Aggregate Markov models in life insurance: estimation via the EM algorithm

In this paper, we consider statistical estimation of time–inhomogeneous aggregate Markov models. Unaggregated models, which corresponds to Markov chains, are commonly used in multi–state life insurance to model the biometric states of an insured. By aggregating microstates to each biometric state, we are able to model dependencies between transitions of the biometric states as well as the distribution of occupancy in these. This allows for non–Markovian modelling in general. Since only paths of the macrostates are observed, we develop an expectation–maximisation (EM) algorithm to obtain maximum likelihood estimates of transition intensities on the micro level. Special attention is given to a semi-Markovian case, known as the reset property, which leads to simplified estimation procedures where EM algorithms for inhomogeneous phase–type distributions can be used as building blocks. We provide a numerical example of the latter in combination with piecewise constant transition rates in a three–state disability model with data simulated from a time–inhomogeneous semi–Markov model. Comparisons of our fits with more classic GLM-based fits as well as true and empirical distributions are provided to relate our model to existing models and their tools.