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:在本文中,我们考虑对时间-非同质综合的Markov模型的统计估计。与Markov链链相对应的未合并模型,通常用于多国人寿保险,以模拟被保险人的生物鉴别状态。通过将微观国家汇总到每个生物鉴别状态,我们可以模拟生物鉴别状态转型与这些状态占用分布之间的依赖性。这样可以进行一般的非马尔科维安建模。由于只观察了宏观国家的路径,我们开发了预期-最大比例算法,以获得微观一级过渡强度的最大可能性估计。特别注意半马尔科维安案例,称为重新设定属性,这导致简化了估算程序,在其中可以使用不相异的相形相形分布的EM算法作为建筑块。我们提供了后者的数字示例,结合了三州残疾模型中的零散态持续过渡率,同时根据时间-不相形半马尔科夫模型模拟了数据。我们与更典型的GLM模型和真实的模拟模型相比,我们与更典型的GLM模型和模拟的模型是真实的。