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Symmetries between life lived and left in finite stationary populations

Author

Listed:
  • Francisco Villavicencio

    (Universitat de Barcelona)

  • Tim Riffe

    (Euskal Herriko Unibertsitatea (University of the Basque Country))

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.

Suggested Citation

  • Francisco Villavicencio & Tim Riffe, 2016. "Symmetries between life lived and left in finite stationary populations," Demographic Research, Max Planck Institute for Demographic Research, Rostock, Germany, vol. 35(14), pages 381-398.
    • Handle: RePEc:dem:demres:v:35:y:2016:i:14
    DOI: 10.4054/DemRes.2016.35.14
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    References listed on IDEAS

    as
    1. Joshua R. Goldstein, 2012. "Historical Addendum to "Life lived equals life left in stationary populations"," Demographic Research, Max Planck Institute for Demographic Research, Rostock, Germany, vol. 26(7), pages 167-172.
    2. Joshua R. Goldstein, 2009. "Life lived equals life left in stationary populations," Demographic Research, Max Planck Institute for Demographic Research, Rostock, Germany, vol. 20(2), pages 3-6.
    3. Maxim Finkelstein & James Vaupel, 2015. "On random age and remaining lifetime for populations of items," Applied Stochastic Models in Business and Industry, John Wiley & Sons, vol. 31(5), pages 681-689, September.
    Full references (including those not matched with items on IDEAS)

    Citations

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    Cited by:

    1. James W. Vaupel & Francisco Villavicencio, 2018. "Life lived and left: Estimating age-specific survival in stable populations with unknown ages," Demographic Research, Max Planck Institute for Demographic Research, Rostock, Germany, vol. 39(37), pages 991-1008.
    2. Tim Riffe & Pil H. Chung & Jeroen Spijker & John MacInnes, 2016. "Time-to-death patterns in markers of age and dependency," Vienna Yearbook of Population Research, Vienna Institute of Demography (VID) of the Austrian Academy of Sciences in Vienna, vol. 14(1), pages 229-254.

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    More about this item

    Keywords

    stationary population; Lexis diagram; life lived; life left; Brouard-Carey equality; symmetries;
    All these keywords.

    JEL classification:

    • J1 - Labor and Demographic Economics - - Demographic Economics
    • Z0 - Other Special Topics - - General

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