Fitting bivariate losses with phase-type distributions

Abstract

Maximum likelihood estimation and (parametric bootstrap) goodness-of-fit test are considered for bivariate phase-type distributions introduced by Assaf and Colleagues. In a special case, the dependence structure of bivariate phase-type distributions is revealed. The results are used to fit a real bi-dimensional data set related to insurance losses (LOSS) and allocated loss adjustment expenses (ALAE). The fitted bivariate phase-type is used to obtain conditional quantiles and mean of ALAE for a given value of LOSS. The bivariate phase-type distribution meets all the requirements listed in the study by Klugman and Parsa.

Keywords:

• bivariate phase-type distributions
• dependent losses
• continuous-time Markov chain
• copulas
• EM algorithm
• parametric bootstrap goodness-of-fit test