Multivariate Modelling Using Copulas
Copulas are a family of multivariate distributions whose marginal distributions are uniform. At the end of reserving problems, we need to aggregate the outstanding liability distribution of each line of business or each type of benefit to get the total outstanding liability distribution. The dependence between them must be considered. In the Bayesian copulas framework, all the uncertainties and correlations are considered during the inferential process which is an advantage compared with the likelihood-based frequentist inference. In Sect. 6.1, the elements of copulas are reviewed, including Sklar’s theorem, parametric copulas, inference methods, etc. In Sect. 6.2, we discuss the usefulness of copulas in risk modelling generally. The copula is used to model the empirical dependence between risks while the marginal regression model is used to model the structural dependence. In Sect. 6.3, a bivariate Gaussian copula is used to aggregate the liabilities of the doctor benefit and the hospital benefit in WorkSafe Victoria. These two benefits are correlated positively even after removing the structural effects of the development periods.
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