Discretizing and validating Keyfitz' entropy for any demographic classification

Abstract

Abstract Keyfitz's entropy originally is a shape measure of senescence for continuous‐time models. But a later formula of Keyfitz's entropy for discrete time also exists. de Vries et al. (2023) showed that this discrete‐time formula is not a genuine shape measure of senescence and proposed a new discrete‐time formula for entropy that qualifies as a shape measure. However, these authors obtained their new formula by discretizing an alternative version of Keyfitz' continuous‐time formula and not by following his original derivation, that is not by computing the elasticity of life expectancy to an age‐uniform mortality change. Hence, they regarded as a loose end of their work whether discretizing Keyfitz' original derivation would also lead to their new formula. These authors also deemed lack of direct applicability to stage‐classified models a major downside of the new formula they proposed. Here we do two separate things: we discretize Keyfitz' original derivation of entropy and we generalize the new discrete‐time formula for this quantity proposed by de Vries et al. (2023). We show that the discrete‐time formula for entropy that de Vries et al. (2023) propose to supersede, while potentially problematic for studying senescence, is, as Keyfitz' original formula, an elasticity. Generalizing the work of de Vries et al. (2023) leads to a formula for entropy that can be directly applied as a shape measure of senescence to models with any demographic classification. We also prove convergence properties that validate the application of this general formula to models classified by stage. We thus support and generalize the successful approach by de Vries et al. (2023) of obtaining new discrete‐time formulas for Keyfitz' entropy from a shape perspective: merits of shape measures of senescence, which capture the overall age‐pattern of mortality, should be based on their desirable properties and not necessarily on their means of derivation. Moreover, our work gives one more example of how in ecological and evolutionary modelling the passage from continuous to discrete time may be non‐obvious.