On inferring the number of phases in a coxian phase-type distribution

Abstract

Coxian phase-type distributions of increasing order were fitted to simulated exponential data by maximum likelihood, and the distributions of the resulting likelihood ratio statistic considered. These distributions resembled mixed chi-squared distributions with mixing probabilities related to parameter redundancies apparent from the estimates of the transition rates that make up the matrix describing the phase-type distributions. Such redundancies involved combinations of zero transition rates and equalities among the eigenvalues of this transition rate matrix. An example of fitting a more structured high order phase-type distribution is also described, where the structure was identified from the data and the number of phases required then estimated