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A distribution function F is a generalized distorted distribution of the distribution functions $$F_1,\ldots ,F_n$$ if $$F=Q(F_1,\ldots ,F_n)$$ for an increasing continuous distortion function Q such that $$Q(0,\ldots ,0)=0$$ and $$Q(1,\ldots ,1)=1$$ . In this paper, necessary and sufficient conditions for the stochastic (ST) and the hazard rate (HR) orderings of generalized distorted distributions are provided when the distributions $$F_1,\ldots ,F_n$$ are ordered. These results are used to obtain distribution-free ordering properties for coherent systems with heterogeneous components. In particular, we determine all the ST and HR orderings for coherent systems with 1-3 independent components. We also compare systems with dependent components. The results on distorted distributions are also used to get comparisons of finite mixtures. [ABSTRACT FROM AUTHOR]

The copula representations for conditionally independent random variables and the distribution properties of order statistics of these random variables are studied. [ABSTRACT FROM AUTHOR]

We derive best-possible bounds on the class of copulas with known values at several points, under the assumption that the points are either in “increasing order” or in “decreasing order”. These bounds may be used to establish best-possible bounds on Kendall's τ and Spearman's ρ, for such copulas. An important special case is when the values of a copula are known at several diagonal points. We also use our results to establish best-possible bounds on the distribution function of the sum of two random variables with known marginal distributions when the values of the joint distribution function are known at several points. [ABSTRACT FROM AUTHOR]

In this study, we show an application of Archimedean copulas for meteorological data, which may be of interest in its own right. Hence we take monthly lowest and highest air temperature records from 1951 to 2005 in Tehran, and then investigate a suitable family of Archimedean copulas to fit this data. Also characterize the interval of suitable, and also the best copula parameter. Moreover basic properties of Archimedean copulas will be presented. [ABSTRACT FROM AUTHOR]

Recently, Liebscher [Construction of asymetric multivariate copulas, J. Multivariate Anal. 99 (2008), pp. 2234–2250] introduced a general construction scheme of d-variate copulas which generalizes the Archimedean family. Similarly, Morillas [A method to obtain new copulas from a given one, Metrika 61 (2005), pp. 169–184] proposed a method to obtain a variety of new copulas from a given d-copula. Both approaches coincide only for the particular subclass of Archimedean copulas. Within this work, we present a unifying framework which includes both Liebscher and Morillas copulas as special cases. Above that, more general copulas may be constructed. First examples are given. [ABSTRACT FROM AUTHOR]

Summary The goal of mediation analysis is to assess direct and indirect effects of a treatment or exposure on an outcome. More generally, we may be interested in the context of a causal model as characterized by a directed acyclic graph (DAG), where mediation via a specific path from exposure to outcome may involve an arbitrary number of links (or 'stages'). Methods for estimating mediation (or pathway) effects are available for a continuous outcome and a continuous mediator related via a linear model, while for a categorical outcome or categorical mediator, methods are usually limited to two-stage mediation. We present a method applicable to multiple stages of mediation and mixed variable types using generalized linear models. We define pathway effects using a potential outcomes framework and present a general formula that provides the effect of exposure through any specified pathway. Some pathway effects are nonidentifiable and their estimation requires an assumption regarding the correlation between counterfactuals. We provide a sensitivity analysis to assess the impact of this assumption. Confidence intervals for pathway effect estimates are obtained via a bootstrap method. The method is applied to a cohort study of dental caries in very low birth weight adolescents. A simulation study demonstrates low bias of pathway effect estimators and close-to-nominal coverage rates of confidence intervals. We also find low sensitivity to the counterfactual correlation in most scenarios. [ABSTRACT FROM AUTHOR]

Multivariate count data occur in several different disciplines. However, existing models do not offer great flexibility for dependence modeling. Models based on copulas nowadays are widely used for continuous data dependence modeling. Modeling count data via copulas is still in its infancy; see the recent article of Genest and Neslehova (2007). A series of different copula models providing various residual dependence structures are considered for vectors of count response variables whose marginal distributions depend on covariates through negative binomial regressions. A real data application related to the number of purchases of different products is provided. [ABSTRACT FROM AUTHOR]

Abstract: We describe several analytical and numerical procedures to obtain bounds on the distribution function of a sum of dependent risks having fixed overlapping marginals. As an application, we produce bounds on quantile-based risk measures for portfolios of financial and actuarial interest. [Copyright &y& Elsevier]

The article presents a model which yields maximum likelihood (ML) estimates of regression parameters when the response is discrete and spatially dependent. This proposed model employs a spatial Gaussian copula, bringing the discrete distribution into the Gaussian geostatistical framework, where correlation completely describes dependence. It is used to estimate the relationship between Japanese beetle grub counts and soil organic matter.

