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Two-dimensional distributions whose marginal distributions are uniform are called bivariate copulas. Among them, the one that satisfies given constraints on expectation and is closest to being an independent distribution in the sense of Kullback–Leibler divergence is called the minimum information bivariate copula. The density function of the minimum information copula contains a set of functions called the normalizing functions, which are often difficult to compute. Although a number of proper scoring rules for probability distributions having normalizing constants such as exponential families have been proposed, these scores are not applicable to the minimum information copulas due to the normalizing functions. In this paper, we propose the conditional Kullback–Leibler score, which avoids computation of the normalizing functions. The main idea of its construction is to use pairs of observations. We show that the proposed score is strictly proper in the space of copula density functions and therefore the estimator derived from it has asymptotic consistency. Furthermore, the score is convex with respect to the parameters and can be easily optimized by the gradient methods. [ABSTRACT FROM AUTHOR]

A framework for quantifying dependence between random vectors is introduced. Using the notion of a collapsing function, random vectors are summarized by single random variables, referred to as collapsed random variables. Measures of association computed from the collapsed random variables are then used to measure the dependence between random vectors. To this end, suitable collapsing functions are presented. Furthermore, the notion of a collapsed distribution function and collapsed copula are introduced and investigated for certain collapsing functions. This investigation yields a multivariate extension of the Kendall distribution and its corresponding Kendall copula for which some properties and examples are provided. In addition, non-parametric estimators for the collapsed measures of association are provided along with their corresponding asymptotic properties. Finally, data applications to bioinformatics and finance are presented along with a general graphical assessment of independence between groups of random variables. [ABSTRACT FROM AUTHOR]

We present a mixed-type multivariate distributional model generalizing the model of k -independent identically distributed random variables, and give necessary and sufficient conditions for a distribution of a random vector to be the joint distribution of the numbers of successes of q categories in n such dependent trials. The model includes, as particular cases, mixtures of multivariate extremal distributions and models corresponding to ordinal sums of independence and comonotonicity copulas. As an application, we study the extremal properties of order statistics based on random vectors with known multivariate marginals. [ABSTRACT FROM AUTHOR]

A common assumption in pair-copula constructions is that the copula of the conditional distribution of two random variables given a covariate does not depend on the value of that covariate. Two conflicting intuitions arise about the best possible rate of convergence attainable by nonparametric estimators of that copula. On the one hand, the best possible rates for estimating the marginal conditional distribution functions are slower than the parametric one. On the other hand, the invariance of the conditional copula given the value of the covariate suggests the possibility of parametric convergence rates. The more optimistic intuition is shown to be correct, confirming a conjecture supported by extensive Monte Carlo simulations by Hobæk Haff and Segers (2015) and improving upon the nonparametric rate obtained theoretically by Gijbels et al. (2015). The novelty of the proposed approach lies in a double smoothing procedure for the estimator of the marginal conditional distribution functions. The copula estimator itself is asymptotically equivalent to an oracle empirical copula, as if the marginal conditional distribution functions were known. [ABSTRACT FROM AUTHOR]

The empirical copula has proved to be useful in the construction and understanding of many statistical procedures related to dependence within random vectors. The empirical beta copula is a smoothed version of the empirical copula that enjoys better finite-sample properties. At the core lie fundamental results on the weak convergence of the empirical copula and empirical beta copula processes. Their scope of application can be increased by considering weighted versions of these processes. In this paper we show weak convergence for the weighted empirical beta copula process. The weak convergence result for the weighted empirical beta copula process is stronger than the one for the empirical copula and its use is more straightforward. The simplicity of its application is illustrated for weighted Cramér–von Mises tests for independence and for the estimation of the Pickands dependence function of an extreme-value copula. [ABSTRACT FROM AUTHOR]

