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Parameter estimation in Coxian phase-type models can be challenging due totheir non-unique representation leading to a multi-modal likelihood. Since eachrepresentation corresponds to a different underlying data-generating mechanism,it is of interest to identify those supported by given data (i.e., find alllikelihood modes). The standard approach is to simply refit using variousinitial values, but this has no guarantee of working. Thus, we develop newproperties specific to this class of models, and employ these to determine allthe equivalent model representations. The proposed approach only requiresfitting the model once, and is guaranteed to find all representations.

On the Non-uniqueness of Representations of Coxian Phase-Type Distributions