The limiting lower-tail dependence copula (LLTDC) is defined as the copula of random variables which are right-truncated at thresholds tending to their left endpoints. This article shows LLTDCs are truncation-invariant and belong to the Ahmadi-Clayton family. Accordingly, it follows that limiting upper-tail dependence copulas are members of the survival Ahmadi-Clayton family. [ABSTRACT FROM AUTHOR]

In this paper, we study opposite diagonal sections of quasi-copulas and copulas. The best-possible upper bound for the set of copulas with a given opposite diagonal section being known, we focus on the best-possible lower bound, which in general is a quasi-copula. Moreover, it exhibits an interesting type of bivariate symmetry called opposite symmetry. [ABSTRACT FROM AUTHOR]

We consider semiparametric multivariate data models based on copula representation of the common distribution function. A copula is characterized by a parameter of association and marginal distribution functions. This parameter and the marginal distributions are unknown. In this article, we study the estimator of the parameter of association in copulas with the marginal distribution functions assumed as nuisance parameters restricted by the assumption that the components are identically distributed. Results of this work could be used to construct special kinds of tests of homogeneity for random vectors having dependent components. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we prove that any Kendall distribution function is the Kendall distribution function of some associative copula. We use this result to show that each equivalence class of the relation “to have the same Kendall distribution function as” defined on the set of copulas contains a unique associative representative. [Copyright &y& Elsevier]

We present a new way of constructing n-copulas, by scaling and gluing finitely many n-copulas. Gluing for bivariate copulas produces a copula that coincides with the independence copula on some grid of horizontal and vertical sections. Examples illustrate how gluing can be applied to build complicated copulas from simple ones. Finally, we investigate the analytical as well as statistical properties of the copulas obtained by gluing, in particular, the behavior of Spearman's ρ and Kendall's τ. [ABSTRACT FROM AUTHOR]

Copulas and frailty models are important tools to model bivariate survival data. Equivalence between Archimedean copula models and shared frailty models, e.g. between the Clayton-Oakes copula model and the shared gamma frailty model, has often been claimed in the literature. In this note we show that, in both the models, there is indeed a well-known equivalence between the copula functions; the modeling of the marginal survival functions, however, is quite different. The latter fact leads to different joint survival functions. [ABSTRACT FROM AUTHOR]

The article presents information on the finding of goodness-of-fit method in order to test the parametric specification for the function of Archimedean copulas. It informs that copulas are an important tool in multivariate statistics and they capture the pure dependence structure between random variables irrespective of their marginal distributions. Random number generation from Archimedean copulas is stated to be easy because of their simple structure. It is stated that if the dependence structure of the return vector can accurately be described by a parametric Archimedean copula, one could aim at developing credit portfolio models that capture the behavior of joint returns more realistically.

Not only are copula functions joint distribution functions in their own right, they also provide a link between multivariate distributions and their lower-dimensional marginal distributions. Copulas have a structure that allows us to characterize all possible multivariate distributions, and therefore they have the potential to be a very useful statistical tool. Although copulas can be traced back to 1959, there is still much scope for new results, as most of the early work was theoretical rather than practical. We focus on simple practical tools based on conditional expectation, because such tools are not widely available. When dealing with data sets in which the dependence throughout the sample is variable, we suggest that copula-based regression curves may be more accurate predictors of specific outcomes than linear models. We derive simple conditional expectation formulae in terms of copulas and apply them to a combination of simulated and real data. [ABSTRACT FROM AUTHOR]

A family of copulas, called semilinear, is constructed starting with some assumptions about the linearity of the copulas along some segments of the unit square. This family contains some other known families of copulas (e.g., Cuadras–Augé, Fréchet) and has a nice statistical interpretation. Several construction methods are provided, especially concerning aggregation of semilinear copulas, and a special form of ordinal sum construction is introduced. Some results about related families of quasi-copulas and semicopulas are hence given. [Copyright &y& Elsevier]

