Modeling Dependent Risks with Multivariate Erlang Mixtures

In this paper, we introduce a class of multivariate Erlang mixtures and present its desirable properties. We show that a multivariate Erlang mixture could be an ideal multivariate parametric model for insurance modeling, especially when modeling dependence is a concern.When multivariate losses are governed by a multivariate Erlang mixture, many quantities of interest such as joint density and Laplace transform, moments, and Kendall’s tau have a closed form. Further, the class is closed under convolutions and mixtures, which enables us to model aggregate losses in a straightforward way. We also introduce a new concept called quasi-comonotonicity that can be useful to derive an upper bound for individual losses in a multivariate stochastic order and upper bounds for stop-loss premiums of the aggregate loss. Finally, an EM algorithm tailored to multivariate Erlang mixtures is presented and numerical experiments are performed to test the efficiency of the algorithm.

Keywords Erlang mixture, dependent risk, multivariate analysis, quasi-comonotonicity, aggregate losses, EM algorithm.