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DEMOGRAPHIC RESEARCH
A peer-reviewed, open-access journal of population sciences
DEMOGRAPHIC RESEARCH
VOLUME 50, ARTICLE 44, PAGES 1281–1300
PUBLISHED 12 JUNE 2024
https://www.demographic-research.org/Volumes/Vol50/44/
DOI: 10.4054/DemRes.2024.50.44
Formal Relationship
The Average Uneven Mortality index: Building
on the ‘e-dagger’ measure of lifespan inequality
Marco Bonetti
Ugofilippo Basellini
Andrea Nigri
This publication is part of the Special Collection in the Memory of
Professor James W Vaupel (1945–2022), founder and long-time publisher
of Demographic Research. The Special Collection is edited by Jakub Bijak,
Griffith Feeney, Nico Keilman, and Carl Schmertmann.
©2024 Marco Bonetti, Ugofilippo Basellini & Andrea Nigri.
This open-access work is published under the terms of the Creative
Commons Attribution 3.0 Germany (CC BY 3.0 DE), which permits use,
reproduction, and distribution in any medium, provided the original
author(s) and source are given credit.
See https://creativecommons.org/licenses/by/3.0/de/legalcode
Contents
1 Relationship 1282
1.1 The AUM index, a novel mortality indicator 1282
1.2 An upper bound for e†and for H1282
2 Proof 1283
3 Related results 1285
4 Applications 1286
4.1 AUM at age 0 1286
4.2 AUM at all ages 1293
5 Discussion 1294
6 Acknowledgements 1297
7 Author statement 1297
8 Funding 1297
References 1298
Demographic Research: Volume 50, Article 44
Formal Relationship
The Average Uneven Mortality index: Building on the ‘e-dagger’
measure of lifespan inequality
Marco Bonetti1
Ugofilippo Basellini2
Andrea Nigri3
Abstract
BACKGROUND
In recent years, lifespan inequality has become an important indicator of population
health. Uncovering the statistical properties of lifespan inequality measures can provide
novel insights on the study of mortality.
METHODS
We introduce the ‘Average Uneven Mortality’ (AUM) index, a novel mortality indicator
for the study of mortality patterns and lifespan inequality. We prove some new properties
of interest, as well as relationships with the ‘e-dagger’ and entropy measures of lifespan
inequality.
RESULTS
The use of the AUM index is illustrated through an application to observed period and
cohort 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; also,
the index generally increases with age.
CONTRIBUTION
The AUM index is a normalized version of Vaupel and Canudas-Romo’s e-dagger mea-
sure that can be meaningfully compared across countries and over time. Additionally,
we derive an upper bound for both e-dagger and the life-table entropy measures, which
are novel formal results. Finally, we develop novel routines to compute e-dagger and
the standard deviation of lifetimes from death rates, which are often more precise than
available software, particularly for calculations involving older ages.
1Carlo F. Dondena Research Center, Bocconi University, Milan, Italy. Email: marco.bonetti@unibocconi.it.
2Max Planck Institute for Demographic Research, Rostock, Germany.
3University of Foggia, Foggia, Italy.
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1. Relationship
1.1 The AUM index, a novel mortality indicator
Let Tbe a nonnegative random variable denoting time to death, which is distributed
according to the probability density function f(t). Let S(t) = exp[−Rt
0λ(u)du] =
exp[−Λ(t)] denote the survival function, that is the fraction of the population expected to
survive at least tyears, where λ(t)indicates the force of mortality or hazard function of
the population, and Λ(t)the cumulative hazard function of T.
We define the ‘Average Uneven Mortality’ (AUM) index as the correlation coeffi-
cient between the time to death random variable Tand its transformation through its own
cumulative hazard function:
AUM = Corr[T, Λ(T)] with 0 <AUM ≤1 . (1)
The AUM index can be used to study mortality patterns and lifespan inequality. It
ranges between 0 and 1, and it is equal to 1 if and only if Thas an exponential distribution
with parameter θ(T∼Exp(θ); for a proof of this result, see Section 2). As a conse-
quence, the index can help determine whether the hazard rate is constant (AUM = 1) or
varies with age (AUM <1). Note that this can be particularly useful when one considers
the tail of a survival distribution – that is, the distribution of Tconditionally on surviving
up to some age a.