We present new tests of marginal independence for d-valued random vectors. Our tests rely upon weighted Cramér-von Mises-type statistics, which are functionals of the empirical copula process based upon a random sample of size n. We establish a decomposition of this process into asymptotically independent components, and describe the tests which follow from these arguments. [ABSTRACT FROM AUTHOR]

We generalize the extreme value analysis for Archimedean copulas (seeAlink,Löwe andWüthrich, 2003) to the non-Archimedean case: Assume we have d≥2 exchangeable and continuously distributed risks X1,..., X d. Under appropriate assumptions there is a constant q d such that, for all large u, we have . The constant q d describes the asymptotic dependence structure. Typically, q d will depend on more aspects of this dependence structure than the well-known tail dependence coefficient. [ABSTRACT FROM AUTHOR]

Multivariate exchangeable Archimedean copulas are one of the most popular classes of copulas that are used in actuarial science and finance for modeling risk dependencies and for using them to quantify the magnitude of tail dependence. Owing to the increase in popularity of copulas to measure dependent risks, generating from multivariate copulas has become a very crucial exercise. Current methods for generating from multivariate Archimedean copulas could become a very difficult task as the number of dimension increases. The resulting analytical procedures suggested in the existing literature do not offer much guidance for practical implementation. This article presents an algorithm for generating values from the multivariate exchangeable Archimedean copulas based on a multivariate extension of a bivariate result. A procedure for generating values from bivariate Archimedean copulas has been originally proposed in Genest and Rivest (1993) and again later described in Nelsen (1999) and Embrechts et al. (2001a, b). Using a proof that is simply based on fundamental Jacobian techniques for deriving distributions of transformed random variables, we are able to extend the bivariate result into the multivariate case allowing us to develop an interesting algorithm to generate values from exchangeable Archimedean copulas. As auxiliary results, we are able to derive the distribution function of an n-dimensional Archimedean copula, a result already known in Genest and Rivest (2001), but our approach of proving this result is based on a different perspective. This article focuses on this class of copulas that has one generating function and one parameter that characterizes the dependence structure of the joint distribution function. [ABSTRACT FROM AUTHOR]

In multivariate and multi-parameter contexts, new expressions for Fisher Information are derived using the copula representation of the joint distribution of random variables. Invariance of Fisher Information to margins of the joint distribution is then demonstrated. [ABSTRACT FROM AUTHOR]

Abstract: This paper concerns the application of copula functions in VaR valuation. The copula function is used to model the dependence structure of multivariate assets. After the introduction of the traditional Monte Carlo simulation method and the pure copula method we present a new algorithm based on mixture copula functions and the dependence measure, Spearman’s rho. This new method is used to simulate daily returns of two stock market indices in China, Shanghai Stock Composite Index and Shenzhen Stock Composite Index, and then empirically calculate six risk measures including VaR and conditional VaR. The results are compared with those derived from the traditional Monte Carlo method and the pure copula method. From the comparison we show that the dependence structure between asset returns plays a more important role in valuating risk measures comparing with the form of marginal distributions. [Copyright &y& Elsevier]

A new class of copulas, depending on a univariate function, is studied and its properties (dependence, ordering, measures of association, symmetry) are investigated. [ABSTRACT FROM AUTHOR]

Un estimateur copulo-graphique pour la fonction de survie conditionnelle en présence de censure dépendante: Rivest & Wells (2001) ont montré que dans les situations où la dépendance entre une durée de vie et une variable de censure peut être modéliste par une copule archimédienne donnée, I'estimateur co-pulo-graphique de Zheng & Klein (1995) a alors une forme explicite. Les auteurs étendent ces travaux au cas régressif avec plan fixe. Ils montrent que l'estimateur copulo-graphique admet alors une représentation asymptotique et une limite gaussienne. Ils étudient en outre l'effet d'une mauvaise spécification de la copule sur le comportement de l'estimateur. Leurs propos sont illustrés au moyen d'un jeu de données portant sur la survie du flétan de l'Atlantique. [ABSTRACT FROM AUTHOR]

Data on ozone concentrations at four monitoring sites in central Canada are examined. Univariate extreme value methods do not allow for the required inference, and multivariate methods exploiting the joint dependence of the data are necessary. Families of multivariate extreme-value copulas with a flexible dependence structure and closed-form cumulative distribution functions have recently been derived. This article shows that maximum likelihood estimates of these parametric models may be obtained numerically and that their use can lead to increased efficiency in the estimation of margins. We do not, however, have any assurance that all data are well modeled by these distributions, because the space of multivariate extreme-value copulas is infinite-dimensional, and data on ozone levels are prone to outliers. This article addresses the robust methods required for proper analysis. [ABSTRACT FROM AUTHOR]

