The Average Uneven Mortality index: building on the "e-dagger" measure of lifespan inequality
submitted: 03 April 2023; last edited: 07 April 2023 (2023), unpublished
In recent years, lifespan inequality has become an important indicator of population health, alongside more established longevity measures. Uncovering the statistical properties of lifespan inequality measures can provide novel insights on the study of mortality developments.
We revisit the "e-dagger" measure of lifespan inequality, introduced in Vaupel and Canudas-Romo (2003). We note that, conditioning on surviving at least until age a, e-dagger(a) is equal to the covariance between the conditional lifespan random variable Ta and its transformation through its own cumulative hazard function (hence generalizing a result first noted in Schmertmann, 2020). We then derive an upper bound for e-dagger(a). Leveraging this result, we introduce the "Average Uneven Mortality" (AUM) index, a novel relative mortality index that can be used to analyze mortality patterns. We discuss some general features of the index, including its relationship with a constant ("even") force of mortality, and we study how it changes over time.
The use of the AUM index is illustrated through an application to observed period and cohort death rates as well as to period life-table death rates from the Human Mortality Database. We explore the behavior of the index across age and over time, and we study its relationship with life expectancy. The AUM index at birth declined over time until the 1950s, when it reverted its trend. The index generally increases over age and reduces with increasing values of life expectancy, with differences between the period and cohort perspectives.
We elaborate on Vaupel and Canudas-Romo’s e-dagger measure, deriving its upper bound. We exploit this result to introduce a novel mortality indicator, which enlarges the toolbox of available methods for the study of mortality dynamics. We also develop some new routines to compute e-dagger(a) and σ_Ta from death rates, and show that they have higher precision when compared to conventional and available functions, particularly for calculations involving older ages.
Keywords: Italy, Sweden, inequality, life expectancy, mortality