Characterizing joint distributions of random sets by multivariate capacities

By the Choquet theorem, distributions of random closed sets can be characterized by a certain class of set functions called capacity functionals. In this paper a generalization to the multivariate case is presented, that is, it is proved that the joint distribution of finitely many random sets can be characterized by a multivariate set function being completely alternating in each component, or alternatively, by a capacity functional defined on complements of cylindrical sets. For the special case of finite spaces a multivariate version of the Moebius inversion formula is derived. Furthermore, we use this result to formulate an existence theorem for set-valued stochastic processes.     

Tail dependence for multivariate t -copulas and its monotonicity

The tail dependence indexes of a multivariate distribution describe the amount of dependence in the upper right tail or lower left tail of the distribution and can be used to analyse the dependence among extremal random events. This paper examines the tail dependence of multivariate t-distributions whose copulas are not explicitly accessible. The tractable formulas of tail dependence indexes of a multivariate t-distribution are derived in terms of the joint moments of its underlying multivariate normal distribution, and the monotonicity properties of these indexes with respect to the distribution parameters are established. Simulation results are presented to illustrate the results.     

Quasi-arithmetic means of covariance functions with potential applications to space–time data

The theory of quasi-arithmetic means represents a powerful tool in the study of covariance functions across space–time. In the present study we use quasi-arithmetic functionals to make inferences about the permissibility of averages of functions that are not, in general, permissible covariance functions. This is the case, e.g., of the geometric and harmonic averages, for which we obtain permissibility criteria. Also, some important inequalities involving covariance functions and preference relations as well as algebraic properties can be derived by means of the proposed approach. In particular, quasi-arithmetic covariances allow for ordering and preference relations, for a Jensen-type inequality and for a minimal and maximal element of their class. The general results shown in this paper are then applied to the study of spatial and spatio-temporal random fields. In particular, we discuss the representation and smoothness properties of a weakly stationary random field with a quasi-arithmetic covariance function. Also, we show that the generator of the quasi-arithmetic means can be used as a link function in order to build a space–time nonseparable structure starting from the spatial and temporal margins, a procedure that is technically sound for those working with copulas. Several examples of new families of stationary covariances obtainable with this procedure are shown. Finally, we use quasi-arithmetic functionals to generalise existing results concerning the construction of nonstationary spatial covariances, and discuss the applicability and limits of this generalisation.     

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.     

A practical non-parametric copula algorithm for system reliability with correlations

System reliability analysis involving correlated random variables is challenging because the failure probability cannot be uniquely determined under the given probability information. This paper proposes a system reliability evaluation method based on non-parametric copulas. The approximated joint probability distribution satisfying the constraints specified by correlations has the maximal relative entropy with respect to the joint probability distribution of independent random variables. Thus the reliability evaluation is unbiased from the perspective of information theory. The estimation of the non-parametric copula parameters from Pearson linear correlation, Spearman rank correlation, and Kendall rank correlation are provided, respectively. The approximated maximum entropy distribution is then integrated with the first and second order system reliability method. Four examples are adopted to illustrate the accuracy and efficiency of the proposed method. It is found that traditional system reliability method encodes excessive dependence information for correlated random variables and the estimated failure probability can be significantly biased.     

On joint weak reversed hazard rate order under symmetric copulas

In this paper, a weak version of the joint reversed hazard rate order, useful for stochastic comparison of non-independent random variables, has been defined and discussed. In particular, some relationships between the joint weak reversed hazard rate order and the usual reversed hazard rate order are established when the underlying copulas are symmetric.     

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.     

Distribution modeling for reliability analysis: Impact of multiple dependences and probability model selection

Reliability analysis requires modeling of joint probability distribution of uncertain parameters, which can be a challenge since the random variables representing the parameter uncertainties may be correlated. For convenience, a Gaussian data dependence is commonly assumed for correlated random variables. This paper first investigates the effect of multidimensional non-Gaussian data dependences underlying the multivariate probability distribution on reliability results. Using different bivariate copulas in a vine structure, various data dependences can be modeled. The associated copula parameters are identified from available statistical information by moment matching techniques. After the development of the vine copula model for representing the multivariate probability distribution, the reliability involving correlated random variables is evaluated based on the Rosenblatt transformation. The impact of data dependence is significant because a large deviation in failure probability is observed, which emphasizes the need for accurate dependence characterization. A practical method for dependence modeling based on limited data is thus provided. The result demonstrates that the non-Gaussian data dependences can be real in practice, and the reliability can be biased if the Gaussian dependence is used inappropriately. Moreover, the effect of conditioning order on reliability should not be overlooked except that the vine structure contains only one type of copula.     

