New results on multivariate phase-type distributions

Phase-type distributions are used for mathematical modelling within numerous scientific fields of both engineering and commercial interests, such as genetics, transport optimisation, and insurance mathematics. Statistical models based on phase-type distributions are often flexible and well-suited for numerical computations and simulations, but the theoretical foundation of the multivariate phase-type distributions is not yet fully developed. Further research in estimation and characterisation of multivariate phase-type distributions is therefore necessary to learn how the useful properties of phase-type distributions can contribute to the advancement of data science and artificial intelligence.

This thesis concerns univariate and multivariate phase-type distributions, and a manuscript for a scientific paper constitutes the primary research contribution of the thesis. The second chapter of the thesis first describes the construction of the univariate phase-type distributions, which are introduced on the basis of Markov processes. Subsequently, several fundamental theorems regarding the univariate phase-type distributions are derived with a particular emphasis on their closure and denseness properties. Finally, the chapter treats reward structures, which are instrumental in the construction of the multivariate distributions, before some concrete examples of univariate phase-type distributions are presented.

Chapter three of the thesis deals with multivariate phase-type distributions. Like in the previous chapter, multiple theorems and properties are derived, many of which follow immediately from a result on concatenations of Markov processes proved in the second chapter. The proofs of the main theorems on the joint survival function and the cross-moments of the multivariate phase-type distributions contain various technical details and arguments, which cannot be not found elsewhere in the literature. The proofs are followed by a section on phase-type representations of known multivariate exponential and gamma distributions. In said section, phase-type representations are compared to joint density functions and copula representations of different distributions, e.g. Marshall and Olkin’s bivariate exponential distribution and the multivariate gamma distribution by Cheriyan and Ramabhadran.

A manuscript for a scientific paper on necessary and sufficient conditions for the existence of the multivariate gamma distribution by Dussauchoy and Berland comprises the last main chapter of the thesis. In the paper, it is shown that the distribution can be decomposed into a convolution of independent multivariate mixture distributions, which are easily recognised as phase-type distributions. This leads to a phase-type representation of the multivariate gamma distribution by Dussauchoy and Berland, which in conjunction with the abovementioned decomposition give rise to some necessary and sufficient conditions for the existence of the distribution. The conditions are expressed as constraints on the shape parameters of the distribution and the weights defining the previously mentioned mixture distributions. Lastly, the paper discusses the divisibility of the distribution and provides a specific example of parameter values leading to an invalid distribution. Based on this example, the paper concludes that the distribution is not always infinitely divisible and conjectures that the distribution generally cannot exist when the shape parameters take non-integer values.

Furthermore, the thesis contains some sections on related univariate and multivariate distributions as well as a short chapter on future research directions relating to the topics discussed throughout the thesis.