Estimating absorption time distributions of general Markov jump processes

TL;DR

This paper develops a maximum likelihood estimation framework using EM algorithms for absorption time distributions in general Markov jump processes, extending beyond time-homogeneous cases with commuting matrices.

Contribution

It introduces a novel approach to estimate absorption times in time-inhomogeneous Markov jump processes without the commuting matrix restriction, using piecewise constant intensities and matrix analytic methods.

Findings

Effective estimation of complex absorption time distributions.

Application to real data demonstrates practical utility.

Enhanced modeling flexibility for Markov jump processes.

Abstract

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.