Fitting phase-type frailty models

Abstract

Frailty models are survival analysis models which account for heterogeneity and random effects in the data. In these models, the random effect (the frailty) is assumed to have a multiplicative effect on the hazard. In this paper, we present frailty models using phase-type distributions as the frailties. We explore the properties of the proposed frailty models and derive expectation-maximization algorithms for maximum-likelihood estimation. The algorithms' performance is illustrated in several numerical examples of practical significance.

Experience rating in the classic Markov chain life insurance setting

“…To address this, we turn our attention to a more flexible model capable of describing a wide range of dependence structures, namely multivariate phase-type distributions. The use of phase-type distributions as random effects was first explored in Yslas (2021)…”

“…Unfortunately, the posterior distribution is no longer phase-type but rather falls within the more general class of matrix exponential distributions, that is, distributions with rational Laplace transform; for further details, see Yslas (2021). Nevertheless, the model retains a high degree of mathematical tractability and takes full advantage of the modeling capabilities of phase-type distributions, a point we demonstrate in the subsequent discussion of (2.2) with bivariate phase-type mixing.…”

“…This paper builds on and extends the work of Furrer (2019) on group shrinkage estimation of transition rates in Markov chains by allowing for dependence between latent group effects across transitions. Following closely Yslas (2021), but focusing on mixed Poisson regressions rather than frailty models, we study multivariate phasetype distributions as a tractable and flexible class of priors for the latent group effects. The connection to Markov chain models is provided by the classic connection between Poisson regression and parametric inference in multi-state models.…”

“…The downside of this class of priors is that it is only able to capture positive correlation. Second, we adapt the work of Yslas (2021) on phase-type frailty models to the mixed Poisson setting and describe the corresponding expectation-maximization algorithm. In particular, we offer some numerical considerations that are important for an efficient implementation of the algorithm.…”

“…Notable among them are the general MPH* class introduced in Kulkarni (1989) and its more computationally amenable subclasses: the feed-forward type class, confer, for example, with Albrecher et al (2022b), and the mPH class recently detailed in Bladt (2023), which also includes applications towards severity modeling. In this work, we opt as in Yslas (2021) for the feed-forward type class due to its mathematical convenience and its denseness on the set of the distributions on the positive orthant.…”