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In this paper, we consider statistical estimation of time-inhomogeneousaggregate Markov models. Unaggregated models, which corresponds to Markovchains, are commonly used in multi-state life insurance to model the biometricstates of an insured. By aggregating microstates to each biometric state, weare able to model dependencies between transitions of the biometric states aswell as the distribution of occupancy in these. This allows for non--Markovianmodelling in general. Since only paths of the macrostates are observed, wedevelop an expectation-maximization (EM) algorithm to obtain maximum likelihoodestimates of transition intensities on the micro level. Special attention isgiven to a semi-Markovian case, known as the reset property, which leads tosimplified estimation procedures where EM algorithms for inhomogeneousphase-type distributions can be used as building blocks. We provide a numericalexample of the latter in combination with piecewise constant transition ratesin a three-state disability model with data simulated from a time-inhomogeneoussemi-Markov model. Comparisons of our fits with more classic GLM-based fits aswell as true and empirical distributions are provided to relate our model withexisting models and their tools.

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