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Consistency of the GMLE with Mixed Case Interval‐Censored Data

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  • Anton Schick
  • Qiqing Yu

Abstract

In this paper we consider an interval censorship model in which the endpoints of the censoring intervals are determined by a two stage experiment. In the first stage the value k of a random integer is selected; in the second stage the endpoints are determined by a case k interval censorship model. We prove the strong consistency in the L1(μ)‐topology of the non‐parametric maximum likelihood estimate of the underlying survival function for a measure μ which is derived from the distributions of the endpoints. This consistency result yields strong consistency for the topologies of weak convergence, pointwise convergence and uniform convergence under additional assumptions. These results improve and generalize existing ones in the literature.

Suggested Citation

  • Anton Schick & Qiqing Yu, 2000. "Consistency of the GMLE with Mixed Case Interval‐Censored Data," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 27(1), pages 45-55, March.
    • Handle: RePEc:bla:scjsta:v:27:y:2000:i:1:p:45-55
    DOI: 10.1111/1467-9469.00177
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    Cited by:

    1. Guadalupe Gómez & M. Calle & Ramon Oller, 2004. "Frequentist and Bayesian approaches for interval-censored data," Statistical Papers, Springer, vol. 45(2), pages 139-173, April.
    2. Pao-Sheng Shen, 2020. "Nonparametric estimators of survival function under the mixed case interval-censored model with left truncation," Lifetime Data Analysis: An International Journal Devoted to Statistical Methods and Applications for Time-to-Event Data, Springer, vol. 26(3), pages 624-637, July.
    3. Pao-sheng Shen, 2022. "Nonparametric estimation for competing risks survival data subject to left truncation and interval censoring," Computational Statistics, Springer, vol. 37(1), pages 29-42, March.
    4. Michael G. Hudgens & Chenxi Li & Jason P. Fine, 2014. "Parametric likelihood inference for interval censored competing risks data," Biometrics, The International Biometric Society, vol. 70(1), pages 1-9, March.
    5. Peijie Wang & Hui Zhao & Jianguo Sun, 2016. "Regression analysis of case K interval‐censored failure time data in the presence of informative censoring," Biometrics, The International Biometric Society, vol. 72(4), pages 1103-1112, December.
    6. Yuet-Yee Wong, Linda & Yu, Qiqing, 2007. "A bivariate interval censorship model for partnership formation," Journal of Multivariate Analysis, Elsevier, vol. 98(2), pages 370-383, February.
    7. Pao-sheng Shen, 2011. "Nonparametric estimation with doubly censored and truncated data," Computational Statistics, Springer, vol. 26(1), pages 145-157, March.
    8. Yu, Shaohua & Yu, Qiqing & Wong, George Y.C., 2006. "Consistency of the generalized MLE of a joint distribution function with multivariate interval-censored data," Journal of Multivariate Analysis, Elsevier, vol. 97(3), pages 720-732, March.
    9. Wang, Yong, 2008. "Dimension-reduced nonparametric maximum likelihood computation for interval-censored data," Computational Statistics & Data Analysis, Elsevier, vol. 52(5), pages 2388-2402, January.
    10. Ma, Ling & Hu, Tao & Sun, Jianguo, 2016. "Cox regression analysis of dependent interval-censored failure time data," Computational Statistics & Data Analysis, Elsevier, vol. 103(C), pages 79-90.
    11. Sy Han Chiou & Gongjun Xu & Jun Yan & Chiung‐Yu Huang, 2018. "Semiparametric estimation of the accelerated mean model with panel count data under informative examination times," Biometrics, The International Biometric Society, vol. 74(3), pages 944-953, September.
    12. 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.
    13. Marra, Giampiero & Farcomeni, Alessio & Radice, Rosalba, 2021. "Link-based survival additive models under mixed censoring to assess risks of hospital-acquired infections," Computational Statistics & Data Analysis, Elsevier, vol. 155(C).
    14. Qiqing Yu & George Wong & Linxiong Li, 2001. "Asymptotic Properties of Self-Consistent Estimators with Mixed Interval-Censored Data," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 53(3), pages 469-486, September.
    15. Qiqing Yu & Yuting Hsu & Kai Yu, 2014. "A necessary and sufficient condition for justifying non-parametric likelihood with censored data," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 77(8), pages 995-1011, November.
    16. Pao-sheng Shen & Yingwei Peng & Hsin-Jen Chen & Chyong-Mei Chen, 2022. "Maximum likelihood estimation for length-biased and interval-censored data with a nonsusceptible fraction," Lifetime Data Analysis: An International Journal Devoted to Statistical Methods and Applications for Time-to-Event Data, Springer, vol. 28(1), pages 68-88, January.
    17. Li, Chenxi, 2016. "Cause-specific hazard regression for competing risks data under interval censoring and left truncation," Computational Statistics & Data Analysis, Elsevier, vol. 104(C), pages 197-208.
    18. Li, Chenxi, 2016. "The Fine–Gray model under interval censored competing risks data," Journal of Multivariate Analysis, Elsevier, vol. 143(C), pages 327-344.
    19. Chun-Lung Su & Pao-sheng Shen, 2021. "On consistency of the monotone NPMLE of survival function under the mixed case interval-censored model with left truncation," Computational Statistics, Springer, vol. 36(3), pages 1871-1883, September.
    20. Balakrishnan, N. & Zhao, Xingqiu, 2010. "A nonparametric test for the equality of counting processes with panel count data," Computational Statistics & Data Analysis, Elsevier, vol. 54(1), pages 135-142, January.

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