Electronic Thesis and Dissertation Repository

Thesis Format

Integrated Article

Degree

Doctor of Philosophy

Program

Statistics and Actuarial Sciences

Supervisor

Jones, Bruce

2nd Supervisor

Liu, Xiaoming

3rd Supervisor

Mamon, Rogemar

Abstract

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.

Summary for Lay Audience

Ageing is a universal and ever-present biological phenomenon. Yet, describing the ageing mechanism in formal mathematical terms – in particular, capturing the ageing patterns and quantifying the ageing rate - has remained a challenging endeavour. The modelling of the human ageing process, under a spectrum of uncertainty, is critical in the accurate valuation and robust risk management of insurance and pension products. We put forward a general model having a small number of parameters but flexible enough to produce a variety of lifetime distributions. Our research contributions widen the available actuarial and survival analysis techniques in the following way: (i) A phase-type ageing model (PTAM) is constructed in which the ageing’s deteriorating effect and the associated increasing mortality risk are simultaneously taken into account. (ii) A physiological age index is introduced in order to quantify the heterogeneity between individuals. The physiological age index can be used to classify various mortality risk levels. (iii) Some pertinent statistical properties of the PTAM and the physiological age distribution are established. (iv) An efficient method is developed to evaluate the PTAM’s likelihood. Some numerical examples, utilising simulated and actual lifetime data sets, are provided to demonstrate that our proposed calculation technique is faster and more accurate than the traditional method based on matrix exponential. (v) The estimability of the PTAM under different lifetime-information scenarios is examined in the context of improving the parameter estimation. Our modelling of the ageing process through lifetime distributions is also of utmost importance in crafting suitable regulatory requirements and policies that will strengthen further the public confidence in the national or even global financial system.

Erratum

Co-Authorship Statement

Chapter 3 of this thesis has been published. The remaining Chapters consist of materials that delve into the pertinent topics of Chapter 3, dealing with the various aspects of the proposed model. I declare that the research outputs incorporated in this thesis are the direct results of my main research works and efforts during the course of my PhD program.

The research results in Chapter 3 form the main basis of an article with the following citation details:

Cheng, B., Jones, B., Liu, X., and Ren, J.(2020), The mathematical mechanism of biological aging, North American Actuarial Journal, accepted:DOI:10.1080/10920277.2020.1775654.

As the first author of a research paper incorporated in this thesis, I am responsible, with the support and assistance of my supervisors, for the development of the modelling set-up, the implementation of algorithms, and the completion of the manuscript. The research plan, model formulation, and empirical analysis of the results in the published paper are guided and supervised by Dr. Bruce Jones, Dr. Jiandong Ren and Dr. Xiaoming Liu. Some insights in the analysis of the Channing House data set were provided by Dr. Bruce Jones. Dr. Xiaoming Liu oversaw the goodness-of-fit comparison between the proposed model in this thesis and the Lin and Liu’s model.

An integrated-article format is employed in line with Western’s thesis guidelines. I certify that this thesis is fully a product of my own work, completed with the support and assistance of my supervisors. This was conducted from September 2016 to September 2020 under the supervision of Dr. Bruce Jones and Dr. Xiaoming Liu, and from September 2020 to present under the supervision of Dr. Rogemar Mamon at The University of Western Ontario.

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