1.2 An upper bound for e†and for H
Let e†=Rω
0e(u)f(u)du denote the life lost to mortality (Vaupel and Canudas-Romo
2003), where the remaining life expectancy at age uis e(u) = Rω
uS(t)dt/S(u), and ωis
the highest age attained in the population. Further, let σTdenote the standard deviation
of the time to death T,σT=hRω
0(x−e0)2f(x)dxi1/2
, where for brevity we use the
shorthand notation e0=e(0).
The AUM index defined above can also be written as
AUM = e†
σT
so that it can also be interpreted as a normalized version of the e†inequality measure; as
such, the index allows one to conduct lifespan inequality comparisons across countries
and over time due to its fixed (0, 1] support.
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The support of the AUM index implies an upper bound on e†. In particular, it holds
that:
0< e†≤σT;(2)
that is, e†is bounded from above by the standard deviation of the time to death T.
Let
•
AUM denote the partial derivative of AUM with respect to the calendar year.4
In Section 2 we also show that the relative derivative of AUM is such that
•
log(AUM) ⪋0⇐⇒
•
log(e†)⪋•
log(σT)(3)
that is, the relative change of the AUM index is positive (negative) when the relative
change in e†is greater (lower) than the relative change in σT.
Lastly, one may also consider the relationship between the AUM index and the en-
tropy measure H=e†/e0(Leser 1955; Keyfitz 1977; Demetrius 1974). The entropy
can be expressed in terms of the AUM index as H=AUM ·CV , where C V =σT/e0
is the coefficient of variation of the distribution. From this, one obtains the additional
interesting result:
0<H ≤ CV .(4)
Note that all these results also hold more generally when the indices are computed con-
ditionally on surviving until any given age a.
2. Proof
We start by proving that the support of the AUM index is (0, 1], as stated in Equation (1).
Below we note that AUM >0holds; AUM ≤1holds by the correlation inequality. We
now prove that AUM is equal to 1 if and only if Thas an exponential distribution. On
the one hand, if T∼Exp(θ), then Λ(t) = θt and Corr[T, Λ(T)] = AUM = 1. On the
other hand, if AUM = Corr[T, Λ(T)] = 1, then Λ(t) = a+bt. Since Λ(0) = 0, then
a= 0, implying that Λ(t) = bt and S(t) = e−bt – that is, T∼Exp(b).
We now show that 0< e†≤σT. Schmertmann (2020) shows that Vaupel and
Canudas-Romo’s e†equals
e†= Cov[T, Λ(T)] (5)
4In the following, a dot over a function will denote its partial derivative with respect to calendar year y(which
may refer to either a given time period or birth cohort), but we drop the notation yto ease readability.
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(for an alternative proof of that relationship, see Supplementary Materials).
Rewriting (5) in terms of correlation rather than covariance produces
e†= Corr[T, Λ(T)] ·σT·σΛ(T).
We consider the well-known fact that, for any hazard function λ(t),Λ(T)∼Exp(1) (for
a short proof, see Supplementary Materials). This implies that σΛ(T)= 1, yielding the
interesting new expression
e†= Corr[T, Λ(T)] ·σT.
Given the definition AUM = Corr[T, Λ(T)], we have
AUM = Cov[T, Λ(T)]
σT·σΛ(T)
=e†
σT
,
which must therefore be strictly positive since e†>0and σT>0when Tadmits a
probability density function. Hence 0<AUM ≤1, and therefore it follows that 0<
e†≤σT, which proves Equation (2).
Let us now derive the partial derivative of AUM with respect to calendar year y:
•
AUM =
•
e†σT−•
σTe†
σ2
T
,(6)
from which it follows that the partial (and relative) derivative of AUM is positive (neg-
ative) when the numerator of Equation (6) is positive (negative). Since
•
log(AUM) =
•
AUM/AUM, and assuming that both σTand e†are strictly positive, we immediately
have
•
log(AUM) ⪋0⇐⇒
•
AUM ⪋0⇐⇒
•
e†σT−•
σTe†⪋0
⇐⇒
•
e†
e†⪋
•
σT
σT
⇐⇒
•
log(e†)⪋•
log(σT),
which proves Equation (3).
