Search

Nonparametric moment-based goodness-of-fit tests are developed for the family of chi-square copulas of arbitrary dimensions. This class of dependence models allows for tail asymmetries and contains the family of multivariate normal copulas as a special case. The proposed tests are based on two rank correlation coefficients whose population versions are equal, up to a monotone transformation, when the underlying dependence structure is a chi-square copula. The test statistics are computed from natural rank-based estimations of these two correlation coefficients and their large-sample distributions under the null hypothesis of a chi-square copula are derived; the validity of a parametric bootstrap procedure for the computation of p-values is formally established as well. Particular attention is given to tests for the families of normal and centered chi-square copulas. The simulations that are reported indicate that the new tests are reliable alternatives to those based on the empirical copula, both in the bivariate and multivariate cases. The usefulness of the introduced methodology is illustrated on the five-dimensional Nutrient dataset.

Quantile regression, that is the prediction of conditional quantiles, has steadily gained importance in statistical modeling and financial applications. The authors introduce a new semiparametric quantile regression method based on sequentially fitting a likelihood optimal D-vine copula to given data resulting in highly flexible models with easily extractable conditional quantiles. As a subclass of regular vine copulas, D-vines enable the modeling of multivariate copulas in terms of bivariate building blocks, a so-called pair-copula construction (PCC). The proposed algorithm works fast and accurate even in high dimensions and incorporates an automatic variable selection by maximizing the conditional log-likelihood. Further, typical issues of quantile regression such as quantile crossing or transformations, interactions and collinearity of variables are automatically taken care of. In a simulation study the improved accuracy and saved computational time of the approach in comparison with established quantile regression methods is highlighted. An extensive financial application to international credit default swap (CDS) data including stress testing and Value-at-Risk (VaR) prediction demonstrates the usefulness of the proposed method.

Asymmetry of dependence is an inherent property of bivariate probability distributions. Being symmetric, commonly used dependence measures such as Pearson’s r or Spearman’s ρ mask asymmetry and implicitly assume that a random variable Y is equally dependent on a random variable X as vice versa. A copula-based, hence scale-invariant dependence measure called ζ 1 overcoming the just mentioned problem was introduced in 2011. ζ 1 attains values in [ 0 , 1 ] , it is 0 if, and only if X and Y are independent, and 1 if, and only if Y is a measurable function of X . Working with so-called empirical checkerboard copulas allows to construct an estimator ζ 1 n for ζ 1 which is strongly consistent in full generality, i.e., without any smoothness assumptions on the underlying copula. The R-package qad (short for quantification of asymmetric dependence) containing the estimator ζ 1 n is used both, to perform a simulation study illustrating the small sample performance of the estimator as well as to estimate the directed dependence between some global climate variables as well as between world development indicators.

Factor copula models have been recently proposed for describing the joint distribution of a large number of variables in terms of a few common latent factors. A Bayesian procedure is employed in order to make fast inferences for multi-factor and structured factor copulas. To deal with the high dimensional structure, a Variational Inference (VI) algorithm is applied to estimate different specifications of factor copula models. Compared to the Markov Chain Monte Carlo (MCMC) approach, the variational approximation is much faster and could handle a sizeable problem in limited time. Another issue of factor copula models is that the bivariate copula functions connecting the variables are unknown in high dimensions. An automatic procedure is derived to recover the hidden dependence structure. By taking advantage of the posterior modes of the latent variables, the bivariate copula functions are selected by minimizing the Bayesian Information Criterion (BIC). Simulation studies in different contexts show that the procedure of bivariate copula selection could be very accurate in comparison to the true generated copula model. The proposed procedure is illustrated with two high dimensional real data sets.

The Mann–Whitney effect is a nonparametric measure for comparing the distribution between two groups, which can be estimated by right-censored data. However, the traditional estimator of the Mann–Whitney effect based on the Kaplan–Meier estimators can be inconsistent when the independent censoring assumption fails to hold. Investigation is made on the asymptotic bias of the traditional estimator of the Mann–Whitney effect when the independent censoring assumption is violated due to dependence between survival time and censoring time. A new estimator of the Mann–Whitney effect is proposed by applying the copula-graphic estimator to adjust for the effect of dependent censoring. The proposed estimator and test are consistent when the assumed copulas for the two groups are correct. Some consistency properties under misspecified copulas are also given. Simulations are conducted to verify the proposed method under possible misspecification on copulas. The method is illustrated by a real data set. We provide an R function “MW.test” to implement the proposed estimator and test.

