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.

翻译：本文研究时间非齐次聚合马尔可夫模型的统计估计问题。在多人寿险中，对应于马尔可夫链的非聚合模型常用于建模被保险人的生物统计状态。通过将微观状态聚合至各生物统计状态，我们能够建模生物统计状态间转换的依赖性以及各状态的驻留时间分布。这通常可支持非马尔可夫建模。由于仅能观测到宏观状态的路径，我们开发了期望最大化（EM）算法，以在微观层面获得转移强度的最大似然估计。特别关注半马尔可夫情形（即重置特性），该情形可简化估计流程，使非齐次相位型分布的EM算法可作为基本构建模块。我们以含分段常数转移率的三状态残疾模型为例进行数值验证，该模型基于时间非齐次半马尔可夫模型模拟的数据。通过将拟合结果与基于经典广义线性模型（GLM）的拟合结果及真实分布与经验分布进行对比，阐明本模型与现有模型及其分析工具之间的关系。