Copula Construction Research Articles

Bivariate quadratic copula constructions

In this study, we present the concept of quadratic copula constructions using two arbitrary copulas. We characterize all quadratic polynomials of four variables whose composition with any two copulas always results in a copula. We show that these polynomials form a compact convex set in a seven-dimensional vector space. We also apply the result to obtain a new family of copulas.

Recent Developments in Copula Models

Copula models have become very popular and well studied among the scientific community.[...]

Hierarchical Archimedean copulas through multivariate compound distributions

In this paper, we propose a new hierarchical Archimedean copula construction based on multivariate compound distributions. This new imbrication technique is derived via the construction of a multivariate exponential mixture distribution through compounding. The absence of nesting and marginal conditions, contrarily to the nested Archimedean copulas approach, leads to major advantages, such as a flexible range of possible combinations in the choice of distributions, the existence of explicit formulas for the distribution of the sum, and computational ease in high dimensions. A balance between flexibility and parsimony is targeted. After presenting the construction technique, properties of the proposed copulas are investigated and illustrative examples are given. A detailed comparison with other construction methodologies of hierarchical Archimedean copulas is provided. Risk aggregation under this newly proposed dependence structure is also examined.

Vine copula constructions of higher-dimensional dependent reliability systems

ABSTRACTCopulas 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.

Bayesian Inference for Latent Factor Copulas and Application to Financial Risk Forecasting

Factor modeling is a popular strategy to induce sparsity in multivariate models as they scale to higher dimensions. We develop Bayesian inference for a recently proposed latent factor copula model, which utilizes a pair copula construction to couple the variables with the latent factor. We use adaptive rejection Metropolis sampling (ARMS) within Gibbs sampling for posterior simulation: Gibbs sampling enables application to Bayesian problems, while ARMS is an adaptive strategy that replaces traditional Metropolis-Hastings updates, which typically require careful tuning. Our simulation study shows favorable performance of our proposed approach both in terms of sampling efficiency and accuracy. We provide an extensive application example using historical data on European financial stocks that forecasts portfolio Value at Risk (VaR) and Expected Shortfall (ES).

Construction and identification of a D-Vine model applied to the probability distribution of modal parameters in structural dynamics

This study investigates the construction and identification of the probability distribution of random modal parameters (natural frequencies and effective parameters) in structural dynamics. As these parameters present various types of dependence structures, the retained approach is based on pair copula construction (PCC). A literature review leads us to choose a D-Vine model for the construction of modal parameters probability distributions. Identification of this model is based on likelihood maximization which makes it sensitive to the dimension of the distribution, namely the number of considered modes in our context. To this respect, a mode selection preprocessing step is proposed. It allows the selection of the relevant random modes for a given transfer function. The second point, addressed in this study, concerns the choice of the D-Vine model. Indeed, D-Vine model is not uniquely defined. Two strategies are proposed and compared. The first one is based on the context of the study whereas the second one is purely based on statistical considerations. Finally, the proposed approaches are numerically studied and compared with respect to their capabilities, first in the identification of the probability distribution of random modal parameters and second in the estimation of the 99% quantiles of some transfer functions.

A Cognitive Approach to the Copula Construction of Korean

A Cognitive Approach to the Copula Construction of Korean

Defects and transformations of quasi-copulas

Six different functions measuring the defect of a quasi-copula, i. e., how far away it is from a copula, are discussed. This is done by means of extremal non-positive volumes of specific rectangles (in a way that a zero defect characterizes copulas). Based on these defect functions, six transformations of quasi-copulas are investigated which give rise to six different partitions of the set of all quasi-copulas. For each of these partitions, each equivalence class contains exactly one copula being a fixed point of the transformation under consideration. Finally, an application to the construction of so-called imprecise copulas is given.

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.

On the role of ultramodularity and Schur concavity in the construction of binary copulas

On the role of ultramodularity and Schur concavity in the construction of binary copulas

On the role of ultramodularity and Schur concavity in the construction of binary copulas

On the role of ultramodularity and Schur concavity in the construction of binary copulas

The empirical beta copula

Given a sample from a continuous multivariate distribution F , the uniform random variates generated independently and rearranged in the order specified by the componentwise ranks of the original sample look like a sample from the copula of F . This idea can be regarded as a variant on Baker’s [J. Multivariate Anal. 99 (2008) 2312–2327] copula construction and leads to the definition of the empirical beta copula. The latter turns out to be a particular case of the empirical Bernstein copula, the degrees of all Bernstein polynomials being equal to the sample size. Necessary and sufficient conditions are given for a Bernstein polynomial to be a copula. These imply that the empirical beta copula is a genuine copula. Furthermore, the empirical process based on the empirical Bernstein copula is shown to be asymptotically the same as the ordinary empirical copula process under assumptions which are significantly weaker than those given in Janssen, Swanepoel and Veraverbeke [J. Statist. Plann. Inference 142 (2012) 1189–1197]. A Monte Carlo simulation study shows that the empirical beta copula outperforms the empirical copula and the empirical checkerboard copula in terms of both bias and variance. Compared with the empirical Bernstein copula with the smoothing rate suggested by Janssen et al., its finite-sample performance is still significantly better in several cases, especially in terms of bias.