Joint modeling of multiple health related random variables is essential to develop an understanding for the public health consequences of an aging population. This is particularly true for patients suffering from multiple chronic diseases. The contribution is to introduce a novel model for multivariate data where some response variables are discrete and some are continuous. It is based on pair copula constructions (PCCs) and has two major advantages over existing methodology. First, expressing the joint dependence structure in terms of bivariate copulas leads to a computationally advantageous expression for the likelihood function. This makes maximum likelihood estimation feasible for large multidimensional data sets. Second, different and possibly asymmetric bivariate (conditional) marginal distributions are allowed which is necessary to accurately describe the limiting behavior of conditional distributions for mixed discrete and continuous responses. The advantages and the favorable predictive performance of the model are demonstrated using data from the Second Longitudinal Study of Aging (LSOA II).

Both in classical multivariate analysis and in modern copula modeling, correlation matrices are a central concept of dependence modeling using multivariate normal distributions and copulas. Since the number of correlation parameters quadratically increases with the number of variables, parsimonious parameterizations of large correlation matrices in terms of O ( d ) parameters are important. While factor analysis is commonly used for this purpose, the use of vines is an attractive alternative: vines are graphical models based on a sequence of trees, and are based on the decomposition of a correlation matrix in terms of algebraically independent correlations and partial correlations. By limiting the number of trees, with the so-called truncation, parsimonious parameterizations of correlation matrices may be found. Moreover, truncated vines and factor models may be joined to define a combined model, with individual benefits from each of the two approaches. The different parameterizations and how they are estimated for data are discussed. In particular, spanning tree algorithms for truncated vines and a modified EM algorithm for the combined factor–vine model are proposed and evaluated in a simulation study. Three applications to psychometric and finance data sets illustrate the different parsimonious models.

Vine pair-copula constructions (PCCs) provide an important milestone for the usage of multivariate copulas to model dependence. At present time PCCs are recognized to be the most flexible class of multivariate copulas. Vine PCCs and semiparametric copula-based dynamic (SCOMDY) models with ARMA-GARCH margins are combined. As building blocks of the PCCs, bivariate t -copulas are used. Exchange rates are considered as an application and their dependence structure is modelled using regular and canonical vines. A non-nested model comparison of the above SCOMDY models is performed using the adapted Voung's test.

The dependence structure between WTI and Brent crude oil spot log-returns is analysed using modern copula techniques. In a first step, to account for autocorrelation and volatility clustering in the marginals, several single equation models are applied. Second, to select both copulas and tail copulas characterising the joint dynamics between the time series, newly introduced bootstrap-based goodness-of-fit tests are implemented and evaluated. Based on each approach, a comprehensive backtesting is performed by simulating and comparing the risk measures Value-at-Risk and Expected Shortfall with observed values.

The identification of an appropriate multivariate copula for capturing the dependence structure in multivariate data is not straightforward. The reason is because standard multivariate copulas (such as the multivariate Gaussian, Student-t, and exchangeable Archimedean copulas) lack flexibility to model dependence and have other limitations, such as parameter restrictions. To overcome these problems, vine copulas have been developed and applied to many applications. In order to reveal and fully understand the complex and hidden dependence patterns in multivariate data, a mixture of D-vine copulas is proposed incorporating D-vine copulas into a finite mixture model. As a D-vine copula has multiple parameters capturing the dependence through iterative construction of pair-copulas, the proposed model can facilitate a comprehensive study of complex and hidden dependence patterns in multivariate data. The proposed mixture of D-vine copulas is applied to simulated and real data to illustrate its performance and benefits.

It has been shown that vine copulas constructed from bivariate t copulas can provide good fits to multivariate financial asset return data. However, there might be stronger tail dependence of returns in the joint lower tail of assets than the upper tail. To this end, vine copula models with appropriate choices of bivariate reflection asymmetric linking copulas will be used to assess such tail asymmetries. Comparisons of various vine copulas are made in terms of likelihood fit and forecasting of extreme quantiles.

