  The estimation of absorption time distributions of Markov jump processes is
an important task in various branches of statistics and applied probability.
While the time-homogeneous case is classic, the time-inhomogeneous case has
recently received increased attention due to its added flexibility and advances
in computational power. However, commuting sub-intensity matrices are assumed,
which in various cases limits the parsimonious properties of the resulting
representation. This paper develops the theory required to solve the general
case through maximum likelihood estimation, and in particular, using the
expectation-maximization algorithm. A reduction to a piecewise constant
intensity matrix function is proposed in order to provide succinct
representations, where a parametric linear model binds the intensities
together. Practical aspects are discussed and illustrated through the
estimation of notoriously demanding theoretical distributions and real data,
from the perspective of matrix analytic methods.
