Modelling life tables with advanced ages: An extreme value theory approach

Huang, F., Maller, R., and Ning, X.2020, Insurance: Mathematics and Economics93, 95-115.

We propose a new model – we call it a smoothed threshold life table (STLT) model – to generate life tables incorporating information on advanced ages. Our method allows a smooth mortality transition from non-extreme to extreme ages, and provides objectively determined highest attained ages with which to close the life table.

We proceed by modifying the threshold life table (TLT) model developed by Li et al. (2008). In the TLT model, extreme value theory (EVT) is used to make optimal use of the relatively small number of observations at high ages, while the traditional Gompertz distribution is assumed for earlier ages. Our novel contribution is to constrain the hazard function of the two-part lifetime distribution to be continuous at the changeover point between the Gompertz and EVT models. This simple but far-reaching modification not only guarantees a smooth transition from non-extreme to extreme ages, but also provides a better and more robust fit than the TLT model when applied to a high quality Netherlands dataset. We show that the STLT model also compares favourably with other existing methods, including the Gompertz–Makeham model, logistic models, Heligman–Pollard model and Coale–Kisker method, and that a further generalisation, a time-dependent dynamic smooth threshold life table (DSTLT) model, generally has superior in-sample fitting as well as better out-of-sample forecasting performance, compared, for example, with the Cairns et al. (2006) model.