Statistical procedures for the equality of two and k univariate distributions based on samples of dependent observations are proposed in this work. The test statistics are L"2 distances of standard empirical and characteristic function processes. The p-values of the tests are obtained from a version of the multiplier central limit theorem whose asymptotic validity is established. Simple formulas for the test statistics and their multiplier versions in terms of multiplication of matrices are provided. Simulations under many patterns of dependence characterized by copulas show the good behavior of the tests in small samples, both in terms of their power and of their ability to keep their nominal level under the null hypothesis.

While parametric copulas often lack expressive capacity to capture the complex dependencies that are usually found in empirical data, non-parametric copulas can have poor generalization performance because of overfitting. A semiparametric copula method based on the family of bivariate Archimedean copulas is introduced as an intermediate approach that aims to provide both accurate and robust fits. The Archimedean copula is expressed in terms of a latent function that can be readily represented using a basis of natural cubic splines. The model parameters are determined by maximizing the sum of the log-likelihood and a term that penalizes non-smooth solutions. The performance of the semiparametric estimator is analyzed in experiments with simulated and real-world data, and compared to other methods for copula estimation: three parametric copula models, two semiparametric estimators of Archimedean copulas previously introduced in the literature, two flexible copula methods based on Gaussian kernels and mixtures of Gaussians and finally, standard parametric Archimedean copulas. The good overall performance of the proposed semiparametric Archimedean approach confirms the capacity of this method to capture complex dependencies in the data while avoiding overfitting.

Efficient sampling algorithms for both Archimedean and nested Archimedean copulas are presented. First, efficient sampling algorithms for the nested Archimedean families of Ali-Mikhail-Haq, Frank, and Joe are introduced. Second, a general strategy how to build a nested Archimedean copula from a given Archimedean generator is presented. Sampling this copula involves sampling an exponentially tilted stable distribution. A fast rejection algorithm is developed for the more general class of tilted Archimedean generators. It is proven that this algorithm reduces the complexity of the standard rejection algorithm to logarithmic complexity. As an application it is shown that the fast rejection algorithm outperforms existing algorithms for sampling exponentially tilted stable distributions involved, e.g., in nested Clayton copulas. Third, with the additional help of randomization of generator parameters, explicit sampling algorithms for several nested Archimedean copulas based on different Archimedean families are found. Additional results include approximations and some dependence properties, such as Kendall's tau and tail dependence parameters. The presented ideas may also apply in the more general context of sampling distributions given by their Laplace-Stieltjes transforms.

A goodness-of-fit testing procedure for Archimedean copula (AC) models is developed based on right-censored data. The proposed approach extends an existing method, which is suitable for the Clayton model, to general AC models. Asymptotic properties of the proposed test statistics under the true model assumption are derived. Simulation analysis shows that the proposed test has reasonable performance. Finally, two real data examples are analyzed for illustrative purposes.

A new semiparametric dynamic copula model is proposed where the marginals are specified as parametric GARCH-type processes, and the dependence parameter of the copula is allowed to change over time in a nonparametric way. A straightforward two-stage estimation method is given by local maximum likelihood for the dependence parameter, conditional on consistent first stage estimates of the marginals. First, the properties of the estimator are characterized in terms of bias and variance and the bandwidth selection problem is discussed. The proposed estimator attains the semiparametric efficiency bound and its superiority is demonstrated through simulations. Finally, the wide applicability of the model in financial time series is illustrated, and it is compared with traditional models based on conditional correlations.

The agglomerative hierarchical clustering of continuous variables is studied in the framework of the likelihood linkage analysis method proposed by Lerman. The similarity between variables is defined from the process comparing the empirical copula with the independence copula in the spirit of the test of independence proposed by Deheuvels. Unlike more classical similarity coefficients for variables based on rank statistics, the comparison measure considered in this work can also be sensitive to non-monotonic dependencies. As aggregation criteria, besides classical linkages, permutation-based linkages related to procedures for combining dependent p-values are considered. The performances of the corresponding clustering algorithms are compared through thorough simulations. In order to guide the choice of a partition, a natural probabilistic selection strategy, related to the use of the gap statistic in object clustering, is proposed and empirically compared with classical ordinal approaches. The resulting variable clustering procedure can be equivalently regarded as a potentially less computationally expensive alternative to more powerful tests of multivariate independence.