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Given the definition of Hand the newly found range for AUM, Equation (4) follows
immediately.
Lastly, consider the case in which all subjects have the same initial exact age aand
face identical force of mortality λ(t)in future years t≥0beyond a. We can generalize
our results and definitions to this more general setting, so that, for example,
0< e†(a)≤σT(a) ,
where e†(a)is the life lost to deaths after age a, and σT(a)is the standard deviation of
time to death after age a, both conditional on survival to age a. Note that this scenario
can be rewritten in terms of conditional random variable and conditional density func-
tion of the original population, starting from age 0 rather than age a(see Supplementary
Materials).
3. Related results
The closest link to the relationships that we present in this paper is provided by Schmert-
mann (2020) in the framework of revivorship models, where the author derives the equal-
ity between e†and the covariance between Tand its transformation through its own
cumulative hazard function.
A recent interesting use of the cumulative hazard function is provided by Ullrich,
Schmertmann, and Rau (2022): The authors introduce a new longevity measure based on
the cumulative hazard. The proposed death expectancy H1indicator corresponds to the
age at which the cumulative hazard is equal to one (or equivalently, the survival function
is about 36.8%). The authors argue that the H1measure could be used as a dynamic
threshold age for the oldest-old.
In demography, there exists a variety of indicators that are used to summarize the
age-at-death distribution. Several indicators focus on the first moment, or location, of the
distribution – that is, the so-called central longevity indicators (mean, median, and modal
ages at death) (Cheung et al. 2005; Canudas-Romo 2010). Another class of indicators is
used to study the second moment, or scale, of the distribution. Both absolute and rela-
tive measures of the variability of the distribution are used for this purpose. Examples
of absolute indices are the variance and the e†measures, while the life-table entropy, the
coefficient of variation, and the Gini coefficient are examples of relative measures. Typ-
ically, relative measures of variability are computed by dividing an absolute variability
measure by life expectancy. Up to our knowledge, the AUM index is among the very first
mortality indicators that go beyond these two classes of indicators, by analyzing the ratio
of two absolute measures of variation. One recent proposal in this direction is the ratio of
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expansion to compression measure, which considers the e†components before and after
the threshold age at death (Zhang and Li 2020).
The life-table entropy measure can be partly related to the AUM index, since both
measures share the same numerator. The life-table entropy is a relative measure of vari-
ability of the age-at-death distribution compared to life expectancy at birth (Leser 1955;
Keyfitz 1977; Demetrius 1974; Rezaei and Yari 2021). The difference between AUM and
His that while the former standardizes by the standard deviation, the latter standardizes
by the life expectancy at birth e0. The difference in the denominator of the two mea-
sures results into two different interpretations of the indices: The entropy measures the
(relative) variability of the distribution, while the AUM measures how close the distri-
bution is from having constant mortality. Both indices can nonetheless be used for the
study of lifespan inequality, with the AUM providing an innovative perspective in terms
of normalized lifespan inequality. Another recent attempt in this direction is made by
Permanyer and Shi (2022), who introduce normalized lifespan inequality to explicitly
consider that life expectancy has been increasing at a faster pace than maximal length of
life. For any given year, the authors compute normalized lifespan inequality by dividing
lifespan inequality indices by their maximum value under an hypothetical distribution
with life expectancy equal to the observed one.
4. Applications
Here we illustrate the use of the AUM index with an application to observed period and
cohort death rates (obtained by dividing deaths by exposures), as well as to period life-
table death rates. All data were retrieved from the Human Mortality Database (2024,
henceforth HMD). Routines for deriving these results were developed in R(R Core Team
2023) and are available in the Supplementary Materials as well as in an open-access
repository.5The analytical formulas that we derived and employed for the implemen-
tation of our routines are available in the Supplementary Materials. Unless otherwise
specified, all mortality measures are computed from age zero.
4.1 AUM at age 0
We start by investigating the temporal evolution of the AUM index in observed period
death rates for two populations: Swedish females and Italian males. Figure 1 shows the
e†,σT, and AUM measures over time for both populations. In the left panels, we can ob-
serve the well-known reduction of lifespan inequality over most of the period analyzed,
coinciding with the increase in life expectancy (see, e.g., Edwards and Tuljapurkar 2005;
5Available at https://osf.io/fj94p/.