It is well-known that the expected scaled maximum of non-negative random variables with unit mean defines a stable tail dependence function associated with some extreme-value copula. In the special case when these random variables are independent and identically distributed, min-stable multivariate exponential random vectors with the associated survival extreme-value copulas are shown to arise as finite-dimensional margins of an infinite exchangeable sequence in the sense of De Finetti’s Theorem. The associated latent factor is a stochastic process which is strongly infinitely divisible with respect to time, which induces a bijection from the set of distribution functions F of non-negative random variables with finite mean to the set of Lévy measures ν on ( 0 , ∞ ] . Since the Gumbel and the Galambos copula are the most popular examples of this construction, the investigation of this bijection contributes to a further understanding of their well-known analytical similarities. Furthermore, a simulation algorithm based on the latent factor representation is developed, if the support of F is bounded. Especially in large dimensions, this algorithm is efficient because it makes use of the De Finetti structure. [ABSTRACT FROM AUTHOR]

In the present paper we study the problem of how to transform a copula for an arbitrary distribution function into a copula for its conditional distribution function where conditioning is meant with respect to a tail event in which the observations lie below some threshold. To this end, we consider conditioning of copulas as a map which transforms every copula into another one. Besides the general case, which refers to conditioning in all coordinates, we also pay attention to the special case of univariate conditioning, which refers to conditioning in a single coordinate. We investigate the behaviour of conditioning under composition and with respect to certain transformations of copulas, and we show that invariance of a copula under conditioning is equivalent to invariance of a copula under univariate conditioning in each coordinate. Finally, we apply conditioning of copulas to Sklar's Theorem and to measures of concordance. [ABSTRACT FROM AUTHOR]

Multivariate generalized Pareto distributions arise as the limit distributions of exceedances over multivariate thresholds of random vectors in the domain of attraction of a max-stable distribution. These distributions can be parametrized and represented in a number of different ways. Moreover, generalized Pareto distributions enjoy a number of interesting stability properties. An overview of the main features of such distributions is given, expressed compactly in several parametrizations, giving the potential user of these distributions a convenient catalogue of ways to handle and work with generalized Pareto distributions. [ABSTRACT FROM AUTHOR]

An integrated risk is a scene in the future associated with some adverse incident caused by multiple hazards. An integrated probability risk is the expected value of disaster. Due to the difficulty of assessing an integrated probability risk with a small sample, weighting methods and copulas are employed to avoid this obstacle. To resolve the problem, in this paper, we develop the information diffusion technique to construct a joint probability distribution and a vulnerability surface. Then, an integrated risk can be directly assessed by using a small sample. A case of an integrated risk caused by flood and earthquake is given to show how the suggested technique is used to assess the integrated risk of annual property loss. [ABSTRACT FROM AUTHOR]

Liouville copulas introduced in McNeil and Nešlehová (2010) are asymmetric generalizations of the ubiquitous Archimedean copula class. They are the dependence structures of scale mixtures of Dirichlet distributions, also called Liouville distributions. In this paper, the limiting extreme-value attractors of Liouville copulas and of their survival counterparts are derived. The limiting max-stable models, termed here the scaled extremal Dirichlet, are new and encompass several existing classes of multivariate max-stable distributions, including the logistic, negative logistic and extremal Dirichlet. As shown herein, the stable tail dependence function and angular density of the scaled extremal Dirichlet model have a tractable form, which in turn leads to a simple de Haan representation. The latter is used to design efficient algorithms for unconditional simulation based on the work of Dombry et al. (2016) and to derive tractable formulas for maximum-likelihood inference. The scaled extremal Dirichlet model is illustrated on river flow data of the river Isar in southern Germany. [ABSTRACT FROM AUTHOR]

Some recent papers deal with smooth nonparametric estimators for copula functions and copula derivatives. These papers contain results on copula-based Bernstein estimators for conditional distribution functions and related functionals such as regression and quantile functions. The focus in the present paper is on new copula-based smooth Bernstein estimators for the conditional density. Our approach avoids going through separate density estimation of numerator and denominator. Our estimator is defined as a smoother of the copula-based Bernstein estimator of the conditional distribution function. We establish asymptotic properties of bias and variance and discuss the asymptotic mean squared error in terms of the smoothing parameters. We also obtain the asymptotic normality of the new estimator. In a simulation study we show the good performance of the new estimator in comparison with other estimators proposed in the literature. [ABSTRACT FROM AUTHOR]

