Multidimensional smoothing by adaptive local kernel-weighted log-likelihood: Application to long-term care insurance

Abstract

We are interested in modeling the mortality of long-term care (LTC) claimants having the same level of severeness (heavy claimant). Practitioners often use empirical methods that rely heavily on expert opinions. We propose approaches not depending on an expert’s advice. We analyze the mortality as a function of both the age of occurrence of the claim and the duration of the care. LTC claimants are marked by a relatively complex mortality pattern. Hence, rather than using parametric approaches or models with expert opinions, adaptive local likelihood methods allow us to extract the information from the data more pertinently. We characterize a locally adaptive smoothing pointwise method using the intersection of confidence intervals rule, as well as a global method using local bandwidth correction factors. The latter is an extension of the adaptive kernel method proposed by Gavin et al. (1995) to likelihood techniques. We vary the amount of smoothing in a location-dependent manner and allow adjustments based on the reliability of the data. Tests, and single indices summarizing the lifetime probability distribution are used to compare the graduated series obtained by adaptive local kernel-weighted log-likelihoods to p-spline and local likelihood models.