Bivariate phase-type distributions for experience rating in disability insurance

TL;DR

This work addresses experience rating for disability insurance by modeling group heterogeneity through mixed Poisson regressions. It advances the field by proposing three prior structures for the latent effects: independent Gamma, hierarchical Gamma, and multivariate phase-type (PH) mixing, with estimation implemented via EM/ECM algorithms. The PH approach, with its dense representation on the positive orthant and explicit dependence modeling, yields superior and more flexible predictive performance across scenarios, including positive and negative dependence. The findings highlight the practical value of allowing dependence among latent group effects for more accurate disability and recovery rate estimation and pricing in group contracts.

Abstract

In this paper, we consider the problem of experience rating within the classic Markov chain life insurance framework. We begin by establishing a link between mixed Poisson distributions and the problem of pricing group disability insurance contracts that exhibit heterogeneity. We focus on shrinkage estimation of disability and recovery rates, taking into account sampling effects such as right-censoring. We then investigate some specific multivariate mixed Poisson models with mixing distributions encompassing independent Gamma, hierarchical Gamma, and multivariate phase-type. In particular, we demonstrate how maximum likelihood estimation for these models can be performed using expectation-maximization algorithms, which might be of independent interest. Finally, we showcase the practicality of the proposed shrinkage estimators through a numerical study based on simulated yet realistic insurance data. Our findings highlight that by allowing for dependency between latent group effects, estimates of recovery and disability rates mutually improve, leading to enhanced predictive performance.

Figures (12)

• Figure 2.1: The Markov chain $Z$ with values in the set $\mathcal{S}=\{a,i,d\}$ and transition rates $t \mapsto \mu_{jk}(t)$.
• Figure 4.1: Group sizes; groups $16$, $23$, and $68$ have been highlighted.
• Figure 4.2: Mixture density for the initial ages (top), histogram of simulated ages against original density (bottom left), and average initial age per group size with groups $16$, $23$, and $68$ highlighted (bottom right).
• Figure 4.3: True transition rates for the Markov chain.
• Figure 4.4: Marginal densities of the mixing vector $\boldsymbol{\bm \Theta}$.

Theorems & Definitions (6)

• Example 2.1
• Proposition 2.2
• Remark 2.1
• Example 3.1: Gamma mixing
• Remark 3.1
• Remark 3.2