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Probabilistic projections of distributions of kin over the life course

Author

Listed:
  • Joe Butterick

    (University of Southampton)

  • Jason Hilton

    (University of Southampton)

  • Peter W F Smith

    (University of Southampton)

  • Jakub Bijak

    (University of Oxford)

  • Erengul Dodd

    (University of Southampton)

Abstract

Background: Mathematical kinship demography is an expanding area of research. Recent papers have explored the expected number of kin a typical individual should experience. Despite the uncertainty of the future number and distributions of kin, just one paper investigates it. Objective: We aim to develop a new method for obtaining the probability that a typical population member experiences one or more of some kin at any age through the life course. Methods: Combinatorics, matrix algebra, and convolution theory are combined to find discrete probability distributions of kin number. We propose closed form expressions, illustrating the recursive nature of kin replenishment, using composition of matrix operations. Our model requires as inputs age-specific mortality and fertility. Conclusions: We derive probabilities of kin number for fixed age of kin and over all possible ages of kin. From these the expectation, variance, and other moments of kin number can be found. We demonstrate how kinship structures are conditional on familial events. Contribution: The paper presents the first analytic approach allowing the projection of a full probability distribution of the number of kin of arbitrary type that a population member has over the life course.

Suggested Citation

  • Joe Butterick & Jason Hilton & Peter W F Smith & Jakub Bijak & Erengul Dodd, 2026. "Probabilistic projections of distributions of kin over the life course," Demographic Research, Max Planck Institute for Demographic Research, Rostock, Germany, vol. 54(9), pages 263-308.
    • Handle: RePEc:dem:demres:v:54:y:2026:i:9
    DOI: 10.4054/DemRes.2026.54.9
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    References listed on IDEAS

    as
    1. Hal Caswell, 2022. "The formal demography of kinship IV: Two-sex models and their approximations," Demographic Research, Max Planck Institute for Demographic Research, Rostock, Germany, vol. 47(13), pages 359-396.
    2. Hal Caswell, 2020. "The formal demography of kinship II: Multistate models, parity, and sibship," Demographic Research, Max Planck Institute for Demographic Research, Rostock, Germany, vol. 42(38), pages 1097-1146.
    3. Noreen Goldman, 1978. "Estimating the intrinsic rate of increase of population from the average numbers of younger and older sisters," Demography, Springer;Population Association of America (PAA), vol. 15(4), pages 499-507, November.
    4. Samuel Ellis & Daniel W. Franks & Mia Lybkær Kronborg Nielsen & Michael N. Weiss & Darren P. Croft, 2024. "The evolution of menopause in toothed whales," Nature, Nature, vol. 627(8004), pages 579-585, March.
    5. Marta Pittavino & Bruno Arpino & Elena Pirani, 2025. "Kinlessness at Older Ages: Prevalence and Heterogeneity in 27 Countries," The Journals of Gerontology: Series B, The Gerontological Society of America, vol. 80(1), pages 371-392.
    6. Hal Caswell & Xi Song, 2021. "The formal demography of kinship III: Kinship dynamics with time-varying demographic rates," Demographic Research, Max Planck Institute for Demographic Research, Rostock, Germany, vol. 45(16), pages 517-546.
    7. Hal Caswell, 2024. "The formal demography of kinship VI: Demographic stochasticity and variance in the kinship network," Demographic Research, Max Planck Institute for Demographic Research, Rostock, Germany, vol. 51(39), pages 1201-1256.
    8. Hal Caswell & Rachel Margolis & Ashton Verdery, 2023. "The formal demography of kinship V: Kin loss, bereavement, and causes of death," Demographic Research, Max Planck Institute for Demographic Research, Rostock, Germany, vol. 49(41), pages 1163-1200.
    9. Hal Caswell, 2019. "The formal demography of kinship: A matrix formulation," Demographic Research, Max Planck Institute for Demographic Research, Rostock, Germany, vol. 41(24), pages 679-712.
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    JEL classification:

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

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