Nonparametric universal copula modeling

To handle the ubiquitous problem of “dependence learning,” copulas are quickly becoming a pervasive tool across a wide range of data‐driven disciplines encompassing neuroscience, finance, econometrics, genomics, social science, machine learning, healthcare, and many more. At the same time, despite their practical value, the empirical methods of “learning copula from data” have been unsystematic with full of case‐specific recipes. Taking inspiration from modern LP‐nonparametrics, this paper presents a modest contribution to the need for a more unified and structured approach of copula modeling that is simultaneously valid for arbitrary combinations of continuous and discrete variables.     

A generalized beta copula with applications in modeling multivariate long-tailed data

This work proposes a new copula class that we call the MGB2 copula. The new copula originates from extracting the dependence function of the multivariate GB2 distribution (MGB2) whose marginals follow the univariate generalized beta distribution of the second kind (GB2). The MGB2 copula can capture non-elliptical and asymmetric dependencies among marginal coordinates and provides a simple formulation for multi-dimensional applications. This new class features positive tail dependence in the upper tail and tail independence in the lower tail. Furthermore, it includes some well-known copula classes, such as the Gaussian copula, as special or limiting cases.To illustrate the usefulness of the MGB2 copula, we build a trivariate MGB2 copula model of bodily injury liability closed claims. Extended GB2 distributions are chosen to accommodate the right-skewness and the long-tailedness of the outcome variables. For the regression component, location parameters with continuous predictors are introduced using a nonlinear additive function. For comparison purposes, we also consider the Gumbel and t copulas, alternatives that capture the upper tail dependence. The paper introduces a conditional plot graphical tool for assessing the validation of the MGB2 copula. Quantitative and graphical assessment of the goodness of fit demonstrate the advantages of the MGB2 copula over the other copulas.     

Lévy-frailty copulas

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.     

Tail negative dependence and its applications for aggregate loss modeling

Tail order of copulas can be used to describe the strength of dependence in the tails of a joint distribution. When the value of tail order is larger than the dimension, it may lead to tail negative dependence. First, we prove results on conditions that lead to tail negative dependence for Archimedean copulas. Using the conditions, we construct new parametric copula families that possess upper tail negative dependence. Among them, a copula based on a scale mixture with a generalized gamma random variable (GGS copula) is useful for modeling asymmetric tail negative dependence. We propose mixed copula regression based on the GGS copula for aggregate loss modeling of a medical expenditure panel survey dataset. For this dataset, we find that there exists upper tail negative dependence between loss frequency and loss severity, and the introduction of tail negative dependence structures significantly improves the aggregate loss modeling.     

Representing Sparse Gaussian DAGs as Sparse R-Vines Allowing for Non-Gaussian Dependence

Modeling dependence in high-dimensional systems has become an increasingly important topic. Most approaches rely on the assumption of a multivariate Gaussian distribution such as statistical models on directed acyclic graphs (DAGs). They are based on modeling conditional independencies and are scalable to high dimensions. In contrast, vine copula models accommodate more elaborate features like tail dependence and asymmetry, as well as independent modeling of the marginals. This flexibility comes however at the cost of exponentially increasing complexity for model selection and estimation. We show a novel connection between DAGs with limited number of parents and truncated vine copulas under sufficient conditions. This motivates a more general procedure exploiting the fast model selection and estimation of sparse DAGs while allowing for non-Gaussian dependence using vine copulas. By numerical examples in hundreds of dimensions, we demonstrate that our approach outperforms the standard method for vine structure selection. Supplementary material for this article is available online.     

Estimation of nonstrict Archimedean copulas and its application to quantum networks

Bivariate nonstrict Archimedean copulas form a subclass of Archimedean copulas and are able to model the dependence structure of random variables that do not take on low quantiles simultaneously; i.e. their domain includes a set, the so‐called zero set, with positive Lebesgue measure but zero probability mass. Standard methods to fit a parametric Archimedean copula, e.g. classical maximum likelihood estimation, are either getting computationally more involved or even fail when dealing with this subclass. We propose an alternative method for estimating the parameter of a nonstrict Archimedean copula that is based on the zero set and the functional form of its boundary curve. This estimator is fast to compute and can be applied to absolutely continuous copulas but also allows singular components. In a simulation study, we compare its performance to that of the standard estimators. Finally, the estimator is applied when modeling the dependence structure of quantities describing the quality of transmission in a quantum network, and it is shown how this model can be used effectively to detect potential intruders in this network. Copyright © 2014 John Wiley & Sons, Ltd.     