We consider copula modeling of the dependence between two or more random variables in the presence of a multivariate covariate. The dependence parameter of the conditional copula possibly depends on the value of the covariate vector. In this paper we develop a new testing methodology for some important parametric specifications of this dependence parameter: constant, linear, quadratic, etc. in the covariate values, possibly after transformation with a link function. The margins are left unspecified. Our novel methodology opens plenty of new possibilities for testing how the conditional copula depends on the multivariate covariate and also for variable selection in copula model building. The suggested test is based on a Rao-type score statistic and regularity conditions are given under which the test has a limiting chi-square distribution under the null hypothesis. For small and moderate sample sizes, a permutation procedure is suggested to assess significance. In simulations it is shown that the test performs well (even under misspecification of the copula family and/or the dependence parameter structure) in comparison to available tests designed for testing for constancy of the dependence parameter. The test is illustrated on a real data set on concentrations of chemicals in water samples. [ABSTRACT FROM AUTHOR]

In this paper, one studies tests of serial independence using a fixed number p of consecutive observations from a stationary time series, first in the univariate case, and then in the multivariate case, where even high-dimensional vectors can be used. The common distribution function is not assumed to be continuous, and the test statistics are based on the multilinear copula process. A bootstrap procedure based on multipliers is also proposed and shown to be valid. Tests based on Spearman's rho and Kendall's tau statistics are also considered, extending the results known for the case of continuous distributions. Contiguity results are obtained for some specific models and sufficient conditions for consistency of test statistics are stated, as well as a data-driven procedure to select p. Also, numerical experiments are performed to assess the finite sample level and power of the proposed tests. A case study using a time series of Arctic sea ice extent images is used to illustrate the usefulness of the proposed methodologies. The R package MixedIndTests (Nasri et al., 2022) includes all the methodologies proposed in this article. [ABSTRACT FROM AUTHOR]

The modeling of dependence between maxima is an important subject in several applications in risk analysis. To this aim, the extreme value copula function, characterized via the madogram, can be used as a margin-free description of the dependence structure. From a practical point of view, the family of extreme value distributions is very rich and arise naturally as the limiting distribution of properly normalized component-wise maxima. In this paper, we investigate the nonparametric estimation of the madogram where data are completely missing at random. We provide the functional central limit theorem for the considered multivariate madrogram correctly normalized, towards a tight Gaussian process for which the covariance function depends on the probabilities of missing. Explicit formula for the asymptotic variance is also given. Our results are illustrated in a finite sample setting with a simulation study. Our method is also illustrated on a sparse dataset of annual maxima rainfall in Central Eastern Canada. [ABSTRACT FROM AUTHOR]

Calculating Markov kernels of two-dimensional Archimedean copulas allows for very simple and elegant alternative derivations of various important formulas including Kendall's distribution function and the measures of the level curves. More importantly, using Markov kernels we prove the existence of singular Archimedean copulas A φ with full support of the following two types: (i) All conditional distribution functions y ↦ F x A φ ( y ) are discrete and strictly increasing; (ii) all conditional distribution functions y ↦ F x A φ ( y ) are continuous, strictly increasing and have derivative zero almost everywhere. The results show that despite of their simple analytic form Archimedean copulas can exhibit surprisingly singular behavior. [ABSTRACT FROM AUTHOR]

This paper studies convergence properties of multivariate distributions constructed by endowing empirical margins with a copula. This setting includes Latin Hypercube Sampling with dependence, also known as the Iman–Conover method. The primary question addressed here is the convergence of the component sum, which is relevant to risk aggregation in insurance and finance. This paper shows that a CLT for the aggregated risk distribution is not available, so that the underlying mathematical problem goes beyond classic functional CLTs for empirical copulas. This issue is relevant to Monte-Carlo based risk aggregation in all multivariate models generated by plugging empirical margins into a copula. Instead of a functional CLT, this paper establishes strong uniform consistency of the estimated sum distribution function and provides a sufficient criterion for the convergence rate O ( n − 1 / 2 ) in probability. These convergence results hold for all copulas with bounded densities. Examples with unbounded densities include bivariate Clayton and Gauss copulas. The convergence results are not specific to the component sum and hold also for any other componentwise non-decreasing aggregation function. On the other hand, convergence of estimates for the joint distribution is much easier to prove, including CLTs. Beyond Iman–Conover estimates, the results of this paper apply to multivariate distributions obtained by plugging empirical margins into an exact copula or by plugging exact margins into an empirical copula. [ABSTRACT FROM AUTHOR]