The effect on the estimation of the Value at Risk when dealing with multivariate portfolios when there is a misspecification both in the marginals and in the copulas is investigated. It is first shown that, when there is skewness in the data and symmetric marginals are used, the estimated elliptical (normal or t) copula correlations are negatively biased, reaching values as high as 70% of the true values. Besides, the bias almost doubles if negative correlations are considered, compared to positive correlations. As for the t copula degrees of freedom parameter, the use of wrong marginals delivers large positive biases, instead. If the dependence structure is represented by a copula which is not elliptical, e.g. the Clayton copula, the effects of marginal misspecifications on the copula parameter estimation can be rather different, depending on the sign of marginal skewness. Extensive Monte Carlo studies then show that the misspecifications in the marginal volatility equation more than offset the biases in copula parameters when VaR forecasting is of concern, small samples are considered and the data are leptokurtic. The biases in the volatility parameters are much smaller, whereas those ones in the copula parameters remain almost unchanged or even increase when the sample dimension increases. In this case, copula misspecifications do play a role for VaR estimation. However, these effects depend heavily on the sign of the dependence.

In this paper, we propose a method for constructing a new class of copulas. They are called linear B-spline copulas which are a good approximation of a given complicated copula by using finite numbers of values of this copula without the loss of some essential properties. Moreover, rigorous analysis shows that the empirical linear B-spline copulas are more effective than empirical copulas to estimate perfectly dependent copulas. For the cases of nonperfectly dependent copulas, simulations show that the empirical linear B-spline copulas also improve the empirical copulas to estimate the underlying copula structure. Furthermore, we compare the performance of parametric estimation of copulas based on the empirical copulas with that based on the empirical linear B-spline copulas by simulations. In most of the cases, the latter are better than the former.

Copulas are used to model multivariate data as they account for the dependence structure and provide a flexible representation of the multivariate distribution. A great number of copulas has been proposed with various dependence aspects. One important issue is the choice of an appropriate copula from a large set of candidate families to model the data at hand. A large number of copulas are compared via likelihood principle, showing that it is hard to recognize the true underlying copula from real data since copulas with similar dependence properties are very close together. A goodness of fit test based on Mahalanobis squared distance between original and simulated log-likelihoods through parametric bootstrap techniques is also proposed. The advantage of this approach is that it is applicable to all families of copulas.

Copulas enable to specify multivariate distributions with given marginals. Various parametric proposals were made in the literature for these quantities, mainly in the bivariate case. They can be systematically derived from multivariate distributions with known marginals, yielding e.g. the normal and the Student copulas. Alternatively, one can restrict his/her interest to a sub-family of copulas named Archimedean. They are characterized by a strictly decreasing convex function on (0,1) which tends to +~ at 0 (when strict) and which is 0 at 1. A ratio approximation of the generator and of its first derivative using B-splines is proposed and the associated parameters estimated using Markov chains Monte Carlo methods. The estimation is reasonably quick. The fitted generator is smooth and parametric. The generated chain(s) can be used to build ''credible envelopes'' for the above ratio function and derived quantities such as Kendall's tau, posterior predictive probabilities, etc. Parameters associated to parametric models for the marginals can be estimated jointly with the copula parameters. This is an interesting alternative to the popular two-step procedure which assumes that the regression parameters are fixed known quantities when it comes to copula parameter(s) estimation. A simulation study is performed to evaluate the approach. The practical utility of the method is illustrated by a basic analysis of the dependence structure underlying the diastolic and the systolic blood pressures in male subjects.

Within the context of a general bivariate distribution an intuitive method is presented in order to study the dependence structure of the two distributions. A set of points-level curve-which accumulate the same probability for a fixed quadrant is considered. This procedure provides four level curves which can be considered as the boundary of a generalization of the real interquantile interval. It is shown that the accumulated probability among the level curves depends on the dependence structure of the distribution function where the dependence structure is given by the notion of copula. Furthermore, the case when the marginal distributions are independent is investigated. This result is used to find out positive or negative dependence properties for the variables. Finally, a nonparametric test for independence with a local dependence meaning is performed and applied to different data sets.

