Random Reward Phase-Type Distributions with Applications in Latent Severity Modeling

TL;DR

This paper extends discrete Phase-Type distributions with random rewards, enabling modeling of systems with stochastic rewards, and introduces the Inertia-Escalation model for latent severity levels, validated through simulations and real datasets.

Contribution

It introduces random rewards into DPH distributions and proposes the Inertia-Escalation model for latent severity, with methods for parameter inference and validation.

Findings

Validated the model with simulations.

Applied the model to warfare and churn datasets.

Demonstrated increased flexibility in severity modeling.

Abstract

This paper proposes an extension to discrete Phase-Type distributions (DPH) by introducing random rewards. These allow for modeling a system in which a visit to a certain state does not emit a deterministic reward. Instead, the rewards follow either a Bernoulli or a geometric distribution. Utilizing this increased flexibility, we further sketch a possible use case for these random rewards by introducing the Inertia-Escalation model (IEM), a process with latent severity levels characterized through two parameters: Inertia {\nu} and escalation {\eta}. We also discuss parameter inference for such models. To validate and explore random rewards and the IEM, we conducted extensive simulations and applied the model to two datasets: historical warfare and the Telco customer churn dataset.