IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v12y2024i5p668-d1345266.html
   My bibliography  Save this article

Estimation–Calibration of Continuous-Time Non-Homogeneous Markov Chains with Finite State Space

Author

Listed:
  • Manuel L. Esquível

    (Department of Mathematics, NOVA FCT, and NOVA Math, Universidade Nova de Lisboa, Quinta da Torre, 2829-516 Monte de Caparica, Portugal)

  • Nadezhda P. Krasii

    (Department of Higher Mathematics, Don State Technical University, Gagarin Square 1, Rostov-on-Don 344000, Russia)

  • Gracinda R. Guerreiro

    (Department of Mathematics, NOVA FCT, and NOVA Math, Universidade Nova de Lisboa, Quinta da Torre, 2829-516 Monte de Caparica, Portugal)

Abstract

We propose a method for fitting transition intensities to a sufficiently large set of trajectories of a continuous-time non-homogeneous Markov chain with a finite state space. Starting with simulated data computed with Gompertz–Makeham transition intensities, we apply the proposed method to fit piecewise linear intensities and then compare the transition probabilities corresponding to both the Gompertz–Makeham transition intensities and the fitted piecewise linear intensities; the main comparison result is that the order of magnitude of the average fitting error per unit time—chosen as a year—is always less than 1%, thus validating the methodology proposed.

