Gompertzian survivorship ages: modeling and numerical evidence

Age has long been recognized as one of the primary determinants of mortality and the implicit assumption shared by the mortality models is that age flows chronologically. However, due to their own biological features, individuals’ age does not necessarily move in lockstep with calendar time, and different individuals may age at different rates. Then, a so-called non-chronological age can reasonably be acknowledged as a new element in analyzing lifespan randomness. If biological factors are not observed or cannot be observed, the issue of how to determine a non-chronological age consistent with the chronological age-based mortality experience comes up. The present work is set in this context, namely, it aims to discuss how to determine a non-chronological age in the absence of observable biological factors. To this end, we consider the concept of survivorship age introduced in Alvarez and Vaupel (Demography 60(1):327–342), which allows us to analyze mortality in terms of survival. We derive closed-form expressions for survivorship ages and related quantities under both the Gamma-Gompertz and the Gompertz mortality law. Finally, we present a numerical application to test our proposal by exploiting the mortality experience of four countries worldwide, and for both genders, namely Australia, Italy, Japan, and Sweden, discussing strengths and weaknesses of the use of Gompertz-based models in describing mortality as a function of survival.