  We extend the construction principle of phase-type (PH) distributions to
allow for inhomogeneous transition rates and show that this naturally leads to
direct probabilistic descriptions of certain transformations of PH
distributions. In particular, the resulting matrix distributions enable to
carry over fitting properties of PH distributions to distributions with heavy
tails, providing a general modelling framework for heavy-tail phenomena. We
also illustrate the versatility and parsimony of the proposed approach for the
modelling of a real-world heavy-tailed fire insurance dataset.