Copulas have attracted significant attention in the recent literature for modeling multivariate observations. An important feature of copulas is that they enable us to specify the univariate marginal distributions and their joint behavior separately. The copula parameter captures the intrinsic dependence between the marginal variables and it can be estimated by parametric or semiparametric methods. For practical applications, the so called inference function for margins (IFM) method has emerged as the preferred fully parametric method because it is close to maximum likelihood (ML) in approach and is easier to implement. The purpose of this paper is to compare the ML and IFM methods with a semiparametric (SP) method that treats the univariate marginal distributions as unknown functions. In this paper, we consider the SP method proposed by Genest et al. [1995. A semiparametric estimation procedure of dependence parameters in multivariate families of distributions. Biometrika 82(3), 543-552], which has attracted considerable interest in the literature. The results of an extensive simulation study reported here show that the ML/IFM methods are nonrobust against misspecification of the marginal distributions, and that the SP method performs better than the ML and IFM methods, overall. A data example on household expenditure is used to illustrate the application of various data analytic methods for applying the SP method, and to compare and contrast the ML, IFM and SP methods. The main conclusion is that, in terms of statistical computations and data analysis, the SP method is better than ML and IFM methods when the marginal distributions are unknown which is almost always the case in practice.

An n-dimensional joint uniform distribution is defined as a distribution whose one-dimensional marginals are uniform on some interval I. This interval is taken to be [0,1] or, when more convenient [-12,12]. The specification of joint uniform distributions in a way which captures intuitive dependence structures and also enables sampling routines is considered. The question whether every n-dimensional correlation matrix can be realized by a joint uniform distribution remains open. It is known, however, that the rank correlation matrices realized by the joint normal family are sparse in the set of correlation matrices. A joint uniform distribution is obtained by specifying conditional rank correlations on a regular vine and a copula is chosen to realize the conditional bivariate distributions corresponding to the edges of the vine. In this way a distribution is sampled which corresponds exactly to the specification. The relation between conditional rank correlations on a vine and correlation matrix of corresponding distribution is complex, and depends on the copula used. Some results for the elliptical copulae are given.

S-distributions are univariate statistical distributions with four parameters. They have a simple mathematical structure yet provide excellent approximations for many traditional distributions and also contain a multitude of distributional shapes without a traditional analog. S-distributions furthermore have a number of beneficial features, for instance, in terms of data classification and scaling properties. They provide an appealing compromise between generality in data representation and logistic simplicity and have been applied in a variety of fields from applied biostatistics to survival analysis and risk assessment. Given their advantages in the single- variable case, it is desirable to extend S-distributions to several variates. This article proposes such an extension. It focuses on bivariate distributions whose marginals are S-distributions, but it is clear how more than two variates are to be addressed. The construction of bivariate S- distributions utilizes copulas, which have been developed quite rapidly in recent years. It is demonstrated here how one may generate such copulas and employ them to construct and analyze bivariate-and, by extension, multivariate-S-distributions. Particular emphasis is placed on Archimedean copulas, because they are easy to implement, yet quite flexible in fitting a variety of distributional shapes. It is illustrated that the bivariate S-distributions thus constructed have considerable flexibility. They cover a variety of marginals and a wide range of dependences between the variates and facilitate the formulation of relationships between measures of dependence and model parameters. Several examples of marginals and copulas illustrate the flexibility of bivariate S-distributions.

In recent years, the use of copulas has grown extremely fast and with it, the need for a simple and reliable method to choose the right copula family. Existing methods pose numerous difficulties and none is entirely satisfactory. We propose a Bayesian method to select the most probable copula family among a given set. The copula parameters are treated as nuisance variables, and hence do not have to be estimated. Furthermore, by a parameterization of the copula density in terms of Kendall's @t, the prior on the parameter is replaced by a prior on @t, conceptually more meaningful. The prior on @t, common to all families in the set of tested copulas, serves as a basis for their comparison. Using simulated data sets, we study the reliability of the method and observe the following: (1) the frequency of successful identification approaches 100% as the sample size increases, (2) for weakly correlated variables, larger samples are necessary for reliable identification.

A copula based statistical method for fitting joint cumulative returns between a market index and a single stock to daily data is proposed. Modifying the method of inference functions for margins (IFM method), one performs two separate maximum likelihood estimations of the univariate marginal distributions, assumed to be normal inverse gamma mixtures with kurtosis parameter equal to 6, followed by a minimization of the bivariate chi-square statistic associated to an adequate bivariate version of the usual Pearson goodness-of-fit test. The copula fitting results for daily cumulative returns between the Swiss Market Index and a stock in the index family for an approximate 1-year period are quite satisfactory. The best overall fits are obtained for the new linear Spearman copula, as well as for the Frank and Gumbel–Hougaard copulas. Finally, a significant application to covariance estimation for the linear Spearman copula is discussed.