It is well known that life expectancy can be expressed as an integral of the survival curve. The reverse - that the survival function can be expressed as an integral of life expectancy - is also true.force of mortality, life expectancy, life table, stationary population, survival function

James W. Vaupel

Maxim Finkelstein

English

Research Papers in Economics

Demographic Research a free, expedited, online journal
of peer-reviewed research and commentary
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DEMOGRAPHIC RESEARCH
VOLUME 21, ARTICLE 29, PAGES 879-884
PUBLISHED 11 DECEMBER 2009
http://www.demographic-research.org/Volumes/Vol21/29/
DOI: 10.4054/DemRes.2009.21.29
Formal Relationships 6
Survival as a function of life expectancy
Maxim Finkelstein
James W. Vaupel
This article is part of the Special Collection “Formal Relationships”.
Guest Editors are Joshua R. Goldstein and James W. Vaupel.
c ° 2009 Maxim Finkelstein & James W. Vaupel.
This open-access work is published under the terms of the Creative
Commons Attribution NonCommercial License 2.0 Germany, which permits
use, reproduction & distribution in any medium for non-commercial
purposes, provided the original author(s) and source are given credit.
See http://creativecommons.org/licenses/by-nc/2.0/de/Table of Contents
1 Relationship 879
2 Proof 880
3 History and related results 880
4 Applications 881
5 Acknowledgements 883
References 884Demographic Research: Volume 21, Article 29
Formal Relationships 6
Survival as a function of life expectancy
Maxim Finkelstein 1
James W. Vaupel 2
Abstract
It is well known that life expectancy can be expressed as an integral of the survival curve.
The reverse - that the survival function can be expressed as an integral of life expectancy
- is also true.
1. Relationship
Let e(a) denote remaining life expectancy at age a and let `(a) denote the proportion
surviving to age a (the survival function). Then
(1) l(a) =
e(0)
e(a)
exp
µ
¡
Z a
0
1
e(x)
dx
¶
:
1 Department of Mathematical Statistics University of the Free State, PO Box 339, 9300 Bloemfontein,
Republic of South Africa. E-mail: FinkelM.SCI@ufs.ac.za
2 Max Planck Institute for Demographic Research, Konrad-Zuse-Strasse 1, 18057 Rostock, Germany.
E-mail: jwv@demogr.mpg.de
http://www.demographic-research.org 879Finkelstein & Vaupel: Survival as a function of life expectancy
2. Proof
Thepopulationcompositionofalifetablepopulation, i.e., theprobabilitydensityfunction
(pdf), is given by
(2) c(a) = `(a)=e(0):
The proportion of the population age a and older is given by
(3) C(a) =
Z 1
a
c(x)dx:
The intensity of change in C with age is given by the relative derivative
(4) °(a) =
dC(a)
da
C(a)
=
¡c(a)
C(a)
=
¡`(a)=e(0)
R 1
a `(x)dx=e(0)
=
¡1
R 1
a `(x)dx=`(a)
= ¡
1
e(a)
:
Since C(a) = C(0)exp
¡R a
0 °(x)dx
¢
and C(0) = 1, equation (4) implies that
(5) C(a) = exp
µ
¡
Z a
0
1
e(x)
dx
¶
:
It follows from (2) and (3), as shown in (4), that
(6) C(a) = e(a)`(a)=e(0):
Substituting (6) in (5) and rearranging terms yields (1).
Q.E.D.
A slightly more compact proof can be based on T(a) =
R 1
a `(x)dx = e(a)`(a)
instead of equation (3).
3. History and related results
This elegant proof based on the pdf in (2) is due to Meilijson (1972). Finkelstein (2008)
providesrelatedinformationaboutit. Thepdfin(2), whichiswellknowntodemographers,
plays an important role in different branches of probability and statistics. For instance,
it describes the duration of the ﬁrst cycle in [0;1] for a renewal process that started at
t = ¡1 (Ross 1996). It also provides asymptotic (as t ! 1) expressions for the dis-
880 http://www.demographic-research.orgDemographic Research: Volume 21, Article 29
tributions of backward and forward recurrent times for renewal processes. This result
has an important demographic interpretation for stationary populations, namely, that the
distribution of age in a life table population is equal to the distribution of remaining life-
times (Goldstein 2009; Vaupel 2009).
Because
(7) `(a) =
Z 1
a
d(x)dx =
Z 1
a
¹(x)`(x)dx;
where d(x) is the pdf of deaths and ¹(x) denotes the force of mortality, it follows from (4)
that
(8)
1
e(a)
=
`(a)
R 1
a `(x)dx
=
R 1
a ¹(x)`(x)dx
R 1
a `(x)dx
= ¹ ¹(a);
where ¹ ¹(a) is the weighted average of the force of mortality above age a. Because ¹ ¹(a)
is the usual deaths/person-years measure for an open interval, equation (8) has a nice
intuitive interpretation: among those surviving to age a, there will eventually be one
death for every e(a) person-years lived and therefore the average force of mortality above
age a must be 1=e(a).
4. Applications
As discussed by Gupta and Akman (1995), Mi (1995), Gupta and Kirmani (2000), and
Finkelstein (2002), relationships derived from (1) can be used to analyze how the shape
of the age-trajectory of mortality and the shape of the age-trajectory of remaining life
expectancyinﬂuenceeachother. Weprovidehereasingleillustration, notinthesearticles,
of a demographic application.
Taking the relative derivative of both sides of (1) and re-arranging terms leads to
(9a)
de(a)
da
e(a)
´ ¶ e(a) = ¹(a) ¡
1
e(a)
and, using (8), to
