Life expectancy: Lower and upper bounds from surviving fractions and remaining life expectancy

Joel E. Cohen

Volume 24 - Article 11 | Pages 251–256

Life expectancy: Lower and upper bounds from surviving fractions and remaining life expectancy

By Joel E. Cohen

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Date received:	06 Dec 2010
Date published:	08 Feb 2011
Word count:	881
Keywords:	inequalities, life expectancy, life table, stationary population
DOI:	10.4054/DemRes.2011.24.11
Weblink:	All publications in the ongoing Special Collection 8 "Formal Relationships" can be found at http://www.demographic-research.org/special/8/

Abstract

We give simple upper and lower bounds on life expectancy. In a life-table population, if e(0) is the life expectancy at birth, M is the median length of life, and e(M) is the expected remaining life at age M, then (M+e(M))/2≤e(0)≤M+e(M)/2. In general, for any age x, if e(x) is the expected remaining life at age x, and ℓ(x) is the fraction of a cohort surviving to age x at least, then (x+e(x))≤l(x)≤e(0)≤x+l(x)∙e(x). For any two ages 0≤w≤x≤ω, (x-w+e(x))∙ℓ(x)/ℓ(w)≤e(w)≤x-w+e(x)∙ℓ(x)/ℓ(w) . These inequalities give bounds on e(0) without detailed knowledge of the course of mortality prior to age x, provided ℓ(x) can be estimated. Such bounds could be useful for estimating life expectancy where the input of eggs or neonates can be estimated but mortality cannot be observed before late juvenile or early adult ages.