Copulas are a useful tool to model multivariate distributions. While there exist various families of bivariate copulas, much less work has been done when the dimension is higher. We propose a class of multivariate copulas based on products of transformed bivariate copulas. The analytical forms of the copulas within this class allow to naturally associate a graphical structure which helps to visualize the dependencies and to compute the full joint likelihood even in high dimension. Numerical experiments are conducted both on simulated and real data thanks to a dedicated R package. [ABSTRACT FROM AUTHOR]

One standard way of considering a probability distribution on the unit n -cube, [ 0 , 1 ] n , due to Sklar (1959), is to decompose it into its marginal distributions and a copula, i.e. a probability distribution on [ 0 , 1 ] n with uniform marginals. The definition of copula was extended by Jones et al. (2014) to probability distributions on products of circles. This paper defines a copula as a probability distribution on a product of compact Riemannian manifolds that has uniform marginals. Basic properties of such copulae are established. Two fairly general constructions of copulae on products of compact homogeneous manifolds are given; one is based on convolution in the isometry group, the other using equivariant functions from compact Riemannian manifolds to their spaces of square integrable functions. Examples illustrate the use of copulae to analyse bivariate spherical data and bivariate rotational data. [ABSTRACT FROM AUTHOR]

Conditional quantile estimation is a crucial step in many statistical problems. For example, the recent work on systemic risk relies on estimating risk conditional on an institution being in distress or conditional on being in a crisis (Adrian and Brunnermeier, 2010; Brownlees and Engle, 2011). Specifically, the CoVaR systemic risk measure is based on a conditional quantile when one of the variable is in the tail of the distribution. In this paper, we study properties of conditional quantiles and how they relate to properties of the copula. In particular, we provide a new graphical characterization of tail dependence and intermediate tail dependence from plots of conditional quantiles with normalized marginal distributions (probit scale). A popular method to estimate conditional quantiles is the quantile regression (Koenker, 2005; Koenker and Bassett, 1978). We discuss the properties and pitfalls of this estimation approach. [ABSTRACT FROM AUTHOR]

We study properties of a local dependence function of Wang for copulas. In this paper this dependence function is called the mixed derivative measure of interactions as it is a mixed derivative of a log of a density function. It is stressed that this measure is not margin free in the sense that the interaction function of a density and the corresponding copula are not equal. We show that there is no Archimedean copula with constant interactions. The interaction function is positive (negative) for an Archimedean copula density whose second derivative of the generator is log convex (log concave). Moreover, the only Archimedean copula with interactions proportional to its density is Frank’s copula. We obtain some preliminary results concerning the connection between the behaviour of the interaction function and the tail dependence of the distribution. Moreover, the notion of an interaction function has been extended to more than two dimensional case, and we study its properties for a canonical Archimedean copula. [ABSTRACT FROM AUTHOR]

Abstract: We establish best-possible bounds on the set of quasi-copulas with given degree of non-exchangeability. These bounds are shown to be best-possible bounds as well for the set of copulas with given degree of non-exchangeability, and, consequently, also on the set of bivariate distribution functions of continuous random variables with given margins and given degree of non-exchangeability. Non-exchangeability of a (quasi-)copula is measured in the sense of Nelsen, i.e. proportional to the maximal absolute difference between this (quasi-)copula and its transpose. [Copyright &y& Elsevier]