(9b) ¶ e(a) = ¹(a) ¡ ¹ ¹(a):
Instead of deriving these formulas from (1), they can be derived from the following
http://www.demographic-research.org 881Finkelstein & Vaupel: Survival as a function of life expectancy
important result:
(10)
de(a)
da
= ¹(a)e(a) ¡ 1:
This result is easily proven by differentiating both sides of e(a) =
R 1
a `(x)dx=`(a).
Hence, instead of deriving (9a) from (1), an alternative proof of (1) can be based on
integrating both sides of (9a) derived from (10).
In any case, (9a) indicates that remaining life expectancy will rise, stay constant or fall
depending on whether the force of mortality exceeds, equals, or is less than the inverse
of remaining life expectancy. Similarly, (9b) implies that remaining life expectancy will
rise, stay constant or fall depending on whether the force of mortality exceeds, equals,
or is less than the average force of mortality after age a. It might be naively thought that
remaining life expectancy always falls when the force of mortality rises, or that remaining
life expectancy always rises when the force of mortality falls, but (9a) and (9b) show that
this is not necessarily the case. What is critical in determining whether remaining life
expectancy rises or falls with age is not whether the force of mortality is rising or falling
but whether the force of mortality exceeds or is less than the inverse of remaining life
expectancy.
If the force of mortality is ultimately monotone after some age am , this observation
can be made more explicit. Suppose ¹(a) is monotonically increasing for a ¸ am ¸ 0.
Then its weighted average in [am;1) is obviously larger than ¹(am) and in accordance
with (9b), ¶ e(a) is monotonically decreasing.
In many human populations as well as populations of various non-human species
and some kinds of equipment, the force of mortality falls from birth to a minimum at
some age (for humans and other mammals often around the age of reproductive maturity)
and then rises afterwards. Equation (9a) indicates that remaining life expectancy will
rise after birth if and only if the force of mortality at age 0 exceeds the inverse of life
expectancy at age 0. If it does, and if the force of mortality decreases until some age
and then increases, remaining life expectancy will initially increase and eventually start
to decline. (See Theorem 2.6 in Finkelstein (2008:35) for the formal proof of a general
result). Consequently, there will be an age at which remaining life expectancy reaches
a maximum. This age, (9a) implies, is the age when the force of mortality equals the
inverse of remaining life expectancy. In general, it will not be the age when the force of
mortality reaches a minimum.
The Swedish female life table for 1751 available at www.mortality.com provides
central death rates at various ages. They can be used as approximations to the force
of mortality. At age 0, the central death rate is given as 0.2122, and is much greater than
the inverse of life expectancy at birth, 1/39.88=0.0251. Remaining life expectancy rises
882 http://www.demographic-research.orgDemographic Research: Volume 21, Article 29
at ages 1, 2, 3 and 4 and reaches a maximum at age 5 of 50.44, with an inverse value of
0.0198. At age 4 the central death rate is 0.0237 > 0.0199, the inverse of remaining life
expectancy at age 4, whereas at age 5, the central death rate is 0.0188 < 0.0198, the inverse
of remaining life expectancy at age 5. This is consistent with remaining life expectancy
reaching a maximum around age 5.
5. Acknowledgements
We thank Annette Baudisch, Joshua Goldstein, Trifon Missov, Peter Wagner and two
anonymous referees for helpful comments.
http://www.demographic-research.org 883Finkelstein & Vaupel: Survival as a function of life expectancy
References
Finkelstein, M.S. (2002). On the shape of the mean residual life function. Applied
Stochastic Models in Business and Industry 18(2): 135–146. doi:10.1002/asmb.461.
Finkelstein, M.S. (2008). Failure Rate Modelling for Reliability and Risk. London:
Springer.
Goldstein, J.R. (2009). Life lived equals life left in stationary populations. Demographic
Research 20(2): 3–6. doi:10.4054/DemRes.2009.20.2.
Gupta, R.C. and Akman, H.O. (1995). Mean residual life functions for certain
types of non-monotonic ageing. Stochastic Models 11(1): 219–225. doi:10.1080/
15326349508807340.
Gupta, R.C. and Kirmani, S.N.U.A. (2000). Residual coefﬁcient of variation and some
characterization results. Journal of Statistical Planning and Inference 91(1): 23–31.
doi:10.1016/S0378-3758(00)00134-8.
Meilijson, I. (1972). Limiting properties of the mean residual lifetime function. The
Annals of Mathematical Statistics 43(1): 354–357. doi:10.1214/aoms/1177692731.
Mi, J. (1995). Bathtub failure rate and upside-down bathtub mean residual life. IEEE
Transactions on Reliability 44(3): 388–391. doi:10.1109/24.406570.
Ross, S.M. (1996). Stochastic Processes. New York: John Wiley & Sons.
Vaupel, J.W. (2009). Life lived and left: Carey’s equality. Demographic Research 20(3):
7–10. doi:10.4054/DemRes.2009.20.3.
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