Multivariate Hierarchical Copulas with Shocks

A transformation to obtain new multivariate hierarchical copulas, starting with an arbitrary copula, is introduced. In addition to the hierarchical structure, the presented construction principle explicitly supports singular components. These may be interpreted as the effect of local or global shocks to the underlying random variables. A large spectrum of dependence patterns can be achieved by the presented transformation, which seems promising for practical applications. Moreover, copulas arising from this construction are similarly admissible with respect to analytical tractability and sampling routines as the original copula. Finally, several well-known families of copulas may be interpreted as special cases.     

Distorted Mix Method for constructing copulas with tail dependence

This paper introduces a method for constructing copula functions by combining the ideas of distortion and convex sum, named Distorted Mix Method. The method mixes different copulas with distorted margins to construct new copula functions, and it enables us to model the dependence structure of risks by handling the central and tail parts separately. By applying the method we can modify the tail dependence of a given copula to any desired level measured by tail dependence function and tail dependence coefficients of marginal distributions. As an application, a tight bound for asymptotic Value-at-Risk of order statistics is obtained by using the method. An empirical study shows that copulas constructed by this method fit the empirical data of SPX 500 Index and FTSE 100 Index very well in both central and tail parts.     

Constructing hierarchical Archimedean copulas with Lévy subordinators

A probabilistic interpretation for hierarchical Archimedean copulas based on Lévy subordinators is given. Independent exponential random variables are divided by group-specific Lévy subordinators which are evaluated at a common random time. The resulting random vector has a hierarchical Archimedean survival copula. This approach suggests an efficient sampling algorithm and allows one to easily construct several new parametric families of hierarchical Archimedean copulas.     

基于倾斜-t Copula的半参数马尔可夫链

在不指定时间序列结构的情况下,我们的分布模型是基于多变量离散时间的相应马尔可夫族和相关变量一维的边际分布.这样的模型可以同时处理时间序列之间的相互依赖和每个时间序列沿时间方向的依赖.具体的参数copula被指定为倾斜-t. 倾斜-t Copla能够处理不对称,偏斜和粗尾的数据分布.三个股票指数日均收益的实证研究表明,倾斜-t copula的马尔可夫模型要比以下模型更好:倾斜正态Copula马可夫, t-copula马可夫, 倾斜-t copula但无马尔可夫特性.     

On the construction of minimum information bivariate copula families

Copulas have become very popular as modelling tools in probability applications. Given a finite number of expectation constraints for functions defined on the unit square, the minimum information copula is that copula which has minimum information (Kullback–Leibler divergence) from the uniform copula. This can be considered the most “independent” copula satisfying the constraints. We demonstrate the existence and uniqueness of such copulas, rigorously establish the relation with discrete approximations, and prove an unexpected relationship between constraint expectation values and the copula density formula.     

A clone-based graphical modeler and mathematical model generator for optimal production planning in process industries

This paper outlines a visually interactive graphical modeling approach for process type production systems, with hidden generation of complex optimization models for production planning. The proposed system lets the users build a graphical model of the production system with one-to-one clones of its production units through its interactive visual interface, accepts production-specific data for its components, and finally, internally generates and solves its mathematical programming model without any interaction from the user. This “clone-based” modeling approach allows the continued use of optimization models with minimal mathematical programming understanding, as generation of mathematical model by clones is hidden and automatic, therefore maintenance-free: Updating graphical production system models is enough for renewing internal optimization models. The concept is demonstrated in this paper with a linear programming prototype developed for a petroleum refinery.     

Modeling defaults with nested Archimedean copulas

In 2001, Schönbucher and Schubert extended Li’s well-known Gaussian copula model for modeling dependent defaults to allow for tail dependence. Instead of the Gaussian copula, Schönbucher and Schubert suggested to use Archimedean copulas. These copulas are able to capture tail dependence and therefore allow a standard intensity-based default model to have a positive probability of joint defaults within a short time period. As can be observed in the current financial crisis, this is an indispensable feature of any realistic default model. Another feature, motivated by empirical observations but rarely taken into account in default models, is that modeled portfolio components affected by defaults show significantly different levels of dependence depending on whether they belong to the same industry sector or not. The present work presents an extension of the model suggested by Schönbucher and Schubert to account for this fact. For this, nested Archimedean copulas are applied. As an application, the pricing of collateralized debt obligations is treated. Since the resulting loss distribution is not analytical tractable, fast sampling algorithms for nested Archimedean copulas are developed. Such algorithms boil down to sampling certain distributions given by their Laplace-Stieltjes transforms. For a large range of nested Archimedean copulas, efficient sampling techniques can be derived. Moreover, a general transformation of an Archimedean generator allows to construct and sample the corresponding nested Archimedean copulas.     