Abstract: The selection of a copula that appropriately describes the dependence structure of a bivariate distribution function is an issue of a prime interest in multivariate statistical analysis. The most popular methods of testing are those based on the empirical copula and on Kendall’s process; these tests require the use of the parametric bootstrap, which is computationally heavy. This paper introduces the concept of -power functions associated to bivariate copulas. In the first step, their theoretical properties are explored and computations for many popular dependence models are provided. Then, empirical counterparts of these functions are proposed and the weak convergence of suitably normalized versions is obtained from an application of the functional delta method. In addition, a multiplier bootstrap method for these -power processes is described and validated. While of an independent interest to treat many hypotheses related to copulas, these newly introduced tools are used to develop a goodness-of-fit test for bivariate multi-parameter copula models. The recursive nature of the -power functions induces easily computable formulas for the test statistics and suggests a sequential method of testing. The sample properties of this goodness-of-fit test are investigated and compared to those of competing procedures, and its usefulness is illustrated on some pairs of the multivariate Uranium exploration data set. [Copyright &y& Elsevier]

Abstract: The joint distribution of order statistics is characterized without reference to a parent distribution. To this end, the possible univariate margins of such a distribution are first determined. The class of possible copulas is then characterized under the assumption of continuous margins through a description of their minimal support. A truncation-based construction of copulas is also proposed. In the bivariate case, conditions are given for the existence and uniqueness of copulas in this class having maximal support set. Algorithms and examples also show the effectiveness of this construction. [Copyright &y& Elsevier]

Abstract: Using Iterated Function Systems induced by so-called modifiable transformation matrices T and tools from Symbolic Dynamical Systems we first construct mutually singular copulas with identical (possibly fractal or full) support that are at the same time singular with respect to the Lebesgue measure on . Afterwards the established results are utilized for a simple proof of the existence of singular copulas with full support for which all conditional distribution functions are continuous, strictly increasing and have derivative zero λ-almost everywhere. This result underlines the fact that conditional distribution functions of copulas may exhibit surprisingly irregular analytic behavior. Finally, we extend the notion of empirical copula to the case of non-i.i.d. data and prove uniform convergence of the empirical copula corresponding to almost all orbits of a Markov process usually referred to as chaos game to the singular copula . Several examples and graphics illustrate both the chosen approach and the main results. [Copyright &y& Elsevier]

Abstract: We give a purely probabilistic proof of Sklar’s theorem by using a simple continuation technique and sequential arguments. We then consider the case where the distribution function is unknown but one observes instead a sample of i.i.d. copies distributed according to : we construct a sequence of copula representers associated with the empirical distribution function of the sample which convergences a.s. to the representer of the copula function associated with . Eventually, we are surprisingly able to extend the last theorem to the case where the marginals of are discontinuous. [Copyright &y& Elsevier]

Abstract: This paper deals with the problem of estimating the tail of a bivariate distribution function. To this end we develop a general extension of the POT (peaks-over-threshold) method, mainly based on a two-dimensional version of the Pickands–Balkema–de Haan Theorem. We introduce a new parameter that describes the nature of the tail dependence, and we provide a way to estimate it. We construct a two-dimensional tail estimator and study its asymptotic properties. We also present real data examples which illustrate our theoretical results. [Copyright &y& Elsevier]

Abstract: So-called pair copula constructions (PCCs), specifying multivariate distributions only in terms of bivariate building blocks (pair copulas), constitute a flexible class of dependence models. To keep them tractable for inference and model selection, the simplifying assumption, that copulas of conditional distributions do not depend on the values of the variables which they are conditioned on, is popular. We show that the only Archimedean copulas in dimension which are of the simplified type are those based on the Gamma Laplace transform or its extension, while the Student-t copulas are the only one arising from a scale mixture of Normals. Further, we illustrate how PCCs can be adapted for situations where conditional copulas depend on values which are conditioned on, and demonstrate a technique to assess the distance of a multivariate distribution from a nearby distribution that satisfies the simplifying assumption. [Copyright &y& Elsevier]

