Volume 35 - Article 14 | Pages 381–398

Symmetries between life lived and left in finite stationary populations

By Francisco Villavicencio, Timothy Riffe

Print this page  Facebook  Twitter

 

 
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
DOI:10.4054/DemRes.2016.35.14
Weblink:All publications in the ongoing Special Collection 8 "Formal Relationships" can be found at http://www.demographic-research.org/special/8/
 

Abstract

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 - Syddansk Universitet, Denmark [Email]
Timothy Riffe - Max-Planck-Institut für Demografische Forschung, Germany [Email]

Other articles by the same author/authors in Demographic Research

» Life lived and left: Estimating age-specific survival in stable populations with unknown ages
Volume 39 - Article 37

» The force of mortality by life lived is the force of increment by life left in stationary populations
Volume 32 - Article 29

Most recent similar articles in Demographic Research

» The force of mortality by life lived is the force of increment by life left in stationary populations
Volume 32 - Article 29    | Keywords: stationary population

» Stationary populations with below-replacement fertility
Volume 26 - Article 14    | Keywords: stationary population

» Historical Addendum to "Life lived equals life left in stationary populations"
Volume 26 - Article 7    | Keywords: stationary population

» Life expectancy: Lower and upper bounds from surviving fractions and remaining life expectancy
Volume 24 - Article 11    | Keywords: stationary population

» Total daily change with age equals average lifetime change
Volume 22 - Article 36    | Keywords: stationary population