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Kaplan-Meier Estimator under Association

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  • Cai, Zongwu
  • Roussas, George G.

Abstract

Consider a long term study, where a series of possibly censored failure times is observed. Suppose the failure times have a common marginal distribution functionF, but they exhibit a mode of dependence characterized by positive or negative association. Under suitable regularity conditions, it is shown that the Kaplan-Meier estimatorFnofFis uniformly strongly consistent; rates for the convergence are also provided. Similar results are established for the empirical cumulative hazard rate function involved. Furthermore, a stochastic process generated byFnis shown to be weakly convergent to an appropriate Gaussian process. Finally, an estimator of the limiting variance of the Kaplan-Meier estimator is proposed and it is shown to be weakly convergent.

Suggested Citation

  • Cai, Zongwu & Roussas, George G., 1998. "Kaplan-Meier Estimator under Association," Journal of Multivariate Analysis, Elsevier, vol. 67(2), pages 318-348, November.
    • Handle: RePEc:eee:jmvana:v:67:y:1998:i:2:p:318-348
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    References listed on IDEAS

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    1. Stute, Winfried, 1994. "Convergence of the Kaplan-Meier estimator in weighted sup-norms," Statistics & Probability Letters, Elsevier, vol. 20(3), pages 219-223, June.
    2. Roussas, G. G., 1994. "Asymptotic Normality of Random Fields of Positively or Negatively Associated Processes," Journal of Multivariate Analysis, Elsevier, vol. 50(1), pages 152-173, July.
    3. Roussas, George G., 1991. "Kernel estimates under association: strong uniform consistency," Statistics & Probability Letters, Elsevier, vol. 12(5), pages 393-403, November.
    4. Ying, Z. & Wei, L. J., 1994. "The Kaplan-Meier Estimate for Dependent Failure Time Observations," Journal of Multivariate Analysis, Elsevier, vol. 50(1), pages 17-29, July.
    5. Bagai, Isha & Prakasa Rao, B. L. S., 1991. "Estimation of the survival function for stationary associated processes," Statistics & Probability Letters, Elsevier, vol. 12(5), pages 385-391, November.
    6. Burton, Robert M. & Dabrowski, AndréRobert & Dehling, Herold, 1986. "An invariance principle for weakly associated random vectors," Stochastic Processes and their Applications, Elsevier, vol. 23(2), pages 301-306, December.
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    Citations

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    Cited by:

    1. Nikolai Leonenko & Ludmila Sakhno, 2001. "On the Kaplan–Meier Estimator of Long-Range Dependent Sequences," Statistical Inference for Stochastic Processes, Springer, vol. 4(1), pages 17-40, January.
    2. BOUEZMARNI, Taoufik & ROMBOUTS, Jeroen V. K., 2006. "Density and hazard rate estimation for censored and a-mixing data using gamma kernels," LIDAM Discussion Papers CORE 2006118, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    3. Masry, Elias, 2002. "Multivariate probability density estimation for associated processes: strong consistency and rates," Statistics & Probability Letters, Elsevier, vol. 58(2), pages 205-219, June.
    4. Taoufik Bouezmarni & Jeroen Rombouts, 2008. "Density and hazard rate estimation for censored and α-mixing data using gamma kernels," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 20(7), pages 627-643.
    5. Nassira Menni & Abdelkader Tatachak, 2018. "A note on estimating the conditional expectation under censoring and association: strong uniform consistency," Statistical Papers, Springer, vol. 59(3), pages 1009-1030, September.
    6. Roussas, George G., 2000. "Asymptotic normality of the kernel estimate of a probability density function under association," Statistics & Probability Letters, Elsevier, vol. 50(1), pages 1-12, October.
    7. Masry, Elias, 2003. "Local polynomial fitting under association," Journal of Multivariate Analysis, Elsevier, vol. 86(2), pages 330-359, August.
    8. Liang, Han-Ying & Jing, Bing-Yi, 2005. "Asymptotic properties for estimates of nonparametric regression models based on negatively associated sequences," Journal of Multivariate Analysis, Elsevier, vol. 95(2), pages 227-245, August.
    9. Yi Wu & Wei Yu & Xuejun Wang, 2022. "Strong representations of the Kaplan–Meier estimator and hazard estimator with censored widely orthant dependent data," Computational Statistics, Springer, vol. 37(1), pages 383-402, March.
    10. Ioannides, D. A. & Roussas, G. G., 1999. "Exponential inequality for associated random variables," Statistics & Probability Letters, Elsevier, vol. 42(4), pages 423-431, May.
    11. Ming Yuan & Chun Su & Taizhong Hu, 2003. "A Central Limit Theorem for Random Fields of Negatively Associated Processes," Journal of Theoretical Probability, Springer, vol. 16(2), pages 309-323, April.
    12. Zohra Guessoum & Abdelkader Tatachak, 2020. "Asymptotic Results for Truncated-censored and Associated Data," Sankhya B: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 82(1), pages 142-164, May.
    13. Cai, Zongwu, 2001. "Estimating a Distribution Function for Censored Time Series Data," Journal of Multivariate Analysis, Elsevier, vol. 78(2), pages 299-318, August.
    14. Yinxiao Huang & Stanislav Volgushev & Xiaofeng Shao, 2015. "On Self-Normalization For Censored Dependent Data," Journal of Time Series Analysis, Wiley Blackwell, vol. 36(1), pages 109-124, January.
    15. Guessoum, Zohra & Ould Saïd, Elias & Sadki, Ourida & Tatachak, Abdelkader, 2012. "A note on the Lynden-Bell estimator under association," Statistics & Probability Letters, Elsevier, vol. 82(11), pages 1994-2000.

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