基于双参数Copula沪港股市的相关性分析

通过双参数Copula分析上证指数和恒生指数的尾部相关性,并与单参数Copula及混合Copula进行比较分析,参数估计使用半参数估计法,结果表明:与单参数Clayton Copula、Gumbel-Hougaard Copula以及由两者组成的混合Copula相比,双参数BB1 Copula对数据具有更好的拟合效果;且通过分析发现两股市的上尾相关性大于下尾相关性.     

Estimating standard errors in regular vine copula models

We describe a new algorithm for the computation of the score function and observed information in regular vine (R-vine) copula models. R-vine copulas are constructed hierarchically from bivariate copulas as building blocks only, and the algorithm exploits this hierarchical nature for subsequent computation of log-likelihood derivatives. This allows to routinely estimate standard errors of parameter estimates, and overcomes reliability and accuracy issues associated with numerical differentiation in multidimensional models. Results obtained using the proposed methods are discussed in the context of the asymptotic efficiency of different estimation methods and of an application to exchange rate data.     

Tails of correlation mixtures of elliptical copulas

Correlation mixtures of elliptical copulas arise when the correlation parameter is driven itself by a latent random process. For such copulas, both penultimate and asymptotic tail dependence are much larger than for ordinary elliptical copulas with the same unconditional correlation. Furthermore, for Gaussian and Student t-copulas, tail dependence at sub-asymptotic levels is generally larger than in the limit, which can have serious consequences for estimation and evaluation of extreme risk. Finally, although correlation mixtures of Gaussian copulas inherit the property of asymptotic independence, at the same time they fall in the newly defined category of near asymptotic dependence. The consequences of these findings for modeling are assessed by means of a simulation study and a case study involving financial time series.     

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.     

A note on conjugate distributions for copulas

A family of conjugated distributions for a given type of copulas is defined in this paper. Those copulas can be written as a mixture of d‐dimensional parameter exponential functions. The generalized Farlie–Gumbel–Morgenstern copula is an example of this representation. This family is used to illustrate the estimation technique with real data. Also, the applicability of Bayesian predictive approach is shown in an education policy issue by defining goals for the number of students per class that leads to improve their performance at school. 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.     

A class of multivariate copulas with bivariate Fréchet marginal copulas

In this paper, we present a class of multivariate copulas whose two-dimensional marginals belong to the family of bivariate Fréchet copulas. The coordinates of a random vector distributed as one of these copulas are conditionally independent. We prove that these multivariate copulas are uniquely determined by their two-dimensional marginal copulas. Some other properties for these multivariate copulas are discussed as well. Two applications of these copulas in actuarial science are given.     

Modelling of extreme wave heights and periods through copulas

Optimal design of coastal or offshore structures requires the estimation of extreme quantiles of oceanographic data such as wave heights and wave periods. Since there are strong correlations between oceanographic variables, it is necessary to use multivariate models in order to capture its dependencies. To achieve this, an approach based on copulas is proposed and is compared to a model based on the physical behaviour of waves.     

Intervals of 1-Lipschitz aggregation operators, quasi-copulas, and copulas with given affine section

Best lower and upper bounds for 1-Lipschitz aggregation operators with a given affine section are given. These are used to determine best bounds for quasi-copulas and copulas with a given affine section. However, in general there is no greatest copula with a given non-decreasing affine section. These results are used to study (quasi-)copulas with arbitrary affine sections.     

Discussion of “Copulas: Tales and facts”, by Thomas Mikosch

As Prof. Mikosch correctly points out, there exists very little sound statistical theory on modelling dependence using copulas. In this contribution, an open problem is presented concerning the efficient estimation of the parameter of a copula when no parametric assumptions are made regarding the marginal distributions.     

Robust portfolio optimization with copulas

Conditional Value at Risk (CVaR) is widely used in portfolio optimization as a measure of risk. CVaR is clearly dependent on the underlying probability distribution of the portfolio. We show how copulas can be introduced to any problem that involves distributions and how they can provide solutions for the modeling of the portfolio. We use this to provide the copula formulation of the CVaR of a portfolio. Given the critical dependence of CVaR on the underlying distribution, we use a robust framework to extend our approach to Worst Case CVaR (WCVaR). WCVaR is achieved through the use of rival copulas. These rival copulas have the advantage of exploiting a variety of dependence structures, symmetric and not. We compare our model against two other models, Gaussian CVaR and Worst Case Markowitz. Our empirical analysis shows that WCVaR can asses the risk more adequately than the two competitive models during periods of crisis.     

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.     

Construction of asymmetric multivariate copulas

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.     

Heavy-tailed longitudinal data modeling using copulas

In this paper, we consider “heavy-tailed” data, that is, data where extreme values are likely to occur. Heavy-tailed data have been analyzed using flexible distributions such as the generalized beta of the second kind, the generalized gamma and the Burr. These distributions allow us to handle data with either positive or negative skewness, as well as heavy tails. Moreover, it has been shown that they can also accommodate cross-sectional regression models by allowing functions of explanatory variables to serve as distribution parameters.The objective of this paper is to extend this literature to accommodate longitudinal data, where one observes repeated observations of cross-sectional data. Specifically, we use copulas to model the dependencies over time, and heavy-tailed regression models to represent the marginal distributions. We also introduce model exploration techniques to help us with the initial choice of the copula and a goodness-of-fit test of elliptical copulas for model validation. In a longitudinal data context, we argue that elliptical copulas will be typically preferred to the Archimedean copulas. To illustrate our methods, Wisconsin nursing homes utilization data from 1995 to 2001 are analyzed. These data exhibit long tails and negative skewness and so help us to motivate the need for our new techniques. We find that time and the nursing home facility size as measured through the number of beds and square footage are important predictors of future utilization. Moreover, using our parametric model, we provide not only point predictions but also an entire predictive distribution.     

多元Copula密度估计的小波局部阈值方法

为了量化资产之间相依结构的局部特征,本文将小波阈值规则引入Copula参数估计,提出多元Copula密度的小波局部阈值估计量,发现Copula密度的光滑度指数、维数和采样容量是影响估值精度的重要因素,这一点也得到了以正态Copula为仿真算例的支持。本方法增强了参数Copula建模的局部自适应能力,进而有助于改进资产的市场风险估值与最优化配置。     

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.     

具有n次多项式截面的Copula函数

本文构造一类具有n次多项式截面的Copula函数C(x,y),并给出了C(x,y)是Copula函数的两个充分必要条件,同时推导出此类Copula函数的若干性质,包括对称性和相关度量等.     

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.     

Bivariate extreme-value copulas with discrete Pickands dependence measure

A two-parametric family of bivariate extreme-value copulas (EVCs), which corresponds to precisely the bivariate EVCs whose Pickands dependence measure is discrete with at most two atoms, is introduced and analyzed. It is shown how bivariate EVCs with arbitrary discrete Pickands dependence measure can be represented as the geometric mean of such basis copulas. General bivariate EVCs can thus be represented as the limit of this construction when the number of involved basis copulas tends to infinity. Besides the theoretical value of such a representation, it is shown how several properties of the represented copula can be deduced from properties of the involved basis copulas. An algorithm for the computation of the representation is given.