Nonparametric testing for no covariate effects in conditional copulas

ABSTRACTIn dependence modelling using conditional copulas, one often imposes the working assumption that the covariate influences the conditional copula solely through the marginal distributions. This so-called (pairwise) simplifying assumption is almost standardly made in vine copula constructions. However, in recent literature evidence was provided that such an assumption might not be justified. Among the first issues is thus to test for its appropriateness. In this paper nonparametric tests for the null hypothesis of the simplifying assumption are proposed, and their asymptotic behaviours, under the null hypothesis and under some local alternatives, are established. The tests are fully nonparametric in nature: not requiring choices of copula families nor knowledge of the marginals. In a simulation study, the finite-sample size and power performances of the tests are investigated, and compared with these of the few available tests. A real data application illustrates the use of the tests.

Some alternative bivariate Kumaraswamy-type distributions via copula with application in risk management

ABSTRACTIn this article we discuss various strategies for constructing bivariate Kumaraswamy distributions via the copula approach. The copula methods and construction studied here are different from those briefly discussed in Arnold and Ghosh (2016). In this article, bivariate normal copula, AMH (Ali–Mikhail–Haq), and Marshall–Olkin copula generators were assumed to construct bivariate Kumaraswamy models. Additionally, we consider here a few different types of copula generators, which subsume Clayton copula type generators. Various structural properties of the derived copulas including tail dependence, correlation coefficient, and others are discussed.

Modeling high‐dimensional time‐varying dependence using dynamic D‐vine models

We consider the problem of modeling the dependence among many time series. We build high‐dimensional time‐varying copula models by combining pair‐copula constructions with stochastic autoregressive copula and generalized autoregressive score models to capture dependence that changes over time. We show how the estimation of this highly complex model can be broken down into the estimation of a sequence of bivariate models, which can be achieved by using the method of maximum likelihood. Further, by restricting the conditional dependence parameter on higher cascades of the pair copula construction to be constant, we can greatly reduce the number of parameters to be estimated without losing much flexibility. Applications to five MSCI stock market indices and to a large dataset of daily stock returns of all constituents of the Dax 30 illustrate the usefulness of the proposed model class in‐sample and for density forecasting. Copyright © 2016 John Wiley & Sons, Ltd.

Floating agreement and information structure

This paper investigates the morphosyntactic and pragmatic properties of floating person agreement in Sanzhi Dargwa (Nakh-Daghestanian, Russia). Person agreement enclitics can occur on the verb or on other constituents (NPs, adverbs, or pronouns). In the latter case, they seem to function like constituent focus markers because they emphasize their host, but this effect is limited to elicited sentences. Floating agreement in Sanzhi Dargwa is compared to similar constructions in other Nakh-Daghestanian languages (Udi, Lak) which have been analyzed as synchronic in situ clefts or as diachronically arising from clefts. The paper shows that a synchronic cleft analysis for floating agreement in Sanzhi must be rejected, and it is argued that diachronically they originate from identificational copula constructions.

A multivariate volatility vine copula model

ABSTRACTThis article proposes a dynamic framework for modeling and forecasting of realized covariance matrices using vine copulas to allow for more flexible dependencies between assets. Our model automatically guarantees positive definiteness of the forecast through the use of a Cholesky decomposition of the realized covariance matrix. We explicitly account for long-memory behavior by using fractionally integrated autoregressive moving average (ARFIMA) and heterogeneous autoregressive (HAR) models for the individual elements of the decomposition. Furthermore, our model incorporates non-Gaussian innovations and GARCH effects, accounting for volatility clustering and unconditional kurtosis. The dependence structure between assets is studied using vine copula constructions, which allow for nonlinearity and asymmetry without suffering from an inflexible tail behavior or symmetry restrictions as in conventional multivariate models. Further, the copulas have a direct impact on the point forecasts of the realized covariances matrices, due to being computed as a nonlinear transformation of the forecasts for the Cholesky matrix. Beside studying in-sample properties, we assess the usefulness of our method in a one-day-ahead forecasting framework, comparing recent types of models for the realized covariance matrix based on a model confidence set approach. Additionally, we find that in Value-at-Risk (VaR) forecasting, vine models require less capital requirements due to smoother and more accurate forecasts.

Review of dependence modeling in hydrology and water resources

Various methods have been developed over the past five decades for dependence modeling of multivariate variables in hydrology and water resources, but there has been no overall review of techniques commonly used in the field. This paper, therefore, introduces several methods focusing on dependence structure modeling, including parametric distribution, entropy, copula, and nonparametric. Recent advances in modeling dependences mainly reside in nonlinear dependence modeling (including extreme dependence) with flexible marginal distributions, and in high-dimension dependence modeling via the vine copula construction with flexible dependence structures. Strengths and limitations of different methods and avenues for future research, such as dependence modeling in a changing climate, are discussed to aid water resource planners and managers in the selection and application of suitable techniques.

Dependent Defaults and Losses with Factor Copula Models

We introduce a class of flexible and tractable static factor models for the joint term structure of default probabilities, the factor copula models. These high dimensional models remain parsimonious with pair copula constructions, and nest numerous standard models as special cases. With finitely supported random losses, the loss distributions of credit portfolios and derivatives can be exactly and efficiently computed. Numerical examples on collateral debt obligation (CDO), CDO squared, and credit index swaption illustrate the versatility of our framework. An empirical exercise shows that a simple model specification can fit credit index tranche prices.

Pair copula constructions to determine the dependence structure of treasury bond yields

Pair copula constructions to determine the dependence structure of treasury bond yields