Semiparametric bivariate Archimedean copulas

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.     

Efficiently sampling nested Archimedean copulas

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.     

GeD spline estimation of multivariate Archimedean copulas

A new multivariate Archimedean copula estimation method is proposed in a non-parametric setting. The method uses the so-called Geometrically Designed splines (GeD splines) to represent the cdf of a random variable W_θ, obtained through the probability integral transform of an Archimedean copula with parameter θ. Sufficient conditions for the GeD spline estimator to possess the properties of the underlying theoretical cdf, K(θ,t), of W_θ, are given. The latter conditions allow for defining a three-step estimation procedure for solving the resulting non-linear regression problem with linear inequality constraints. In the proposed procedure, finding the number and location of the knots and the coefficients of the unconstrained GeD spline estimator and solving the constraint least-squares optimisation problem are separated. Thus, the resulting spline estimator is used to recover the generator and the related Archimedean copula by solving an ordinary differential equation. The proposed method is truly multivariate, it brings about numerical efficiency and as a result can be applied with large volumes of data and for dimensions d≥2, as illustrated by the numerical examples presented.     

Linear B-spline copulas with applications to nonparametric estimation of copulas

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.     

On copulas and their diagonals

We formulate the necessary and sufficient conditions for a function δ:[0,1]→[0,1] to be the diagonal section of a multivariate absolutely continuous copula. Moreover we provide some simple analytic formulas for copulas having given diagonal section or given distribution functions of order statistics.     

Construction of bivariate S-distributions with copulas

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.     

New constructions of diagonal patchwork copulas

We present a method for constructing symmetric copulas which generalizes the diagonal patchwork construction of copulas procedure. We also show how it is related to a new construction of a generalized Farlie-Gumbel-Morgenstern distribution and to the copula transforms.     

Quasi-copulas and copulas on a discrete scale

In the paper the structure of quasi-copulas and copulas on a finite discrete scale is studied. The possibility of construction of quasi-copulas (copulas) from given values at diagonal points is investigated. Moreover, the problem of the uniqueness of the existence of a quasi–copula (copula) with given diagonal section is solved. Several examples are included.     

Bayesian spatial modeling and interpolation using copulas

Classical Bayesian spatial interpolation methods are based on the Gaussian assumption and therefore lead to unreliable results when applied to extreme valued data. Specifically, they give wrong estimates of the prediction uncertainty. Copulas have recently attracted much attention in spatial statistics and are used as a flexible alternative to traditional methods for non-Gaussian spatial modeling and interpolation. We adopt this methodology and show how it can be incorporated in a Bayesian framework by assigning priors to all model parameters. In the absence of simple analytical expressions for the joint posterior distribution we propose a Metropolis-Hastings algorithm to obtain posterior samples. The posterior predictive density is approximated by averaging the plug-in predictive densities. Furthermore, we discuss the deficiencies of the existing spatial copula models with regard to modeling extreme events. It is shown that the non-Gaussian χ²-copula model suffers from the same lack of tail dependence as the Gaussian copula and thus offers no advantage over the latter with respect to modeling extremes. We illustrate the proposed methodology by analyzing a dataset here referred to as the Helicopter dataset, which includes strongly skewed radioactivity measurements in the city of Oranienburg, Germany.     

On copulas, quasicopulas and fuzzy logic

We investigate (quasi)copulas as possible truth functions of fuzzy conjunction which is not necessarily associative and present some axiom systems for such fuzzy logics. In particular, we study an expansion of Łukasiewicz (infinite valued propositional) logic by a new connective interpreted as an arbitrary quasicopula (and also by a new connective interpreted as the residuum of the copula). Main results concern standard completeness.     

Archimedean copula estimation using Bayesian splines smoothing techniques

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.     

Conditional copulas, association measures and their applications

One way to model a dependence structure is through the copula function which is a mean to capture the dependence structure in the joint distribution of variables. Association measures such as Kendall’s tau or Spearman’s rho can be expressed as functionals of the copula. The dependence structure between two variables can be highly influenced by a covariate, and it is of real interest to know how this dependence structure changes with the value taken by the covariate. This motivates the need for introducing conditional copulas, and the associated conditional Kendall’s tau and Spearman’s rho association measures. After the introduction and motivation of these concepts, two nonparametric estimators for a conditional copula are proposed and discussed. Then nonparametric estimates for the conditional association measures are derived. A key issue is that these measures are now looked at as functions in the covariate. The performances of all estimators are investigated via a simulation study which also includes a data-driven algorithm for choosing the smoothing parameters. The usefulness of the methods is illustrated on two real data examples.     

Comparison of semiparametric and parametric methods for estimating copulas

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.     

Semiparametric multivariate density estimation for positive data using copulas

The estimation of density functions for positive multivariate data is discussed. The proposed approach is semiparametric. The estimator combines gamma kernels or local linear kernels, also called boundary kernels, for the estimation of the marginal densities with parametric copulas to model the dependence. This semiparametric approach is robust both to the well-known boundary bias problem and the curse of dimensionality problem. Mean integrated squared error properties, including the rate of convergence, the uniform strong consistency and the asymptotic normality are derived. A simulation study investigates the finite sample performance of the estimator. The proposed estimator performs very well, also for data without boundary bias problems. For bandwidths choice in practice, the univariate least squares cross validation method for the bandwidth of the marginal density estimators is investigated. Applications in the field of finance are provided.     

Efficiently sampling exchangeable Cuadras-Augé copulas in high dimensions

An n-dimensional random vector is constructed whose survival copula is given by a copula that was first presented in Cuadras and Augé [C.M. Cuadras, J. Augé, A continuous general multivariate distribution and its properties, Communications in Statistics - Theory and Methods 10 (4) (1981) 339-353]. This construction adds a Poisson subordinator as mixing variable to initially independent exponentially distributed random variables. It is shown how the choice of Poisson process relates to the parameter of the induced Cuadras-Augé copula. Based on this construction, a sampling algorithm for this multivariate distribution is presented which has average computational efficiency O(nloglogn).     

On a family of multivariate copulas for aggregation processes

We introduce a family of multivariate copulas - a special type of n-ary aggregation operations - depending on a univariate function. This family is used in the construction of a special aggregation operation that satisfies a Lipschitz condition. Several examples are provided and some statistical properties are studied.     

频谱无混叠采样和信号完全可重构采样

通过对采样定理的分析,指出频谱无混叠采样和信号可完全重构采样并不等同,并给出和证明了实现这两个采样的条件,从而全面地回答了在何种条件下,经典采样定理中的最低采样频率可以等于信号最高频率的二倍.作为最低采样频率问题的最典型例子,本文对正弦信号的采样和重构问题作了进一步的分析,揭示了二倍信号频率的采样导致正弦信号相位信息丢失的原因.本文对如何在应用中正确地确定最低采样频率,提供了有益的参考.