Doctor of Philosophy
Statistics and Actuarial Sciences
Provost, Serge B
This thesis studies the estimability and the estimation methods for two models based on Markov processes: the phase-type aging model (PTAM), which models the human aging process, and the discrete multivariate phase-type model (DMPTM), which can be used to model multivariate insurance claim processes.
The principal contributions of this thesis can be categorized into two areas. First, an objective measure of estimability is proposed to quantify estimability in the context of statistical models. Existing methods for assessing estimability require the subjective specification of thresholds, which potentially limits their usefulness. Unlike these methods, the proposed measure of estimability is objective. In particular, this objectivity is achieved via a carefully designed distribution function sensitivity measure, under which the threshold will become an experiment-based quantity. The proposed measure which is validated to be innately sound, is then applied to assess and improve the estimability of several statistical models, the focus being placed on the PTAM.
Secondly, Markov chain Monte Carlo (MCMC) algorithms are proposed for inference on the PTAM and the DMPTM. Up to now, the MCMC algorithms for continuous phase-type distributions have been applied via the Gibbs sampler which consists of two iterative steps: a data augmentation step and a posterior sampling step. However, owing to unique structures of the PTAM and the DMPTM, this Gibbs sampler turns out to be inadequate, giving rise to problems occurring in either the data augmentation step or the posterior sampling step. To circumvent these difficulties, we methodologically extend the existing Gibbs sampling methodology in terms of rejection sampling and data cloning. The proposed algorithms are then applied to calibrate the PTAM and the DMPTM based on simulated and real-life data. Experimental results show that the proposed MCMC algorithms, as a stochastic approximation technique, achieve estimation results that are comparable to those obtained by deterministic approximation techniques, which can also be seen as a contribution made to the field of approximate inference.
Summary for Lay Audience
This thesis principally contributes to two areas of studies.
In statistics, it is well-known that a statistical model is identifiable if parameters can be uniquely inferred from data. However, identifiability does not imply estimability. For example, if the number of observations is low or the numerical algorithm is not sufficiently accurate, then the parameters can only be roughly estimated, even if the model is identifiable. Identifiability has a rigorous mathematical definition. However, estimability is usually measured subjectively and an objective measure appears to be lacking in the context of statistical models. Accordingly, the first contribution of this thesis is to propose an objective measure to quantify estimability in the context of statistical models.
Secondly, Markov chain Monte Carlo (MCMC) algorithms are proposed for inference on two actuarial models: the phase-type aging model (PTAM) and the discrete multivariate phase-type model (DMPTM), where the former models aging processes and the latter models multivariate insurance claim processes. MCMC is a methodology for sampling complicated distributions, where one constructs a carefully designed Markov chain whose stationary distribution coincides with the target distribution. Then, sampling from the target distribution is replaced with sampling from the designed Markov chain. In this thesis, we develop MCMC algorithms for the PTAM and DMPTM, where the models' special structures are utilized.
Nie, Cong, "New Developments on the Estimability and the Estimation of Phase-Type Actuarial Models" (2022). Electronic Thesis and Dissertation Repository. 8667.
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