Asymmetry of copulas arising from shock models

Journal of Computational and Applied Mathematics

Copula is a useful tool that captures the dependence structure among random variables. In practice, it is an important question which copula to choose depending on the given data and stochastic assumptions on the model in order to achieve an appropriate interpretation of the data at hand. This paper intends to help a practitioner to make a better decision about that. We concentrate on the study of the lack of exchangeability, a copulas' attribute closely studied only recently. The main non-exchangeability measure µ ∞ for a family of copulas is the supremum of the differences |C(x, y) − C(y, x)| over all (x, y) and all copulas C in the family. We give the sharp bound of µ ∞ for the families of Marshall copulas, maxmin and reflected maxmin copulas (i.e. the main shock-model based copulas) as well as the families of positively and of negatively quadrant dependent copulas. A major contribution of this paper is also exact calculation of the maximal asymmetry function on each of the particular families of copulas. When restricted to special families of copulas considered, it helps us finding the sharp bound of µ ∞ for each of the given families. And even more importantly, it helps us giving a stochastic interpretation of the extremal copulas and examples of shock models where the maximal asymmetry is attained.

Fuzzy Sets and Systems, 2022

Bivariate imprecise copulas have recently attracted substantial attention. However, the multivariate case seems still to be a "blank slate". It is then natural that this idea be tested first on shock model induced copulas, a family which might be the most useful in various applications. We investigate a model in which some of the shocks are assumed imprecise and develop the corresponding set of copulas. In the Marshall's case we get a coherent set of distributions and a coherent set of copulas, where the bounds are naturally corresponding to each other. The situation with the other two groups of multivariate imprecise shock model induced copulas, i.e., the maxmin and the the reflected maxmin (RMM) copulas, is substantially more involved, but we are still able to produce their properties. These are the main results of the paper that serves as the first step into a theory that should develop in this direction. In addition, we unfold the theory of bivariate imprecise RMM copulas that has not yet been done before.

International Journal of Approximate Reasoning

The omnipotence of copulas when modeling dependence given marginal distributions in a multivariate stochastic situation is assured by the Sklar's theorem. Montes et al. (2015) suggest the notion of what they call an imprecise copula that brings some of its power in bivariate case to the imprecise setting. When there is imprecision about the marginals, one can model the available information by means of p-boxes, that are pairs of ordered distribution functions. By analogy they introduce pairs of bivariate functions satisfying certain conditions. In this paper we introduce the imprecise versions of some classes of copulas emerging from shock models that are important in applications. The so obtained pairs of functions are not only imprecise copulas but satisfy an even stronger condition. The fact that this condition really is stronger is shown in Omladič and Stopar (2019) thus raising the importance of our results. The main technical difficulty

Fuzzy Sets and Systems, 2019

Copula models have become popular in different applications, including modeling shocks, in view of their ability to describe better the dependence concepts in stochastic systems. The class of maxmin copulas was recently introduced by Omladič and Ružić [21]. It extends the well known classes of Marshall-Olkin and Marshall copulas by allowing the external shocks to have different effects on the two components of the system. By a reflection (flip) in one of the variables we introduce a new class of bivariate copulas called reflected maxmin (RMM) copulas. We explore their properties and show that symmetric RMM copulas relate to general RMM copulas similarly as semilinear copulas relate to Marshall copulas. We transfer that relation also to maxmin copulas. We also characterize possible diagonal functions of symmetric RMM copulas.

Fuzzy Sets and Systems

In this paper we introduce some new copulas emerging from shock models. It was shown in [21] that reflected maxmin copulas (RMM for short) are not just some specific singular copulas; they contain many important absolutely continuous copulas including the negative quadrant dependent "half" of the Eyraud-Farlie-Gumbel-Morgenstern class. The main goal of this paper is to develop the RMM copulas with dependent endogenous shocks and give evidence that RMM copulas may exhibit some characteristics better than the original maxmin copulas (MM for short): (1) An important evidence for that is the iteration procedure of the RMM transformation which we prove to be always convergent and we give many properties of it that are useful in applications. (2) Using this result we find also the limit of the iteration procedure of the MM transformation thus answering a question proposed in [10]. (3) We give the multivariate dependent RMM copula that compares to the MM version given in [10]. In all our copulas the idiosyncratic and systemic shocks are combined via asymmetric linking functions as opposed to Marshall copulas where symmetric linking functions are used.

2018

Copula models have become popular in different applications, including modeling shocks, in view of their ability to describe better the dependence concepts in stochastic systems. The class of maxmin copulas was recently introduced by Omladič and Ružić. It extends the well known classes of Marshall-Olkin and Marshall copulas by allowing the external shocks to have different effects on the two components of the system. By a reflection (flip) in one of the variables we introduce a new class of bivariate copulas called reflected maxmin (RMM) copulas. We explore their properties and show that symmetric RMM copulas relate to general RMM copulas similarly as do semilinear copulas relate to Marshall copulas. We transfer that relation also to maxmin copulas. We also characterize possible diagonal functions of symmetric RMM copulas.

Risks

This paper explores the properties of a family of bivariate copulas based on a new approach using the counter-monotonic shock method. The resulting copula covers the full range of negative dependence induced by one parameter. Expressions for the copula and density are derived and many theoretical properties are examined thoroughly, including explicit expressions for prominent measures of dependence, namely Spearman’s rho, Kendall’s tau and Blomqvist’s beta. The convexity properties of this copula are presented, together with explicit expressions of the mixed moments. Estimation of the dependence parameter using the method of moments is considered, then a simulation study is carried out to evaluate the performance of the suggested estimator. Finally, an application of the proposed copula is illustrated by means of a real data set on air quality in New York City.

Fuzzy Sets and Systems, 2013

In 2004, Rodríguez-Lallena and Úbeda-Flores have introduced a class of bivariate copulas which generalizes some known families such as the Farlie-Gumbel-Morgenstern distributions. In 2006, Dolati and Úbeda-Flores presented multivariate generalizations of this class, also they investigated symmetry, dependence concepts and measuring the dependence among the components of each classes. In this paper, a new method of constructing binary copulas is introduced, extending the original study of Rodríguez-Lallena and Úbeda-Flores to new families of symmetric/asymmetric copulas. Several properties and parameters of newly introduced copulas are included. Among others, relationship of our construction method with several kinds of ordinal sums of copulas is clarified.

2015

Under a mild condition we give closed-form expressions for copulas of systems that consist of maxima and of minima of subvectors of a given random vector X with continuous marginals. Said expressions appear explicit in the copula of X and the mentioned condition is for example met when the law of X admits a strictly positive density with respect to Lebesgue measure. In the i.i.d. case these "maxmin" copulae become universal and the conditions on their validity can be dropped entirely. Our main motivation comes from applications to shock models that arise in multivariate survival theory, and indeed the maxmin copulas presented herein are connected to/extend the Marshall-Olkin copulas [9] going through to the copulas given in [12]. Another application is to order statistics copulas.

Stochastic Models, 2006

In this review paper we outline some recent contributions to copula theory. Several new author's investigations are presented brie°y, namely: order statistics copula, copulas with given multivariate marginals, copula representation via a local dependence measure and applications of extreme value copulas.