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Bayesian nonparametric estimation in the current status continuous mark model

Author

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  • Geurt Jongbloed
  • Frank H. van der Meulen
  • Lixue Pang

Abstract

We consider the current status continuous mark model where, if an event takes place before an inspection time T a “continuous mark” variable is observed as well. A Bayesian nonparametric method is introduced for estimating the distribution function of the joint distribution of the event time (X) and mark variable (Y). We consider two histogram‐type priors on the density of (X,Y). Our main result shows that under appropriate conditions, the posterior distribution function contracts pointwisely at rate n/logn−ρ3(ρ+2) if the true density is ρ‐Hölder continuous. In addition to our theoretical results we provide efficient computational methods for drawing from the posterior relying on a noncentered parameterization and Crank–Nicolson updates. The performance of the proposed methods is illustrated in several numerical experiments.

Suggested Citation

  • Geurt Jongbloed & Frank H. van der Meulen & Lixue Pang, 2022. "Bayesian nonparametric estimation in the current status continuous mark model," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 49(3), pages 1329-1352, September.
    • Handle: RePEc:bla:scjsta:v:49:y:2022:i:3:p:1329-1352
    DOI: 10.1111/sjos.12562
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    References listed on IDEAS

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    1. Marloes H. Maathuis & Jon A. Wellner, 2008. "Inconsistency of the MLE for the Joint Distribution of Interval‐Censored Survival Times and Continuous Marks," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 35(1), pages 83-103, March.
    2. Shota Gugushvili & Frank van der Meulen & Moritz Schauer & Peter Spreij, 2018. "Nonparametric Bayesian volatility estimation," Papers 1801.09956, arXiv.org, revised Mar 2019.
    3. Ghosal,Subhashis & van der Vaart,Aad, 2017. "Fundamentals of Nonparametric Bayesian Inference," Cambridge Books, Cambridge University Press, number 9780521878265.
    4. Hanson, Timothy E., 2006. "Inference for Mixtures of Finite Polya Tree Models," Journal of the American Statistical Association, American Statistical Association, vol. 101, pages 1548-1565, December.
    5. Piet Groeneboom & Geurt Jongbloed & Birgit Witte, 2012. "A maximum smoothed likelihood estimator in the current status continuous mark model," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 24(1), pages 85-101.
    6. Michael G. Hudgens & Marloes H. Maathuis & Peter B. Gilbert, 2007. "Nonparametric Estimation of the Joint Distribution of a Survival Time Subject to Interval Censoring and a Continuous Mark Variable," Biometrics, The International Biometric Society, vol. 63(2), pages 372-380, June.
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