Lévy copulas are the most general concept to capture jump dependence in multivariate Lévy processes. They translate the intuition and many features of the copula concept into a time series setting. A challenge faced by both, distributional and Lévy copulas, is to find flexible but still applicable models for higher dimensions. To overcome this problem, the concept of pair-copula constructions has been successfully applied to distributional copulas. In this paper, we develop the pair Lévy copula construction (PLCC). Similar to pair constructions of distributional copulas, the pair construction of a -dimensional Lévy copula consists of bivariate dependence functions. We show that only of these bivariate functions are Lévy copulas, whereas the remaining functions are distributional copulas. Since there are no restrictions concerning the choice of the copulas, the proposed pair construction adds the desired flexibility to Lévy copula models. We discuss estimation and simulation in detail and apply the pair construction in a simulation study. To reduce the complexity of the notation, we restrict the presentation to Lévy subordinators, i.e., increasing Lévy processes. [ABSTRACT FROM AUTHOR]

Conditional copulas describe the conditional dependence and the influence that covariates have on the dependence structure between two (or more) variables. Of interest is to test the null hypothesis that the covariates have a specific effect. This paper proposes an omnibus test for testing the null hypothesis of a specified effect of the covariates. The test statistic is designed for having power against many alternatives, and can be used to test for a variety of covariate effects (no effects, linear effects, partial effects, etc.). A special case is the testing problem that the covariates do not affect the dependence structure. In this semiparametric framework the marginal distribution functions are estimated using nonparametric kernel techniques and the parametric dependence model is estimated using maximum likelihood estimation. We establish the asymptotic distribution of the test statistic under the null hypothesis, and evaluate the finite-sample performance of the test via a simulation study, which also includes comparisons with alternative tests. A real data analysis illustrates the practical use of the test. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we provide a nonparametric estimator of the distribution of bivariate censored lifetimes, in a model where the two censoring variables differ only through an additional observed variable. This situation is motivated by a particular application to insurance, where the supplementary variable corresponds to the age difference between two individuals. Asymptotic results for our estimator are provided. The new tools that we develop are used to perform goodness-of-fit tests for survival copula models. The practical performance is illustrated through simulations and a real data analysis. [Copyright &y& Elsevier]

Abstract: Archimedean copulas are popular in the world of multivariate modelling as a result of their breadth, tractability, and flexibility. McNeil and Nešlehová (2009) [12] showed that the class of Archimedean copulas coincides with the class of positive multivariate -norm symmetric distributions. Building upon their results, we introduce a class of multivariate Markov processes that we call ‘Archimedean survival processes’ (ASPs). An ASP is defined over a finite time interval, is equivalent in law to a vector of independent gamma processes, and its terminal value has an Archimedean survival copula. There exists a bijection from the class of ASPs to the class of Archimedean copulas. We provide various characterisations of ASPs, and a generalisation. [Copyright &y& Elsevier]

Abstract: The extremal dependence of a random vector describes the tail behaviors of joint probabilities of the random vector with respect to that of its margins, and has been often studied by using the tail dependence function of its copula. A tail density approach is introduced in this paper to analyze extremal dependence of the copulas that are specified only by densities. The relation between the copula tail densities and regularly varying densities are established, and the tail densities of Archimedean and copulas are derived explicitly. The tail density approach becomes especially effective for extremal dependence analysis on a vine copula, for which the tail density can be written recursively in the product form of tail densities of bivariate baseline copulas and densities of bivariate linking copulas. [Copyright &y& Elsevier]

Abstract: Dependence coefficients have been widely studied for Markov processes defined by a set of transition probabilities and an initial distribution. This work clarifies some aspects of the theory of dependence structure of Markov chains generated by copulas that are useful in time series econometrics and other applied fields. The main aim of this paper is to clarify the relationship between the notions of geometric ergodicity and geometric -mixing; namely, to point out that for a large number of well known copulas, such as Clayton, Gumbel or Student, these notions are equivalent. Some of the results published in the last years appear to be redundant if one takes into account this fact. We apply this equivalence to show that any mixture of Clayton, Gumbel or Student copulas generates both geometrically ergodic and geometric -mixing stationary Markov chains, answering in this way an open question in the literature. We shall also point out that a sufficient condition for -mixing, used in the literature, actually implies Doeblin recurrence. [Copyright &y& Elsevier]