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Wilmoth and Horiuchi 1999; Aburto et al. 2020). The two graphs show that e†never
exceeds σT, in agreement with our derived upper bound of e†. The right panels show that
the AUM index declined rather consistently from 1751 until the 1950s, when it reached
a minimum value. Thereafter, a reversal of the decreasing trend is observable. Sudden
increases in the index are visible in correspondence with the Spanish flu (for both popu-
lations) and the two World Wars (for males only). From the 2000s, the index displays a
rather constant behavior.
Figure 1: Evolution of the e†(points) and σT(triangles) lifespan variability
measures (left panels), and of the AUM index (right panels) over
time for Swedish females and Italian males, 1751–2022. Colors
correspond to different levels of life expectancy at birth (e0)
σ
σ
Source (all figures): Authors’ own elaborations on data from the HMD (2024).
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Analyzing the temporal trend of the AUM index across the 41 populations available
in the HMD by sex provides additional insights. Figure 2 shows a rather substantial
overlap in the decrease of the AUM index across sexes until the 1950s (except during
the two World Wars). From the 1950s onward, as the AUM index stopped declining
and started its increase, a marked departure from the overlap between sexes occurred,
with male populations generally characterized by greater values of the AUM. It is further
worth noticing that, for several female populations, the decline of the AUM halted and
reversed somewhat later than the 1950s.
Figure 2: Evolution of the AUM index over time for 41 female (purple) and
male (orange) populations, 1751–2023
females
males
0.7
0.8
0.9
1750 1800 1850 1900 1950 2000
Year
AUM
How can we interpret these trends of the AUM index? Formally, as we have shown,
the AUM decreases (increases) when the relative change in e†is lower (greater) than the
relative change in σT. This means that throughout most of the period analyzed (1751–
2023), the relative change (generally, reduction) in σTwas greater than the one in e†;
however, a reversal of this trend occurred around the 1950s–60s. This period is often
identified with a transition to a new mortality regime, characterized by an acceleration of
mortality improvements at older ages (Kannisto et al. 1994; Vaupel et al. 1998; Wilmoth
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and Horiuchi 1999) and a more pronounced shifting dynamic of the age-at-death distribu-
tion (Bergeron-Boucher, Ebeling, and Canudas-Romo 2015). From this perspective, the
AUM index provides insights on the transition from mortality compression to mortality
shifting (Janssen and de Beer 2019).
An alternative interpretation of these findings can be made considering that the AUM
is the normalized version of e†. The normalization implies that AUM takes values on a
fixed support (i.e., between 0 and 1) for all years and populations. Conversely, e†can
vary on very different supports. Consider, for example, the top left panel of Figure 1. Un-
til the 1900s, e†could take values up to approximately 33 years; in 2000, its maximum
value would have been approximately 13 years. The fact that e†(and several other lifes-
pan inequality measures) has a time-varying support may bias our assessment of lifespan
inequality trends. Conversely, the fixed support of the AUM index allows for a more
meaningful comparison of the index across countries and over time. Our findings suggest
that the normalized years of life lost have not continued to decrease throughout the time
period analysis (as suggested instead from the historical evolution of lifespan inequality
measures, see, e.g., Edwards and Tuljapurkar 2005; Wilmoth and Horiuchi 1999). As
lifespan inequality continued to decrease, normalized lifespan inequality started to in-
crease around the 1950s–60s. This is further illustrated by Figure 3, which shows, for the
41 female populations of the HMD, the relationship between the AUM index and two in-
dices of (absolute and relative) lifespan variability: the e†and the entropy of the life table.
The figure clearly shows that, as lifespan inequality continued to reduce over time, the
AUM index declined for most of the period considered, reaching a minimum for values
of e†and the entropy around 13.5 and 0.2, respectively, when it then started to increase.
