71 research outputs found

    Constructing copulas from shock models with imprecise distributions

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    The omnipotence of copulas when modeling dependence given marg\-inal 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 \emph{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 pp-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 Omladi\v{c} and Stopar (2019) thus raising the importance of our results. The main technical difficulty in developing our imprecise copulas lies in introducing an appropriate stochastic order on these bivariate objects

    Using rotations to build non symmetric extensions of Amblard-Girard copulas

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    A copula is a function that completely describes the dependence structure between the marginal distributions. One of the most important para-metric family of copulas is the Farlie-Gumbel-Morgenstern (FGM) family. In practical applications this copula has been shown to be somewhat limited and a symmetric extension of this family, known as the Amblard-Girard copula, has been introduced. Basing on rotations, we propose a new non symmetric extension of this family

    Survival Amblard-Girard copulas

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    A copula is a function that completely describes the dependence structure between the marginal distributions. One of the most important para-metric family of copulas is the Farlie-Gumbel-Morgenstern (FGM) family. In practical applications this copula has been shown to be somewhat limited and a symmetric extension of this family, known as the Amblard-Girard copula, has been introduced. The goal of this note is to prove that the survival copula associated with an Amblard-Girard copula still is an Amblard-Girard copula

    An extension of FGM distributions based on an univariate function

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    A copula is a function that completely describes the dependence structure between the marginal distributions. One of the most important para-metric family of copulas is the Farlie-Gumbel-Morgenstern (FGM) family. In practical applications this copula has been shown to be somewhat limited. We propose a new extension of this family based on the introduction of an univariate function

    Exchangeable min-id sequences: Characterization, exponent measures and non-decreasing id-processes

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    We establish a correspondence between exchangeable sequences of random variables whose finite-dimensional distributions are min- (or max-) infinitely divisible and non-negative, non-decreasing, infinitely divisible stochastic processes. The exponent measure of a min-id sequence is shown to be the sum of a very simple "drift measure" and a mixture of product probability measures, which corresponds uniquely to the L\'evy measure of a non-decreasing infinitely divisible process. The latter is shown to be supported on non-negative and non-decreasing functions. Our results provide an analytic umbrella which embeds the de Finetti subfamilies of many classes of multivariate distributions, such as exogenous shock models, exponential and geometric laws with lack-of-memory property, min-stable multivariate exponential and extreme-value distributions, as well as reciprocal Archimedean copulas with completely monotone generator and Archimedean copulas with log-completely monotone generator.Comment: 53 pages, 3 Figure

    Common Poisson Shock Models: Applications to Insurance and Credit Risk Modelling

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    The idea of using common Poisson shock processes to model dependent event frequencies is well known in the reliability literature. In this paper we examine these models in the context of insurance loss modelling and credit risk modelling. To do this we set up a very general common shock framework for losses of a number of different types that allows for both dependence in loss frequencies across types and dependence in loss severities. Our aims are threefold: to demonstrate that the common shock model is a very natural way of approaching the modelling of dependent losses in an insurance or risk management context; to provide a summary of some analytical results concerning the nature of the dependence implied by the common shock specification; to examine the aggregate loss distribution that results from the model and its sensitivity to the specification of the model parameter
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