Abstract
AbstractThis paper addresses the task of modeling severity losses using segmentation when the data distribution does not fall into the usual regression frameworks. This situation is not uncommon in lines of business such as third-party liability insurance, where heavy-tails and multimodality often hamper a direct statistical analysis. We propose to use regression models based on phase-type distributions, regressing on their underlying inhomogeneous Markov intensity and using an extension of the expectation–maximization algorithm. These models are interpretable and tractable in terms of multistate processes and generalize the proportional hazards specification when the dimension of the state space is larger than 1. We show that the combination of matrix parameters, inhomogeneity transforms, and covariate information provides flexible regression models that effectively capture the entire distribution of loss severities.
Highlights
The task of modeling claim severities is a well known and difficult challenge in actuarial science, and their correct description is of large interest for the practicing actuary or risk manager
We introduce a regression model based on inhomogeneous phase-type (IPH) distributions and use it to provide a segmentation model for claim severities
We have presented a novel claim severities model based on PH distributions, which implicitly assumes an underlying and unobservable multistate Markov structure
Summary
The task of modeling claim severities is a well known and difficult challenge in actuarial science, and their correct description is of large interest for the practicing actuary or risk manager. PH distributions are roughly defined as the time it takes for a purejump Markov processes on a finite state space to reach an absorbing state Their interpretation in terms of traversing several states before finalizing is appealing actuarial science, since one may think of claim sizes as the finalized “time” after having traversed some unobserved states to reach such magnitude, for instance, legal cases, disabilities, unexpected reparation costs, etc. Transforming PH distributions parametrically has been considered in Albrecher and Bladt (2019) and Albrecher et al (2020b), resulting in inhomogeneous phase-type (IPH) distributions, which leads to non-exponential tail behaviors by allowing for the underlying process to be time-inhomogeneous (see Bladt and Rojas-Nandayapa, 2017; Albrecher et al, 2020a for alternative heavy-tailed PH specifications) The latter development allows to model heavy- or light-tailed data with some straightforward adaptations to the usual PH statistical methods.
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