Volume 35 - Article 14 | Pages 381–398

Symmetries between life lived and left in finite stationary populations

By Francisco Villavicencio, Tim Riffe

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Date received:07 Nov 2015
Date published:19 Aug 2016
Word count:4302
Keywords:Brouard-Carey equality, Lexis diagram, life left, life lived, stationary population, symmetries
Weblink:All publications in the ongoing Special Collection 8 "Formal Relationships" can be found at http://www.demographic-research.org/special/8/


Background: The Brouard-Carey equality describes the symmetries between the distribution of life lived and life left in stationary populations. This result was formally proved for populations of infinite size and continuous time, and a subsequent attempt to prove it for populations of finite size is invalid.

Objective: We attempt to provide a formal mathematical proof of the Brouard-Carey equality for finite stationary populations.

Conclusions: The symmetries between life lived and life left in finite stationary populations can only be proved if time is explicitly discretized. The proof is more complex than in a continuous-time framework, but it conforms with the kinds of data usually available to researchers.

Contribution: The main contribution of this paper is to offer a complete and formal proof of the symmetries between life lived and life left for finite stationary populations in a discrete-time framework. This result is a useful tool for the study of human and non-human populations when the assumption of stationarity is acceptable, especially when subject ages are unknown, but individuals are followed-up until death.

Author's Affiliation

Francisco Villavicencio - Johns Hopkins Bloomberg School of Public Health, United States of America [Email]
Tim Riffe - Max-Planck-Institut für Demografische Forschung, Germany [Email]

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» Life lived and left: Estimating age-specific survival in stable populations with unknown ages
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» The force of mortality by life lived is the force of increment by life left in stationary populations
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