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

TL;DR

This paper develops an EM algorithm for estimating aggregate Markov models in life insurance, enabling non-Markovian modeling of biometric states with observed macrostate paths, and demonstrates its application with a disability model.

Contribution

It introduces an EM-based estimation method for aggregate Markov models, including semi-Markov cases, in life insurance, extending traditional Markov models to handle dependencies and non-Markovian features.

Findings

The EM algorithm effectively estimates transition intensities in aggregate Markov models.

Application to a disability model shows good fit compared to classical methods.

Numerical example illustrates the method's practicality with simulated data.

Abstract

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-maximization (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 with existing models and their tools.