Abstract: This survey reviews the large and growing literature on copula-based models for economic and financial time series. Copula-based multivariate models allow the researcher to specify the models for the marginal distributions separately from the dependence structure that links these distributions to form a joint distribution. This allows for a much greater degree of flexibility in specifying and estimating the model, freeing the researcher from considering only existing multivariate distributions. The author surveys estimation and inference methods and goodness-of-fit tests for such models, as well as empirical applications of these copulas for economic and financial time series. [Copyright &y& Elsevier]

Abstract: We compare two of the most used estimators for the parameters of a pair-copula construction (PCC), namely the semiparametric (SP) and the stepwise semiparametric (SSP) estimators. By construction, the computational speed of the SSP estimator is considerably higher, at the expense of its asymptotic efficiency. Based on an extensive simulation study, we find that the performance of the SSP estimator is overall satisfactory compared to its contender. SSP loses some efficiency with respect to SP with increasing dependence, especially in the top levels of the PCC. On the other hand, the SSP estimator may suffer less under reduced sample sizes and misspecification of the model. Finally, it is the only real alternative for large-dimensional problems. Though it struggles with the top level parameters, the lower order dependences of the resulting estimated PCC mimic the true distribution well. All in all, this study supports the use of SSP in most applications. [Copyright &y& Elsevier]

Abstract: Functions operating on multivariate distribution and survival functions are characterized, based on a theorem of Morillas, for which a new proof is presented. These results are applied to determine those classical mean values on which are distribution functions of probability measures on . As it turns out, the arithmetic mean plays a universal rôle for the characterization of distribution as well as survival functions. Another consequence is a far reaching generalization of Kimberling’s theorem, tightly connected to Archimedean copulas. [Copyright &y& Elsevier]

Abstract: We consider the problem of estimating the marginals in the case where there is knowledge on the copula. If the copula is smooth, it is known that it is possible to improve on the empirical distribution functions: optimal estimators still have a rate of convergence , but a smaller asymptotic variance. In this paper we show that for non-smooth copulas it is sometimes possible to construct superefficient estimators of the marginals: we construct both a copula and, exploiting the information our copula provides, estimators of the marginals with the rate of convergence . [Copyright &y& Elsevier]

Abstract: We derive for the first time the limiting distribution of maxima of skew- random vectors and we show that its limiting case, as the degree of freedom goes to infinity, is the skewed version of the well-known Hüsler–Reiss model. The advantage of the new families of models is that they are particularly flexible, allowing for both symmetric and asymmetric dependence structures and permitting the modelling of multivariate extremes with dimensions greater than two. [Copyright &y& Elsevier]

Abstract: The definition of vectors of dependent random probability measures is a topic of interest in applications to Bayesian statistics. They represent dependent nonparametric prior distributions that are useful for modelling observables for which specific covariate values are known. In this paper we propose a vector of two-parameter Poisson–Dirichlet processes. It is well-known that each component can be obtained by resorting to a change of measure of a -stable process. Thus dependence is achieved by applying a Lévy copula to the marginal intensities. In a two-sample problem, we determine the corresponding partition probability function which turns out to be partially exchangeable. Moreover, we evaluate predictive and posterior distributions. [ABSTRACT FROM AUTHOR]

Abstract: The monotonicity properties of multivariate distribution functions are definitely more complicated than in the univariate case. We show that they fit perfectly well into the general theory of completely monotone and alternating functions on abelian semigroups. This allows us to prove a correspondence theorem which generalizes the classical version in two respects: the function in question may be defined on rather arbitrary product sets in , and it need not be grounded, i.e. disappear at the lower-left boundary. In 2009 a greatly interesting class of copulas was discovered by Mai and Scherer (cf. Mai and Scherer (2009) ), connecting in a very surprising way complete monotonicity with respect to the maximum operation on and with respect to ordinary addition on . Based on the preceding results, we give another proof of this result. [ABSTRACT FROM AUTHOR]

Abstract: Conditions are given under which the empirical copula process associated with a random sample from a bivariate continuous distribution has a smaller asymptotic covariance function than the standard empirical process based on observations from the copula. Illustrations are provided and consequences for inference are outlined. [Copyright &y& Elsevier]

