Modeling dependent losses under censoring and truncation using multivariate mixtures of Erlangs

We study the estimation and use of multivariate mixtures of Erlangs (MME) to model dependent multivariate censored and truncated data. MME form a highly flexible class of distributions as they are dense in the space of positive continuous multivariate distributions. Moreover, the class is analytically tractable. Many quantities of interest such as the joint density and distribution function, the Laplace transform, moments, Kendall's tau and Spearman's rho have a closed form. Moreover, the class enjoys appealing closure properties such as the facts that any uni- or multivariate marginal or conditional distribution is a uni-or multivariate Erlang mixture, the distribution of the sum of the component random variables is a univariate Erlang mixture and the distribution of the excess-of-losses is a again a multivariate Erlang mixture. The use of MME should be regarded as semiparametric density estimation technique to model the dependence directly and hence forms a suitable alternative to the use of copulas. We present an estimation technique for fitting MME using the EM algorithm to data that can be censored and/or truncated, which is often the case with claim severity data in actuarial science due to policy modifications such as deductibles and policy limits. We demonstrate the effectiveness of the proposed algorithm and the practical use of MME on simulated data as well as on real-world data sets.status: publishe