  This paper develops a new class of conditional Markov jump processes with
regime switching and paths dependence. The key novel feature of the developed
process lies on its ability to switch the transition rate as it moves from one
state to another with switching probability depending on the current state and
time of the process as well as its past trajectories. As such, the transition
from current state to another depends on the holding time of the process in the
state. Distributional properties of the process are given explicitly in terms
of the speed regimes represented by a finite number of different transition
matrices, the probabilities of selecting regime membership within each state,
and past realization of the process. In particular, it has distributional
equivalent stochastic representation with a general mixture of Markov jump
processes introduced in Frydman and Surya (2020). Maximum likelihood estimates
(MLE) of the distribution parameters of the process are derived in closed form.
The estimation is done iteratively using the EM algorithm. Akaike information
criterion is used to assess the goodness-of-fit of the selected model. An
explicit observed Fisher information matrix of the MLE is derived for the
calculation of standard errors of the MLE. The information matrix takes on a
simplified form of the general matrix formula of Louis (1982). Large sample
properties of the MLE are presented. In particular, the covariance matrix for
the MLE of transition rates is equal to the Cram\'er-Rao lower bound, and is
less for the MLE of regime membership. The simulation study confirms these
findings and shows that the parameter estimates are accurate, consistent, and
have asymptotic normality as the sample size increases.