If H denotes the joint distribution function of n random variables X1, X2, …, Xn whose margins are F1, F2, …, Fn, respectively, then the fundamental best-possible bounds inequality for H is max(0, Σi=1n Fi(xi) - n + 1) ≤ H(x1,x2, …,xn) ≤ min (F1(x1), F2(x2), …,Fn(xn)) for all x1, x2, …,xn in [-∞, ∞]. In this paper we employ n-copulas and n-quasi-copulas to find similar bounds on arbitrary sets of multivariate distribution functions with given margins. We discuss bounds for an n-quasi-copula Q when a value of Q at a single point is known. As an application, we investigate about bounds for a multivariate distribution function H with given univariate margins when the value of H is known at a single point whose coordinates are percentiles of the variables X1, X2, …,Xn, respectively. [ABSTRACT FROM AUTHOR]

We present a method which reduces the Kolmogorov goodness-of-test of independence to the Kolmogorov-Smimov one sample test. The null distribution of test statistic is the same as the Kolmogorov-Smimov test statistic in this test of independence. [ABSTRACT FROM AUTHOR]

A bivariate distribution function H(x, y) with marginals F(x) and G(y) is said to be generated by an Archimedean copula if it can be expressed in the form H(x, y) = φ-1[φ{F(x)} + φ{G(y)}] for some convex, decreasing function φ defined on (0, 1] in such a way that φ(1) = 0. Many well-known systems of bivariate distributions belong to this class, including those of Gumbel, Ali-Mikhail-Haq-Thélot, Clayton, Frank, and Hougaard. Frailty models also fall under that general prescription. This article examines the problem of selecting an Archimedean copula providing a suitable representation of the dependence structure between two variates X and Y in the light of a random sample (X1, Y1), ,(Xn, Yn) The key to the estimation procedure is a one-dimensional empirical distribution function that can be constructed whether the uniform representation of X and Y is Archimedean or not, and independently of their marginals. This semiparametric estimator, based on a decomposition of Kendall's tau statistic, is seen to be √n-consistent, and an explicit formula for its asymptotic variance is provided. This leads to a strategy for selecting the parametric family of Archimedean copulas that provides the best possible fit to a given set of data. To illustrate these procedures, a uranium exploration data set is reanalyzed. Although the presentation is restricted to problems revolving a random sample from a bivariate distribution, extensions to situations involving multivariate or censored data could be envisaged. [ABSTRACT FROM AUTHOR]

The article presents the author's comments on the paper "Model Selection and Semiparametric Interference for Bivariate Failure-Time Data," by Weijing Wang and Martin T. Wells. Wang and Wells have presented an interesting method for selecting the Archimedean copula (AC) model for bivariate failure time data in the presence of right cursoring. The intent of these comments is to call attention to the issue of whether one should standardize statistics in model selection procedures when nuisance parameters are estimated. The model selection procedure proposed by Wang and Wells is but one example in which this issue arises. This situation occurs in other statistical models, such as in parametric model discrimination with right-censored data where the nuisance parameters are the parametric model parameters and the discrimination statistics are based on generalized residuals. The question of whether or not to standardize the statistics before using them for selecting the model, which is an analogous problem to that encountered by Wang and Wells, arises. It is thus of prime interest to further study this aspect more rigorously and substantively so that a definitive resolution on whether or not to standardize can be made.

The dependence between random variables may be measured by mutual information. However, the estimation of mutual information is difficult since the estimation of the joint probability density function (PDF) of non-Gaussian distributed data is a hard problem. Copulas offer a natural approach for estimating mutual information, since the joint probability density function of random variables can be expressed as the product of the associated copula density function and marginal PDFs. The experiment demonstrates that the proposed copulas-based mutual information is much more accurate than conventional methods such as the joint histogram and Parzen window based mutual information that are widely used in image processing. [ABSTRACT FROM AUTHOR]

Copulas offer interesting insights into the dependence structures between the distributions of random variables. This report introduces new copulas, and provides an analysis for copulas, associated with bivariate exponential and Rayleigh distributions that have relevance to signal processing. [ABSTRACT FROM AUTHOR]