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Figure 3: Relationship between the AUM index and absolute and relative
lifespan variability, measured with the e†(left) and the life-table
entropy (right), respectively, for 41 female populations from 1751
to 2023. Colors correspond to different calendar years
The reduction of the AUM index throughout most of the analyzed period can also
be interpreted with respect to how close the distribution is to an exponential one (i.e.,
one with constant mortality). On the one hand, the density of the exponential distribution
is monotonically decreasing, having its maximum at age zero. On the other hand, the
age-at-death distribution is typically bimodal, with one mode in infancy and another in
late life (Kannisto 2001). It is probably not surprising that the highest values of the AUM
index were observed in the past, when a significant number of deaths were occurring at
infant and childhood ages: Indeed, the corresponding distribution of deaths was char-
acterized by high and monotonically decreasing values at the youngest ages, somewhat
closer to the shape (at least in those early ages) of the exponential distribution. Mortality
improvements at these ages throughout subsequent decades decreased the relative impor-
tance of deaths at these ages, with more and more deaths occurring at older ages. These
improvements in infant mortality are reflected in the reduction of the AUM.
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Next, we analyze the relationship between the AUM index and period life expectancy
at birth. Figure 4 shows this relationship for the 41 populations in the HMD, by sex. For
low levels of life expectancy, there is a linear negative relationship between the two mea-
sures. However, there appears to exist a threshold level of life expectancy (around age 70
and 60 for females and males, respectively) at which the negative relationship ceases to
hold. For females, a positive relationship emerges above the threshold, whereas for males,
the relationship appears to be more erratic. It should be noted that, due to data limitations
regarding historical data, few data points are available for older periods, which are charac-
terized by lower levels of life expectancy; as such, the strong linear relationship observed
for low levels of life expectancy could be partially due to data limitations.
Figure 4: Relationship between the AUM index and period life expectancy at
birth for 41 female (left) and male (right) populations from 1751 to
2023. Colors correspond to different calendar years. The black
line corresponds to smoothing the observed data using a cubic
regression spline (using the gam function of the mgcv package
(Wood 2017))
Females
Males
25 50 75 25 50 75
0.7
0.8
0.9
1.0
Period Life Expectancy
AUM
1800
1850
1900
1950
2000
Year
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We now move to the same analysis using cohort instead of period observed death
rates. Figure 5 shows the relationship between the AUM index and cohort life expectancy
at birth by sex for all ten HMD populations with cohort life-table data.6Unlike the period
analysis, one observes a lack of reversal of the linear relationship between the two mea-
sures. For cohorts, the AUM tends to linearly decrease over increasing values of cohort
life expectancy (typically belonging to the more recent cohorts). It is worth noting that
this linear relationship is linked to the cohort nature of the data analyzed and not a result
of country selection: Indeed, the period analysis of this relationship for the ten countries
shows a very similar pattern to that reported in Figure 4, with a break in the linear relation-
ship at higher levels of life expectancy (see Figure S1 in the Supplementary Materials).
Figure 5: Relationship between the AUM index and cohort life expectancy at
birth for ten female (left) and male (right) populations from 1751
to 1932. Colors correspond to different birth cohorts. The black
line corresponds to smoothing the observed data using a cubic
regression spline (using the gam function of the mgcv package
(Wood 2017))
Females
Males
20 40 60 80 20 40 60 80
0.6
0.7
0.8
0.9
Cohort Life Expectancy
AUM
1800
1850
1900
Cohort
6The ten populations are those of France, Italy, Finland, Denmark, Sweden, the Netherlands, Iceland, Norway,
Switzerland, and England and Wales.
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4.2 AUM at all ages
Finally, we turn to study the behavior of the AUM index at all ages, focusing on a single
population only, namely, Swedish females. In addition to computing the AUM using
observed death rates, we also employ period life-table death rates. The reason for doing
so is that observed and life-table death rates differ at the oldest ages, since the latter are
smoothed at older ages using the Kannisto model of mortality (see Wilmoth et al. 2021:
p. 34).
Figure 6 shows the results of the age analysis. In general, we observe that the AUM
index tends to increase with age (declining in the first few years of life in the oldest
periods considered). Importantly, the figure highlights the difference of the AUM index
at the oldest ages. When the index is computed on observed rates (left panel), there
is high variability of the AUM estimate at the oldest ages, reflecting the variability of
the underlying rates; conversely, when life-table death rates are used as input, the AUM
index approaches the value of 1. This was an expected result since life-table death rates
are smoothed at older ages according to a logistic pattern (the Kannisto model), which is
characterized by a hazard function that becomes increasingly flatter, thus mimicking an
exponential (constant) hazard (whose AUM value is exactly 1).
