformations in the class of binary copulas and we also show several properties of (a,b)-transforms of the basic copulas, including some dependence parameters.

Copula–Induced Measures of Concordance

Abstract We study measures of concordance for multivariate copulas and copulas that induce measures of concordance. To this end, for a copula A, we consider the maps C → R given by where C denotes the collection of all d–dimensional copulas, M is the Fréchet–Hoeffding upper bound, Π is the product copula, [. , .] : C × C → R is the biconvex form given by [C, D] := ∫ [0,1]d C(u) dQD(u) with the probability measure QD associated with the copula D, and ψΛ C → C is a transformation of copulas. We present conditions on ψΛ and on A under which these maps are measures of concordance. The resulting class of measures of concordance is rich and includes the well–known examples Spearman’s rho and Gini’s gamma.

Semiquadratic copulas based on horizontal and vertical interpolation

We introduce several families of semiquadratic copulas (i.e. copulas that are quadratic in any point of the unit square in at least one coordinate) of which the diagonal and/or opposite diagonal sections are given functions. These copulas are constructed by quadratic interpolation on segments connecting the diagonal, opposite diagonal and sides of the unit square; all interpolations are therefore performed horizontally or vertically. For each family we provide the necessary and sufficient conditions on the given diagonal and/or opposite diagonal functions and two auxiliary real functions to obtain a copula that has these diagonal and/or opposite diagonal functions as diagonal and/or opposite diagonal sections. Just as the product copula is a central member of all families of semilinear copulas based on horizontal and vertical interpolation, it turns out that the Farlie–Gumbel–Morgenstern family of copulas is included in all families of semiquadratic copulas introduced and characterized here.

Baker- Lin-Huang Type Bivariate Distributions Based on Order Statistics

Baker (2008) introduced a new class of bivariate distributions based on distributions of order statistics from two independent samples of size n. Lin and Huang (2010) discovered an important property of Baker’s distribution and showed that the Pearson’s correlation coefficient for this distribution converges to maximum attainable value, i.e., the correlation coefficient of the Fréchet upper bound, as n increases to infinity. Bairamov and Bayramoglu (2013) investigated a new class of bivariate distributions constructed by using Baker’s model and distributions of order statistics from dependent random variables, allowing higher correlation than that of Baker’s distribution. In this article, a new class of Baker’s type bivariate distributions with high correlation are constructed based on distributions of order statistics by using an arbitrary continuous copula instead of the product copula.

Universal integrals based on copulas

A hierarchical family of integrals based on a fixed copula is introduced and discussed. The extremal members of this family correspond to the inner and outer extension of integrals of basic functions, the copula under consideration being the corresponding multiplication. The limits of the members of the family are just copula-based universal integrals as recently introduced in Klement et al. (IEEE Trans Fuzzy Syst 18:178---187, 2010). For the product copula, the family of integrals considered here contains the Choquet and the Shilkret integral, and it belongs to the class of decomposition integrals proposed in Even and Lehrer (Econ Theory, 2013) as well as to the class of superdecomposition integrals introduced in Mesiar et al. (Superdecomposition integral, 2013). For the upper Frechet-Hoeffding bound, the corresponding hierarchical family contains only two elements: all but the greatest element coincide with the Sugeno integral.

A measure of multivariate mutual complete dependence

The authors propose a multivariate version of Siburg and Stoimenov’s measure of mutual complete dependence. This multivariate version is, however, not the distance between a copula and the product copula CI under the modified Sobolev norm since the set of mutual complete dependence copulas does not lie on the sphere centered at CI. To overcome this difficulty, the authors choose another center and define measures of complete dependence based on the modified Sobolev norm and this center. The measure of multivariate mutual complete dependence is then defined as the summation of the (normalized) measures of complete dependence.

Some results on shuffles of two-dimensional copulas

Using the one-to-one correspondence between two-dimensional copulas and special Markov kernels allows to study properties of T-shuffles of copulas, T being a general Lebesgue-measure-preserving transformation on [0,1], in terms of the corresponding operation on Markov kernels. As one direct consequence of this fact the asymptotic behaviour of iterated T-shuffles STn(A) of a copula A∈C can be characterized through mixing properties of T. In particular it is shown that STn(A) ((1/n)∑i=1nSTi(A)) converges uniformly to the product copula Π for every copula A if and only if T is strongly mixing (ergodic). Moreover working with Markov kernels also allows, firstly, to give a short proof of the fact that the mass of the singular component of ST(A) cannot be bigger than the mass of the singular component of A, secondly, to introduce and study another operator UT:C→C fulfilling ST○UT(A)=A for all A∈C, and thirdly to express ST(A) and UT(A) as ⁎-product of A with the completely dependent copula CT induced by T.

On the axiomatization of some classes of discrete universal integrals

Following the ideas of the axiomatic characterization of the Choquet integral due to [D. Schmeidler, Integral representation without additivity, Proc. Amer. Math. Soc. 97 (1986) 255–261] and of the Sugeno integral given in [J.-L. Marichal, An axiomatic approach of the discrete Sugeno integral as a tool to aggregate interacting criteria in a qualitative framework, IEEE Trans. Fuzzy Syst. 9 (2001) 164–172], we provide a general axiomatization of some classes of discrete universal integrals, including the case of discrete copula-based universal integrals (as usual, the product copula corresponds just to the Choquet integral, and the minimum to the Sugeno integral).