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Demographic Research a free, expedited, online journal

of peer-reviewed research and commentary

in the population sciences published by the

Max Planck Institute for Demographic Research

Konrad-Zuse Str. 1, D-18057 Rostock · GERMANY

www.demographic-research.org

DEMOGRAPHIC RESEARCH

VOLUME 22, ARTICLE 5, PAGES 115-128

PUBLISHED 22 JANUARY 2010

http://www.demographic-research.org/Volumes/Vol22/5/

DOI: 10.4054/DemRes.2010.22.5

Formal Relationships 7

Life expectancy is the death-weighted average

of the reciprocal of the survival-specific force of

mortality

Joel E. Cohen

This article is part of the Special Collection “Formal Relationships”.

Guest Editors are Joshua R. Goldstein and James W. Vaupel.

c ? 2010 Joel E. Cohen.

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/

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Table of Contents

1

1.1

1.2

Background and relationships

Background

Relationships

116

116

118

2 Proofs119

3 History and related results120

4

4.1

4.2

4.3

4.4

Applications

Lower bound inequality

Lower bound in the exponential distribution

Discrete actuarial approximations

Example based on life tables of the United States in 2004

121

121

121

122

122

5 Acknowledgements126

References127

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Demographic Research: Volume 22, Article 5

Formal Relationships 7

Life expectancy is the death-weighted average of the reciprocal of the

survival-specific force of mortality

Joel E. Cohen1

Abstract

The hazard of mortality is usually presented as a function of age, but can be defined as a

function of the fraction of survivors. This definition enables us to derive new relationships

for life expectancy. Specifically, in a life-table population with a positive age-specific

forceofmortalityatallages, theexpectationoflifeatagexistheaverageofthereciprocal

of the survival-specific force of mortality at ages after x, weighted by life-table deaths at

each age after x, as shown in (6). Equivalently, the expectation of life when the surviving

fraction in the life table is s is the average of the reciprocal of the survival-specific force of

mortality over surviving proportions less than s, weighted by life-table deaths at surviving

proportions less than s, as shown in (8). Application of these concepts to the 2004 life

tables of the United States population and eight subpopulations shows that usually the

younger the age at which survival falls to half (the median life length), the longer the life

expectancy at that age, contrary to what would be expected from a negative exponential

life table.

1Laboratory of Populations, Rockefeller & Columbia Universities, 1230 York Avenue, Box 20, New York, NY

10065-6399, USA. E-mail: cohen@rockefeller.edu

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Cohen: Life expectancy averages reciprocal force of mortality

1.Background and relationships

1.1Background

The life table ?(x), constant in time, with continuous age x, is the proportion of a cohort

(whether a birth cohort or a synthetic period cohort) that survives to age x or longer. In

probabilistic terms, ?(x) is one minus the cumulative distribution function of length of

life x. The maximum possible age ω may be finite or infinite. If ω = ∞, then some

individuals may live longer than any finite bound. By definition, ?(0) = 1 and ?(ω) = 0.

Assume ?(x) is a continuous, differentiable function of x, 0 ≤ x ≤ ω, and assume life

expectancy at age 0 is finite. The age-specific force of mortality at age x is, by definition,

(1)

µ(x) = −

1

?(x)

d?(x)

dx

.

Assume µ(x) > 0 for all 0 ≤ x ≤ ω. The life table ?(x) is strictly decreasing from

?(0) = 1 to ?(ω) = 0 so there is a one-to-one correspondence between age x in [0,ω] and

the proportion s in [0,1] of the cohort that survives to age x or longer. One direction of

this correspondence is given by the life table function s = ?(x) (illustrated schematically

in Figure 1 and for the United States population in 2004 in Figure 3A).

Figure 1: When the force of mortality is positive at every age x, the pro-

portion surviving s, given by the life table according to s = ?(x),

strictly decreases as age x increases, so there is a one-to-one

correspondence between age x and the proportion surviving s.

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Demographic Research: Volume 22, Article 5

There appears to be no standard demographic term for the inverse function that maps

the proportion surviving s, 0 ≤ s ≤ 1, to the corresponding age x, so I propose to

call it the age function a (illustrated schematically in Figure 2 and for the United States

population in 2004 in Figure 3D). In words, the age a(s) at which the fraction s of the

birth cohort survives is the age x at which the life table function ?(x) is s. By definition,

under the assumption µ(x) > 0 for all 0 ≤ x ≤ ω, a(s) = x if and only if ?(x) = s.

Equivalently, by definition, for every s in 0 ≤ s ≤ 1 and every x in 0 ≤ x ≤ ω,

a(?(x)) = x and ?(a(s)) = s. We define a(1/2) as the median life length, that is, the age

by which half the cohort has died.

Figure 2:The function x = a(s) that expresses the age x at which a fraction

s of a birth cohort survives is the inverse of the life table function

s = ?(x) when the force of mortality is positive at every age. Apart

from a reflection across the diagonal line x = s, the curve in this

figure has the same relative shape as the curve in Figure 1 but the

rescaling of both axes makes the two curves look different.

For every s in 0 ≤ s ≤ 1, we define the survival-specific force of mortality λ(s) in

terms of the age-specific force of mortality µ(x) in (1) in three equivalent ways:

(2)

λ(s) = µ(x) if s = ?(x);

or

λ(s) = µ(a(s));

or

λ(?(x)) = µ(x).

In words, the survival-specific force of mortality λ(s) at surviving proportion s equals the

age-specificforceofmortalityµ(x)attheagexwherethelifetable?(x) = s. Thedomain

of the age-specific force of mortality µ is 0 ≤ x ≤ ω while the domain of the survival-

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