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A generalization of the Kaplan–Meier estimator for analyzing bivariate mortality under right-censoring and left-truncation with applications in model-checking for survival copula models

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  • Lopez, Olivier

Abstract

In this paper we provide a new nonparametric estimator of the joint distribution of two lifetimes under random right censoring and left truncation which can be seen as a bivariate extension of the Kaplan–Meier estimator. We derive asymptotic results for this estimator, including uniform n1/2-consistency, and develop a general methodology for bivariate lifetime modeling, a critical issue in studying reversion conditions that are commonplace in defined benefit pensions and private annuity contracts. An application to goodness-of-fit for survival copula models is discussed. We show that the procedures that we use are consistent, and propose a bootstrap procedure based on our estimator to compute the critical values. The new technique that we propose is tested on the Canadian dataset initially studied by Frees et al. (1996).

Suggested Citation

  • Lopez, Olivier, 2012. "A generalization of the Kaplan–Meier estimator for analyzing bivariate mortality under right-censoring and left-truncation with applications in model-checking for survival copula models," Insurance: Mathematics and Economics, Elsevier, vol. 51(3), pages 505-516.
    • Handle: RePEc:eee:insuma:v:51:y:2012:i:3:p:505-516
    DOI: 10.1016/j.insmatheco.2012.07.009
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    References listed on IDEAS

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    1. César Sánchez Sellero & Wenceslao González Manteiga & Ingrid Van Keilegom, 2005. "Uniform Representation of Product‐Limit Integrals with Applications," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 32(4), pages 563-581, December.
    2. Ross L. Prentice & F. Zoe Moodie & Jianrong Wu, 2004. "Hazard‐based nonparametric survivor function estimation," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 66(2), pages 305-319, May.
    3. Luciano, Elisa & Spreeuw, Jaap & Vigna, Elena, 2008. "Modelling stochastic mortality for dependent lives," Insurance: Mathematics and Economics, Elsevier, vol. 43(2), pages 234-244, October.
    4. Christian Genest & Jean‐François Quessy & Bruno Rémillard, 2006. "Goodness‐of‐fit Procedures for Copula Models Based on the Probability Integral Transformation," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 33(2), pages 337-366, June.
    5. Michael G. Akritas & Ingrid Van Keilegom, 2003. "Estimation of bivariate and marginal distributions with censored data," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 65(2), pages 457-471, May.
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    Cited by:

    1. Ramon Alemany & Catalina Bolancé & Montserrat Guillén, 2012. "Nonparametric estimation of Value-at-Risk," Working Papers XREAP2012-19, Xarxa de Referència en Economia Aplicada (XREAP), revised Oct 2012.
    2. Jackson P. Lautier & Vladimir Pozdnyakov & Jun Yan, 2022. "Pricing Time-to-Event Contingent Cash Flows: A Discrete-Time Survival Analysis Approach," Papers 2201.04981, arXiv.org, revised Jan 2023.
    3. Souto Arias, Luis A. & Cirillo, Pasquale, 2021. "Joint and survivor annuity valuation with a bivariate reinforced urn process," Insurance: Mathematics and Economics, Elsevier, vol. 99(C), pages 174-189.
    4. Lopez, Olivier, 2019. "A censored copula model for micro-level claim reserving," Insurance: Mathematics and Economics, Elsevier, vol. 87(C), pages 1-14.
    5. Spreeuw, Jaap, 2014. "Archimedean copulas derived from utility functions," Insurance: Mathematics and Economics, Elsevier, vol. 59(C), pages 235-242.
    6. Gribkova, Svetlana & Lopez, Olivier & Saint-Pierre, Philippe, 2013. "A simplified model for studying bivariate mortality under right-censoring," Journal of Multivariate Analysis, Elsevier, vol. 115(C), pages 181-192.
    7. Alemany, Ramon & Bolancé, Catalina & Guillén, Montserrat, 2013. "A nonparametric approach to calculating value-at-risk," Insurance: Mathematics and Economics, Elsevier, vol. 52(2), pages 255-262.
    8. Ramon Alemany & Catalina Bolance & Montserrat Guillen, 2014. "Accounting for severity of risk when pricing insurance products," Working Papers 2014-05, Universitat de Barcelona, UB Riskcenter.

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