  A phase-type distribution is the distribution of the time until absorption in
a finite state-space time-homogeneous Markov jump process, with one absorbing
state and the rest being transient. These distributions are mathematically
tractable and conceptually attractive to model physical phenomena due to their
interpretation in terms of a hidden Markov structure. Three recent extensions
of regular phase-type distributions give rise to models which allow for heavy
tails: discrete- or continuous-scaling; fractional-time semi-Markov extensions;
and inhomogeneous time-change of the underlying Markov process. In this paper,
we present a unifying theory for heavy-tailed phase-type distributions for
which all three approaches are particular cases. Our main objective is to
provide useful models for heavy-tailed phase-type distributions, but any other
tail behavior is also captured by our specification. We provide relevant new
examples and also show how existing approaches are naturally embedded.
Subsequently, two multivariate extensions are presented, inspired by the
univariate construction which can be considered as a matrix version of a
frailty model. We provide fully explicit EM-algorithms for all models and
illustrate them using synthetic and real-life data.
