Simulating Coxian phase‐type distributions for patient survival

Abstract

Coxian phase-type distributions are a special type of Markov model that can be used to represent survival times in terms of phases through which an individual may progress until they eventually leave the system completely. Previous research has considered the Coxian phase-type distribution to be ideal in representing patient survival in hospital. However, problems exist in fitting the distributions. This paper investigates the problems that arise with the fitting process by simulating various Coxian phase-type models for the representation of patient survival and examining the estimated parameter values and eigenvalues obtained. The results indicate that numerical methods previously used for fitting the model parameters do not always converge. An alternative technique is therefore considered. All methods are influenced by the choice of initial parameter values. The investigation uses a data set of 1439 elderly patients and models their survival time, the length of time they spend in a UK hospital.

Application of the phase‐type mortality law to life contingencies and risk management

“…If the benefit of forming a portfolio is large (small), the denominator gets small (large) and PDI becomes large (small). A general form of index (24) under tail risk measures is also found in, for example, Tasche [26] in the context of the subadditivity of risk capital allocation. Again, the unconditional variance (19) and (23) can be computed efficiently as the conditional moments given are analytically available for the common products.…”

“…In line with this, we focus on the Coxian PH distribution for illustration purpose. Several parameter estimation methods are available for the Coxian PH distributions; see Marshall and Zenga for the comparison of some popular methods. In this section, we use the EM algorithm proposed by Asmussen et al (1996) to fit the Coxian PH model to the Japanese life table (JLT17, data source: http://mort.soa.org/).…”

“…which would take values less than one in practice as the covariance terms in (23) are positive as reserves are decreasing functions in the interest rates. The PDI essentially measures the degree of risk reduction due to forming a portfolio by comparing the variability of the contracts on a stand-alone basis, which corresponds to the numerator in (24), to that on a portfolio basis, the denominator. If the benefit of forming a portfolio is large (small), the denominator gets small (large) and PDI becomes large (small).…”

“…Again, the unconditional variance (19) and (23) can be computed efficiently as the conditional moments given are analytically available for the common products. We comment that PDI takes both the systematic interest risk and product diversity within the portfolio into account, as indicated from the unconditional moment in (24). The essential difference between IRS and PDI is that the former focuses on how much systematic interest risk is in the portfolio whereas the latter focuses on how diverse the portfolio is, although to some extent these two are always related to each other via the common interest risk shared by all contracts.…”

Observation-Enhanced QoS Analysis of Component-Based Systems

“…OMNI uses phase-type distributions (PHDs) to refine the relevant elements of the analysed high-level abstract CTMC. PHDs model stochastic processes where the event of interest is the time to reach a specific state, and are widely used in the performance modelling of systems from domains ranging from call centres to healthcare [43]- [45]. PHDs support efficient numerical and analytical evaluation [21], and can approximate arbitrarily close any continuous distribution with a strictly positive density in (0, ∞) [46], although PHD fitting of distributions with deterministic delays requires extremely large numbers of states.…”