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Dependence in a background risk model. (English) Zbl 1419.62295

Summary: Many copula families, including the classes of Archimedean, elliptical and Liouville copulas, may be written as the survival copula of a random vector \(R \times(Y_1, Y_2)\), where \(R\) is a strictly positive random variable independent of the random vector \((Y_1, Y_2)\). A unified framework is presented for studying the dependence structure underlying this stochastic representation, which is called the background risk model. Formulas for the copula, Kendall’s tau and tail dependence coefficients are obtained and special cases are detailed. The usefulness of the construction for model building is illustrated with an extension of Archimedean copulas with completely monotone generators, based on the Farlie-Gumbel-Morgenstern copula. In particular, explicit expressions for the distribution and the Tail-Value-at-Risk of the aggregated risk \(R Y_1 + R Y_2\) are available in a generalization of the widely used multivariate Pareto-II model.

MSC:

62P05 Applications of statistics to actuarial sciences and financial mathematics
62H05 Characterization and structure theory for multivariate probability distributions; copulas
62H20 Measures of association (correlation, canonical correlation, etc.)
91B30 Risk theory, insurance (MSC2010)
60E05 Probability distributions: general theory

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