A class of phase-type ageing models and their lifetime distributions

Ageing is a universal and ever-present biological phenomenon. Yet, describing the ageing mechanism in formal mathematical terms — in particular, capturing the ageing pattern and quantifying the ageing rate — has remained a challenging actuarial modelling endeavour. In this thesis, we propose a class of Coxian-type Markovian models. This class enables a quantitative description of the well-known characteristics of ageing, which is a genetically determined, progressive, and essentially irreversible process. The unique structure of our model features the transition rate for the ageing process and a functional form for the relationship between ageing and death with a shape parameter that captures the biologically deteriorating effect of ageing. The force of moving from one state to another in the Markovian process indicates the intrinsic biological ageing force. The associated increasing exit rate captures the external force of stress due to mortality risk on a living organism. We define an index, called physiological age, to quantify the heterogeneity between individuals. The physiological age can be used to compare the death rate between individuals in which an individual with a higher physiological age has higher mortality rate. The probability in each state at any time is calculated, and the distribution of the physiological age at any chronological age is obtained. We also prove that the distribution of the physiological age at a given time can be approximated by a normal distribution as the Phase-Type Ageing Model (PTAM) allows for a large number of states. The approximation can be used to quickly compute the probability in each state at any given time. The lifetime distribution for each individual readily follows from their physiological ages whose distribution is helpful in quantifying the variability of individual health status in the population. We develop an efficient method to evaluate the PTAM’s likelihood utilising a lifetime data set. Our likelihood calculation uses vectorisation to find simultaneously the density function at observed lifetimes. Furthermore,our method uses uniformisation strategy to stabilise the numerical calculation with a guaranteed accuracy for any error tolerance. We demonstrate that our numerical method is more accurate and faster than the traditional method using matrix exponential. Lastly, we investigate the estimability of the PTAM when only the lifetime data is observable along with some conditions that could improve the model’s estimability in terms of parameters’ identification