Application of Phase-Type Distributions to Actuarial Lifetime Data under Left-Truncated and Right-Censored Observations

Estimating lifetime distributions under left-truncated and right-censored (LTRC) data is an important problem in actuarial science because it directly influences premium calculation, reserve valuation, and mortality modeling based on life table data. In insurance practice, delayed entry due to policy enrollment age and incomplete observation due to the end of the study period occur simultaneously, producing observation bias and a complicated likelihood structure. Under this framework, reliable lifetime modeling requires appropriate statistical modeling. Phase-Type (PH) distributions provide a flexible class of lifetime models constructed from absorbing Markov processes and are dense in the class of positive-valued distributions. Their structural flexibility makes them particularly suitable for capturing complex hazard patterns observed in actuarial mortality data. This study applies PH distributions to actuarial life table data under the LTRC framework. In particular, we explicitly formulate and estimate PH distributions using life table data regarded as LTRC observations, which constitutes a novel application in this context. Model parameters are estimated via maximum likelihood. Since increasing the number of phases monotonically improves the likelihood, practical implementation requires careful consideration of model complexity. To address this issue, we employ the Extended Information Criterion (EIC), which incorporates bootstrap-based bias correction of the log-likelihood, to examine phase selection. Furthermore, we compare PH distributions with conventional lifetime distributions using EIC, providing a systematic evaluation of their relative performance under the LTRC setting. The numerical results demonstrate the practical applicability of PH distributions to actuarial lifetime modeling under LTRC data. We observe the trade-off between improved goodness-of-fit and increased model complexity as the number of phases grows, and show that EIC can be used to support practical analysis of model complexity and model comparison in this context.