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Copulas: a review and recent developments. (English) Zbl 1120.60006
This paper provides a survey on some recent contributions to copula theory, with an emphasis on the presentation of several new authors’ investigations, viz. order statistics copula, copulas with given multivariate marginals, copula representation via local dependence measures, and applications of extreme value copulas. Section 2 introduces two extensions of the probability integral transform, Bernstein and Bertino’s family of copulas, notions on quasi-copulas, several copula-like versions related to conditional and pseudo-conditional copulas used in time series context. Copula applications for evaluating quantile risk measures are also considered. Section 3 discusses three topics: order statistics copula, copula with multivariate marginals, and a representation of copula via a local dependence measure. Section 4 gives a review of some known facts and authors’ recent results treating the applications of extreme value copulas. The extreme value multivariate modeling presents special features, viz. the variables involved are usually non-exchangeable, positively associated (if not independent) with nonlinear forms of dependence. In applications, additional complications arise from the usual scarcity of data. The importance of survival copulas in life modeling is mentioned.

MSC:
60E05 Probability distributions: general theory
62E20 Asymptotic distribution theory in statistics
62G30 Order statistics; empirical distribution functions
62G32 Statistics of extreme values; tail inference
62H12 Estimation in multivariate analysis
62H25 Factor analysis and principal components; correspondence analysis
62J05 Linear regression; mixed models
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