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Figure 6: AUM index over ages 0 to 110+ for Swedish females for the years
1751–2022 computed on observed rates (left panel) and life-table
rates (right panel)
observed rates
life−table rates
0 30 60 90 0 30 60 90
0.6
0.7
0.8
0.9
1.0
Age
AUM
1800
1850
1900
1950
2000
Year
It is worth mentioning here that conventional routines generally employed to calcu-
late life-table variability measures (such as those available in the LifeIneq R package,
Riffe, van Raalte, and Dudel 2023) return AUM estimates that exceed its upper bound
at the very oldest ages (i.e., AUM values greater than 1), suggesting some potential bias
in the estimation of e†or σT, or both, at older ages. Please refer to the Supplementary
Materials for a comparison analysis between our AUM estimates and those derived from
conventional routines.
5. Discussion
In this paper, we have elaborated on the e†measure of lifespan inequality introduced by
Vaupel and Canudas-Romo (2003). Leveraging a recent result noted in Schmertmann
(2020), we derived the upper bound of e†. This is, to our knowledge, a novel and im-
portant result. It is intriguing that the upper bound of e†is another absolute measure
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of lifespan inequality – the standard deviation of ages at death in the population (σT).
Even more intriguing, we have shown that e†reaches its upper bound if and only if the
underlying age-at-death distribution is exponential.
The upper bound of e†stimulated us to introduce the ‘Average Uneven Mortality’
(AUM) index, a new mortality index that can be used to study mortality across age and
over time. The index has two closely related interpretations. On the one hand, it measures
the linearity of the relationship between the random variable Tand its cumulative hazard
function; on the other hand, and equivalently, it measures the distance of the age-at-
death distribution from an exponential one, or the distance of the hazard function from a
constant (‘even’) hazard. The AUM is a relative index, bounded between 0 and 1, and it is
obtained by dividing two absolute measures of variation of the age-at-death distribution.
This is one among the very first proposals to build a mortality indicator based on (the ratio
of) two lifespan inequality measures. Other relative indicators of mortality employed in
the literature are computed by dividing an absolute variability measure by life expectancy
(e.g., life-table entropy, coefficient of variation, Gini coefficient).
Introducing a novel mortality index raises the question ‘So what?’. We believe that
the AUM index provides novel insights on the study of human mortality age patterns and
time developments. Importantly, the AUM index takes values on a fixed support, suggest-
ing that comparison across countries and over time are more meaningful than when one
uses an indicator whose range can change over time (such as e†). Indeed, in applied statis-
tics, the correlation coefficient is generally favoured to the covariance, since the latter tells
little about the strength of the dependence between two random variables. The analysis of
the normalized e†suggests that the decrease of lifespan inequality has reversed its secular
decline in most recent decades. Our findings align well with those of Permanyer and Shi
(2022), who also observed that declines of normalized lifespan inequality stopped and
even reversed at high levels of life expectancy. Furthermore, while it is well known that
lifespan inequality measures are highly correlated between each other (see, e.g., Wilmoth
and Horiuchi 1999; Van Raalte and Caswell 2013), the evolution of the AUM index shows
that the relative change of e†and σTdiffered over time: The relative change (typically
reduction) in σTwas greater than the one of e†for most of the period that we analyzed.
The 1950s marked a clear reversal of this trend, likely connected to accelerating mortality
improvements at older ages and related shifting of the age-at-death distribution.
The relationship between the AUM index and life expectancy at birth provides an
interesting perspective on the evolution of the age-at-death distribution. We found a lin-
ear negative relationship between the AUM and life expectancy, which ceased to hold
only at high levels of period life expectancy. Interestingly, this disruption did not occur
in the analysis of cohort mortality data. Due to the limitation of such data, we cannot
know whether this disruption will materialize for more recent birth cohorts, or if it is a
specific feature of the most recent period death rates. If the second hypothesis were to
hold true, it would imply that information on the scale of the age-at-death distribution
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would be predictive of its location, with important consequences for mortality analysis
and forecasting.
