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