Suggested Citation

  • Manuel L. Esquível & Nadezhda P. Krasii & Gracinda R. Guerreiro, 2024. "Estimation–Calibration of Continuous-Time Non-Homogeneous Markov Chains with Finite State Space," Mathematics, MDPI, vol. 12(5), pages 1-21, February.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:5:p:668-:d:1345266
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/12/5/668/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/12/5/668/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Sophie de Mol van Otterloo & Jennifer Alonso-García, 2023. "A multi-state model for sick leave and its impact on partial early retirement incentives: the case of the Netherlands," Scandinavian Actuarial Journal, Taylor & Francis Journals, vol. 2023(3), pages 244-268, March.
    2. Poontavika Naka & María del Carmen Boado-Penas & Gauthier Lanot, 2020. "A multiple state model for the working-age disabled population using cross-sectional data," Scandinavian Actuarial Journal, Taylor & Francis Journals, vol. 2020(8), pages 700-717, September.
    3. Joelle H. Fong & Adam W. Shao & Michael Sherris, 2015. "Multistate Actuarial Models of Functional Disability," North American Actuarial Journal, Taylor & Francis Journals, vol. 19(1), pages 41-59, January.
    4. Renshaw, A. E. & Haberman, S., 1995. "On the graduations associated with a multiple state model for permanent health insurance," Insurance: Mathematics and Economics, Elsevier, vol. 17(1), pages 1-17, August.
    5. Eugene Feinberg & Manasa Mandava & Albert N. Shiryaev, 2022. "Kolmogorov’s equations for jump Markov processes with unbounded jump rates," Annals of Operations Research, Springer, vol. 317(2), pages 587-604, October.
    6. Niels Keiding & Per Kragh Andersen, 1989. "Nonparametric Estimation of Transition Intensities and Transition Probabilities: A Case Study of a Two‐State Markov Process," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 38(2), pages 319-329, June.
    7. Marcus Christiansen, 2012. "Multistate models in health insurance," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 96(2), pages 155-186, June.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Qiqi Wang & Katja Hanewald & Xiaojun Wang, 2022. "Multistate health transition modeling using neural networks," Journal of Risk & Insurance, The American Risk and Insurance Association, vol. 89(2), pages 475-504, June.
    2. Manuel L. Esquível & Gracinda R. Guerreiro & Matilde C. Oliveira & Pedro Corte Real, 2021. "Calibration of Transition Intensities for a Multistate Model: Application to Long-Term Care," Risks, MDPI, vol. 9(2), pages 1-17, February.
    3. Fuino, Michel & Wagner, Joël, 2018. "Long-term care models and dependence probability tables by acuity level: New empirical evidence from Switzerland," Insurance: Mathematics and Economics, Elsevier, vol. 81(C), pages 51-70.
    4. Guibert, Quentin & Planchet, Frédéric, 2018. "Non-parametric inference of transition probabilities based on Aalen–Johansen integral estimators for acyclic multi-state models: application to LTC insurance," Insurance: Mathematics and Economics, Elsevier, vol. 82(C), pages 21-36.
    5. Michael Sherris, 2021. "On Sustainable Aged Care Financing in Australia," Australian Economic Review, The University of Melbourne, Melbourne Institute of Applied Economic and Social Research, vol. 54(2), pages 275-284, June.
    6. Odd O. Aalen & Johan Fosen & Harald Weedon-Fekjær & Ørnulf Borgan & Einar Husebye, 2004. "Dynamic Analysis of Multivariate Failure Time Data," Biometrics, The International Biometric Society, vol. 60(3), pages 764-773, September.
    7. D'Amico, Guglielmo & Singh, Shakti & Selvamuthu, Dharmaraja, 2024. "Optimal investment-disinvestment choices in health-dependent variable annuity," Insurance: Mathematics and Economics, Elsevier, vol. 117(C), pages 1-15.
    8. William Lim & Gaurav Khemka & David Pitt & Bridget Browne, 2019. "A method for calculating the implied no-recovery three-state transition matrix using observable population mortality incidence and disability prevalence rates among the elderly," Journal of Population Research, Springer, vol. 36(3), pages 245-282, September.
    9. Mi, Hong & Fan, Xiaodong & Lu, Bei & Cai, Liming & Piggott, John, 2020. "Preparing for population ageing: Estimating the cost of formal aged care in China," The Journal of the Economics of Ageing, Elsevier, vol. 17(C).
    10. Czado, Claudia & Rudolph, Florian, 2002. "Application of survival analysis methods to long-term care insurance," Insurance: Mathematics and Economics, Elsevier, vol. 31(3), pages 395-413, December.
    11. Oliver Lunding Sandqvist, 2023. "A multistate approach to disability insurance reserving with information delays," Papers 2312.14324, arXiv.org, revised Jan 2025.
    12. Christiansen, Marcus C. & Djehiche, Boualem, 2020. "Nonlinear reserving and multiple contract modifications in life insurance," Insurance: Mathematics and Economics, Elsevier, vol. 93(C), pages 187-195.
    13. Javier Pla-Porcel & Manuel Ventura-Marco & Carlos Vidal-Meliá, 2017. "How do unisex life care annuities embedded in a pay-as-you-go retirement system affect gender redistribution?," Documentos de Trabajo del ICAE 2017-11, Universidad Complutense de Madrid, Facultad de Ciencias Económicas y Empresariales, Instituto Complutense de Análisis Económico.
    14. Maegebier, Alexander, 2013. "Valuation and risk assessment of disability insurance using a discrete time trivariate Markov renewal reward process," Insurance: Mathematics and Economics, Elsevier, vol. 53(3), pages 802-811.
    15. Deng, Yuanyuan & Fang, Hanming & Hanewald, Katja & Wu, Shang, 2023. "Delay the Pension Age or Adjust the Pension Benefit? Implications for Labor Supply and Individual Welfare in China," Journal of Economic Behavior & Organization, Elsevier, vol. 212(C), pages 1192-1215.
    16. Ahmad, Jamaal & Bladt, Mogens & Furrer, Christian, 2023. "Aggregate Markov models in life insurance: Properties and valuation," Insurance: Mathematics and Economics, Elsevier, vol. 113(C), pages 50-69.
    17. Carlos Vidal-Meliá & Manuel Ventura-Marco & Juan Manuel Pérez-Salamero González, 2018. "Actuarial accounting for a notional defined contribution scheme combining retirement and longterm care benefits," Documentos de Trabajo del ICAE 2018-16, Universidad Complutense de Madrid, Facultad de Ciencias Económicas y Empresariales, Instituto Complutense de Análisis Económico.
    18. Nadine Gatzert & Alexander Maegebier, 2015. "Critical Illness Insurances: Challenges and Opportunities for Insurers," Risk Management and Insurance Review, American Risk and Insurance Association, vol. 18(2), pages 255-272, September.
    19. repec:ulb:ulbeco:2013/340821 is not listed on IDEAS
    20. Andreas Niemeyer, 2015. "Safety Margins for Systematic Biometric and Financial Risk in a Semi-Markov Life Insurance Framework," Risks, MDPI, vol. 3(1), pages 1-26, January.
    21. Long Xia & Lulu Chai & Hanyu Zhang & Zhaohui Sun, 2022. "Mapping the Global Landscape of Long-Term Care Insurance Research: A Scientometric Analysis," IJERPH, MDPI, vol. 19(12), pages 1-19, June.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:12:y:2024:i:5:p:668-:d:1345266. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.