The AUM index can also be employed to study the mortality age pattern. In our
analysis, we observed that the AUM index generally increases over age. In the analysis
of period life tables (with modeled death rates), the index always approaches its upper
limit at older ages, signaling that the hazard function resembles a constant exponential
hazard. When observed death rates are employed instead, there is significant variability
in the estimate of the AUM at older ages. Clearly, employing the AUM index for detect-
ing mortality deceleration, and eventually the existence of a mortality plateau, has great
potential; our proposed indicator could thus contribute to the current debate about the
mortality plateau at the oldest ages (see, e.g., Barbi et al. 2018; Dang et al. 2023; Gampe
2021; Newman 2018). The quantification of the statistical uncertainty associated with the
estimated AUM estimate should be a critical aspect to inform such analysis, especially
for small sample sizes, and we plan to pursue this in our future research.
An important contribution of our work relates to the software routines that we have
developed to calculate the AUM index. Initially, we started by computing the AUM using
standard and available routines for calculating lifespan inequality measures from a life
table. The results that we obtained were unexpected as the AUM was exceeding its upper
bounds at older ages – an empirical result that contradicted our theoretical findings. As
such, we decided to implement new routines based on the formulas that we derived in
this paper. The new empirical estimates that we obtained did not present such anoma-
lies. Evidence presented in the Supplementary Materials suggests that our routines for
computing e†and σTimprove estimation precision with respect to conventional routines,
particularly for the older ages. Our routines are publicly available in the Supplementary
Materials accompanying this article as well as in a public repository, and we hope that
further computational efforts will be directed to assess estimation accuracy of lifespan
variability measures at the oldest ages.
Given the definitions of AUM and of the entropy index H, if two populations have
equal e†but different AUM, that information would be equivalent to an examination of
the difference in their standard deviations. This is similar to the conclusion that one would
draw if two populations have same e†but different H, with respect to differences in life
expectancy between the two populations. Combining these two observations yields the
fact that if two populations have different values for AUM but equal H(or the other way
around), that should therefore be attributed to a difference in the coefficients of variation
of the two populations. In general, it seems clear that an understanding of the survival
pattern of a population will benefit from the examination of all these indices.
Lastly, note that measuring e†in standard deviation units through AUM improves the
interpretation of inequality in mortality by allowing one to highlight novel shape behav-
iors. Indeed, the AUM index can be related to the more general study of the shape of the
age-at-death distribution, which has gained increasing attention in most recent decades
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Demographic Research: Volume 50, Article 44
(for a recent review, see, e.g., Bonetti, Gigliarano, and Basellini 2021). In particular, as
we showed, the AUM index approaches the value of one if and only if the tail behavior is
that of an exponential random variable, regardless of the value of the parameter θof the
distribution. Such invariance with respect to θensures that the phenomenon of increasing
AUM to its upper bound is not equivalent to the simple decrease in the standard deviation
of the death ages (which is equal to 1/θ for the exponential model). We believe that this
novel mortality indicator can provide additional insights on human mortality, enlarging
the toolbox of available methods for the analysis of mortality developments.
6. Acknowledgements
The authors would like to thank Jos´
e Manuel Aburto, the Guest Editor Carl Schmert-
mann, and two anonymous reviewers for their comments on a previous version of this
manuscript. UB would like to thank Jim Vaupel for his kindness and generosity, for his
support and encouragement to become a demographer, and for his inestimable contribu-
tions to our discipline – we all miss you very much.
7. Author statement
The first two authors contributed equally to the paper. Conceptualization: MB, UB, AN.
Methodology: MB. Software: MB, UB. Data curation and visualization: UB. Supervi-
sion: MB. Writing original draft: MB, UB. Reviewing and editing: MB, UB, AN.
8. Funding
Marco Bonetti and Andrea Nigri were supported by the MUR-PRIN 2022 project
CARONTE (Prot. 2022KBTEBN), funded by the European Union – Next Generation
EU. MB was also supported by the MUSA – Multilayered Urban Sustainability Action –
project, funded by the European Union – Next Generation EU, under the National Re-
covery and Resilience Plan (NRRP) Mission 4 Component 2 Investment Line 1.5.
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