AbstractThe most accurate estimation of transition probabilities for a multi-state model of health status requires longitudinal data. However, for many countries such data are usually not available. Instead, population level mortality incidence and disability prevalence rates are often all that can be accessed. In this paper, for a three-state no-recovery model (with states healthy, disabled, dead), using simple mathematical derivations, we propose a framework to estimate the age- and gender-specific boundaries within which each of the transition probabilities should fall. We then provide two methods for estimating unique transition probabilities—a least squares procedure and a method based on the ‘extra mortality’ factor proposed by Rickayzen and Walsh (Br Actuarial J 8(2):341–393, 2002, https://doi.org/10.1017/s1357321700003755). We also show the acceptable range for the ‘extra mortality’ factor given the mortality and disability data. Furthermore, we provide a critique of the method proposed by Van der Gaag et al. (Demogr Res 32:75, 2015), as their estimates can fall outside the acceptable boundaries. Finally, we estimate life and health expectancies, as well as premium rates for a life care annuity and a disability annuity using our derived transition probabilities.

中文翻译：

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一种利用可观察的人口死亡率和老年人残障患病率计算隐含的无法恢复的三态过渡矩阵的方法

摘要对健康状况的多状态模型进行转移概率的最准确估计需要纵向数据。但是，对于许多国家来说，此类数据通常不可用。取而代之的是，通常可以访问人口水平的死亡率和残疾发生率。在本文中，对于三状态的无恢复模型（健康，残疾，死亡状态为状态），我们使用简单的数学推导，提出了一个框架来估算年龄和性别特定的边界，每个过渡概率都应在该边界内落下。然后，我们提供了两种用于估计唯一过渡概率的方法-最小二乘法和基于Rickayzen和Walsh提出的“额外死亡率”因子的方法（Br Actuarial J 8（2）：341-393，2002，https：// doi.org/10.1017/s1357321700003755）。考虑到死亡率和残疾数据，我们还显示了“额外死亡率”因子的可接受范围。此外，我们对Van der Gaag等人提出的方法提出了批评。（Demogr Res 32:75，2015），因为他们的估算可能会超出可接受的范围。最后，我们使用得出的过渡概率估算生命和健康预期以及寿险年金和伤残年金的保费率。