Tail order and intermediate tail dependence of multivariate copulas

In order to study copula families that have tail patterns and tail asymmetry different from multivariate Gaussian and t copulas, we introduce the concepts of tail order and tail order functions. These provide an integrated way to study both tail dependence and intermediate tail dependence. Some fundamental properties of tail order and tail order functions are obtained. For the multivariate Archimedean copula, we relate the tail heaviness of a positive random variable to the tail behavior of the Archimedean copula constructed from the Laplace transform of the random variable, and extend the results of Charpentier and Segers [7] [A. Charpentier, J. Segers, Tails of multivariate Archimedean copulas, Journal of Multivariate Analysis 100 (7) (2009) 1521–1537] for upper tails of Archimedean copulas. In addition, a new one-parameter Archimedean copula family based on the Laplace transform of the inverse Gamma distribution is proposed; it possesses patterns of upper and lower tails not seen in commonly used copula families. Finally, tail orders are studied for copulas constructed from mixtures of max-infinitely divisible copulas.     

COPAR—multivariate time series modeling using the copula autoregressive model

The analysis of multivariate time series is a common problem in areas like finance and economics. The classical tools for this purpose are vector autoregressive models. These however are limited to the modeling of linear and symmetric dependence. We propose a novel copula‐based model that allows for the non‐linear and non‐symmetric modeling of serial as well as between‐series dependencies. The model exploits the flexibility of vine copulas, which are built up by bivariate copulas only. We describe statistical inference techniques for the new model and discuss how it can be used for testing Granger causality. Finally, we use the model to investigate inflation effects on industrial production, stock returns and interest rates. In addition, the out‐of‐sample predictive ability is compared with relevant benchmark models. Copyright © 2014 John Wiley & Sons, Ltd.     

On the construction of copulas and quasi-copulas with given diagonal sections

We study a method, which we call a copula (or quasi-copula) diagonal splice, for creating new functions by joining portions of two copulas (or quasi-copulas) with a common diagonal section. The diagonal splice of two quasi-copulas is always a quasi-copula, and we find a necessary and sufficient condition for the diagonal splice of two copulas to be a copula. Applications of this method include the construction of absolutely continuous asymmetric copulas with a prescribed diagonal section, and determining the best-possible upper bound on the set of copulas with a particular type of diagonal section. Several examples illustrate our results.     

Dependence modeling in non-life insurance using the Bernstein copula

This paper illustrates the modeling of dependence structures of non-life insurance risks using the Bernstein copula. We conduct a goodness-of-fit analysis and compare the Bernstein copula with other widely used copulas. Then, we illustrate the use of the Bernstein copula in a value-at-risk and tail-value-at-risk simulation study. For both analyses we utilize German claims data on storm, flood, and water damage insurance for calibration. Our results highlight the advantages of the Bernstein copula, including its flexibility in mapping inhomogeneous dependence structures and its easy use in a simulation context due to its representation as mixture of independent Beta densities. Practitioners and regulators working toward appropriate modeling of dependences in a risk management and solvency context can benefit from our results.     

Copula selection for graphical models in continuous Estimation of Distribution Algorithms

This paper presents the use of graphical models and copula functions in Estimation of Distribution Algorithms (EDAs) for solving multivariate optimization problems. It is shown in this work how the incorporation of copula functions and graphical models for modeling the dependencies among variables provides some theoretical advantages over traditional EDAs. By means of copula functions and two well known graphical models, this paper presents a novel approach for defining new EDAs. Either dependence is modeled by a copula function chosen from a predefined set of six functions that aim to cover a wide range of inter-relations. It is also shown how the use of mutual information in the learning of graphical models implies a natural way of employing copula entropies. The experimental results on separable and non-separable functions show that the two new EDAs, which adopt copula functions to model dependencies, perform better than their original version with Gaussian variables.     

Modelling dependence

A new way of choosing a suitable copula to model dependence is introduced. Instead of relying on a given parametric family of copulas or applying the other extreme of modelling dependence in a nonparametric way, an intermediate approach is proposed, based on a sequence of parametric models containing more and more dependency aspects. In contrast to a similar way of thinking in testing theory, the method here, intended for estimating the copula, often requires a somewhat larger number of steps. One approach is based on exponential families, another on contamination families. An extensive numerical investigation is supplied on a large number of well-known copulas. The method based on contamination families is recommended. A Gaussian start in this approximation looks very promising.     

On spatial contagion and multivariate GARCH models

We propose a method for defining and measuring spatial contagion between two financial markets via conditional copulas. Some theoretical results on monotonicity and asymptotic properties of Gaussian copulas with respect to conditioning are presented. Next, we combine the spatial contagion approach with time series models. We investigate which model from a large family of multivariate GARCH is the best tool for modelling spatial contagion. In an empirical study, we show that among models designed for general fit, a two‐step model fitting procedure reduces the ability to describe the contagion effect. This is a feature of copula‐GARCH models. Copyright © 2013 John Wiley & Sons, Ltd.