Implementing Markovian models for extendible Marshall–Olkin distributions

We derive a novel stochastic representation of exchangeable Marshall–Olkin distributions based on their death-counting processes. We show that these processes are Markov. Furthermore, we provide a numerically stable approximation of their infinitesimal generator matrices in the extendible case. This approach uses integral representations of Bernstein functions to calculate the generator’s first row, and then uses a recursion to calculate the remaining rows. Combining the Markov representation with the numerically stable approximation of corresponding generators allows us to sample extendible Marshall–Olkin distributions with a flexible simulation algorithm derived from known Markov sampling strategies. Finally, we benchmark an implementation of this Markov-based simulation algorithm against alternative simulation algorithms based on the Lévy frailty model, the Arnold model, and the exogenous shock model.