Convex combinations of quadrant dependent copulas

It is well known that quadrant dependent (QD) random variables are also quadrant dependent in expectation (QDE). Recent literature has offered examples rigorously establishing the fact that there are QDE random variables which are not QD. The examples are based on convex combinations of specially chosen QD copulas: one negatively QD and another positively QD. In this paper we establish general results that determine when convex combinations of arbitrary QD copulas give rise to negatively or positively QD/QDE copulas. In addition to being an interesting mathematical exercise, the established results are helpful when modeling insurance and financial portfolios.     

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.     

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.     

Construction of asymmetric copulas and its application in two-dimensional reliability modelling

Copulas offer a useful tool in modelling the dependence among random variables. In the literature, most of the existing copulas are symmetric while data collected from the real world may exhibit asymmetric nature. This necessitates developing asymmetric copulas that can model such data. In the meantime, existing methods of modelling two-dimensional reliability data are not able to capture the tail dependence that exists between the pair of age and usage, which are the two dimensions designated to describe product life. This paper proposes a new method of constructing asymmetric copulas, discusses the properties of the new copulas, and applies the method to fit two-dimensional reliability data that are collected from the real world.     

On the stable containment of two sets

This paper studies the stability of the set containment problem. Given two non-empty sets in the Euclidean space which are the solution sets of two systems of (possibly infinite) inequalities, the Farkas type results allow to decide whether one of the two sets is contained or not in the other one (which constitutes the so-called containment problem). In those situations where the data (i.e., the constraints) can be affected by some kind of perturbations, the problem consists of determining whether the relative position of the two sets is preserved by sufficiently small perturbations or not. This paper deals with this stability problem as a particular case of the maintaining of the relative position of the images of two set-valued mappings; first for general set-valued mappings and second for solution sets mappings of convex and linear systems. Thus the results in this paper could be useful in the postoptimal analysis of optimization problems with inclusion constraints.      

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.     

A Characterization of Quasi-copulas

The notion of quasi-copula was introduced by C. Alsina, R. B. Nelsen, and B. Schweizer (Statist. Probab. Lett.(1993), 85–89) and was used by these authors and others to characterize operations on distribution functions that can or cannot be derived from operations on random variables. In this paper, the concept of quasi-copula is characterized in simpler operational terms and the result is used to show that absolutely continuous quasi-copulas are not necessarily copulas, thereby answering in the negative an open question of the above mentioned authors.     

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.     

Joint cumulative distribution functions for Dempster–Shafer belief structures using copulas

We first introduce the Dempster–Shafer belief structure and highlight its role in the representation of information about a random variable for which our knowledge of the probabilities is interval-valued. We investigate the formation of the cumulative distribution function (CDF) for these types of variables. It is noted that this is also interval-valued and is expressible in terms of plausibility and belief measures. The class of aggregation operators known as copulas are introduced and a number of their properties are provided. We discuss Sklar’s theorem, which provides for the use of copulas in the formulation of joint CDFs from the marginal CDFs of classic random variables. We then look to extend these ideas to the case of joining the marginal CDFs associated with Dempster–Shafer belief structures. Finally we look at the formulation CDFs obtained from functions of multiple D–S belief structures.     

Uncorrelatedness sets of bounded random variables

An uncorrelatedness set of two random variables shows which powers of random variables are uncorrelated. These sets provide a measure of independence: the wider an uncorrelatedness set is, the more independent random variables are. Conditions for a subset of to be an uncorrelatedness set of bounded random variables are studied. Applications to the theory of copulas are given.     

Lower semicontinuous functionals with uniform sublevel sets

In this paper, lower semicontinuous functionals with uniform sublevel sets are investigated, where the sublevel sets are linear shifts of a set in a fixed direction. The extended real-valued functionals are defined on a topological vector space. Conditions are given under which they are proper, finite-valued, continuous, convex, sublinear, strictly quasi-convex, strictly quasi-concave or monotone. We apply the functionals to the separation of not necessarily convex sets.     

On approximating max-stable processes and constructing extremal copula functions

There is an infinite number of parameters in the definition of multivariate maxima of moving maxima (M4) processes, which poses challenges in statistical applications where workable models are preferred. This paper establishes sufficient conditions under which an M4 process with infinite number of parameters may be approximated by an M4 process with finite number of parameters. In statistical inferences, the paper focuses on a family of sectional multivariate extreme value copula (SMEVC) functions which is derived from the joint distribution functions of M4 processes. A new non-standard parameter estimation procedure is introduced, which is based on order statistics of ratios of (transformed) marginal unit Fréchet random variables, and is shown via simulation to be more efficient than a semi-parametric estimation procedure. In real data analysis, empirical results show that SMEVCs are more flexible for modeling various dependence structures, and perform better than the widely used Gumbel-Hougaard copulas.