Abstract: We use a recent characterization of the -dimensional Archimedean copulas as the survival copulas of -dimensional simplex distributions (McNeil and Nešlehová (2009) ) to construct new Archimedean copula families, and to examine the relationship between their dependence properties and the radial parts of the corresponding simplex distributions. In particular, a new formula for Kendall’s tau is derived and a new dependence ordering for non-negative random variables is introduced which generalises the Laplace transform order. We then generalise the Archimedean copulas to obtain Liouville copulas, which are the survival copulas of Liouville distributions and which are non-exchangeable in general. We derive a formula for Kendall’s tau of Liouville copulas in terms of the radial parts of the corresponding Liouville distributions. [Copyright &y& Elsevier]

Abstract: We describe several analytical and numerical procedures to obtain bounds on the distribution function of a sum of dependent risks having fixed overlapping marginals. As an application, we produce bounds on quantile-based risk measures for portfolios of financial and actuarial interest. [Copyright &y& Elsevier]

Abstract: A parametric family of n-dimensional extreme-value copulas of Marshall–Olkin type is introduced. Members of this class arise as survival copulas in Lévy-frailty models. The underlying probabilistic construction introduces dependence to initially independent exponential random variables by means of first-passage times of a Lévy subordinator. Jumps of the subordinator correspond to a singular component of the copula. Additionally, a characterization of completely monotone sequences via the introduced family of copulas is derived. An alternative characterization is given by Hausdorff’s moment problem in terms of random variables with compact support. The resulting correspondence between random variables, Lévy subordinators, and copulas is studied and illustrated with several examples. Finally, it is used to provide a general methodology for sampling the copula in many cases. The new class is shown to share some properties with Archimedean copulas regarding construction and analytical form. Finally, the parametric form allows us to compute different measures of dependence and the Pickands representation. [Copyright &y& Elsevier]

Abstract: We show that every copula that is a shuffle of Min is a special push-forward of the doubly stochastic measure induced by the copula M. This fact allows to generalize the notion of shuffle by replacing the measure induced by M with an arbitrary doubly stochastic measure, and, hence, the copula M by any copula C. [Copyright &y& Elsevier]

Abstract: We develop a test of equality between two dependence structures estimated through empirical copulas. We provide inference for independent or paired samples. The multiplier central limit theorem is used for calculating -values of the Cramér–von Mises test statistic. Finite sample properties are assessed with Monte Carlo experiments. We apply the testing procedure on empirical examples in finance, psychology, insurance and medicine. [Copyright &y& Elsevier]

Abstract: We consider the valuation of CDO tranches with single factor copula model, where the involved distributions are mixtures of Gaussian distribution and distribution. In addition, we consider two cases for stochastic correlation and random factor loadings instead of constant factor loadings. We analyze the unconditional characteristic function of accumulated loss of the reference portfolio, and derive the loss distribution through the fast Fourier transform. Moreover, using the loss distribution and semi-analytic approach, we can get the CDO tranches spreads. [Copyright &y& Elsevier]

Abstract: The orthant tail dependence describes the relative deviation of upper- (or lower-) orthant tail probabilities of a random vector from similar orthant tail probabilities of a subset of its components, and can be used in the study of dependence among extreme values. Using the conditional approach, this paper examines the extremal dependence properties of multivariate extreme value distributions and their scale mixtures, and derives the explicit expressions of orthant tail dependence parameters for these distributions. Properties of the tail dependence parameters, including their relations with other extremal dependence measures used in the literature, are discussed. Various examples involving multivariate exponential, multivariate logistic distributions and copulas of Archimedean type are presented to illustrate the results. [Copyright &y& Elsevier]

Abstract: In this paper we introduce two methods for the construction of asymmetric multivariate copulas. The first is connected with products of copulas. The second approach generalises the Archimedean copulas. The resulting copulas are asymmetric and may have more than two parameters in contrast to most of the parametric families of copulas described in the literature. We study the properties of the proposed families of copulas such as the dependence of two components (Kendall’s tau, tail dependence), marginal distributions and the generation of random variates. [Copyright &y& Elsevier]