Maximum Likelihood Estimation for Scaled Inhomogeneous Phase-Type Distributions from Discrete Observations

Inhomogeneous phase-type (IPH) distributions extend classical phase-type models by allowing transition intensities to vary over time, offering greater flexibility for modeling heavy-tailed or time-dependent absorption phenomena. We focus on the subclass of IPH distributions with time-scaled sub-intensity matrices of the form ${\Lambda}(t) = h_{\beta}(t){\Lambda}$, which admits a time transformation to a homogeneous Markov jump process. For this class, we develop a statistical inference framework for discretely observed trajectories that combines Markov-bridge reconstruction with a stochastic EM algorithm and a gradient-based update. The resulting method yields joint maximum-likelihood estimates of both the baseline sub-intensity matrix ${\Lambda}$ and the time-scaling parameter $\beta$. Through simulation studies for the matrix-Gompertz and matrix-Weibull families, and a real-data application to coronary allograft vasculopathy progression, we demonstrate that the proposed approach provides an accurate and computationally tractable tool for fitting time-scaled IPH models to irregular multi-state data.