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Empirical likelihood bivariate nonparametric maximum likelihood estimator with right censored data

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  • Jian-Jian Ren
  • Tonya Riddlesworth

Abstract

This article considers the estimation for bivariate distribution function (d.f.) $$F_0(t, z)$$ F 0 ( t , z ) of survival time $$T$$ T and covariate variable $$Z$$ Z based on bivariate data where $$T$$ T is subject to right censoring. We derive the empirical likelihood-based bivariate nonparametric maximum likelihood estimator $$\hat{F}_n(t,z)$$ F ^ n ( t , z ) for $$F_0(t,z)$$ F 0 ( t , z ) , which has an explicit expression and is unique in the sense of empirical likelihood. Other nice features of $$\hat{F}_n(t,z)$$ F ^ n ( t , z ) include that it has only nonnegative probability masses, thus it is monotone in bivariate sense. We show that under $$\hat{F}_n(t,z)$$ F ^ n ( t , z ) , the conditional d.f. of $$T$$ T given $$Z$$ Z is of the same form as the Kaplan–Meier estimator for the univariate case, and that the marginal d.f. $$\hat{F}_n(\infty ,z)$$ F ^ n ( ∞ , z ) coincides with the empirical d.f. of the covariate sample. We also show that when there is no censoring, $$\hat{F}_n(t,z)$$ F ^ n ( t , z ) coincides with the bivariate empirical d.f. For discrete covariate $$Z$$ Z , the strong consistency and weak convergence of $$\hat{F}_n(t,z)$$ F ^ n ( t , z ) are established. Some simulation results are presented. Copyright The Institute of Statistical Mathematics, Tokyo 2014

Suggested Citation

  • Jian-Jian Ren & Tonya Riddlesworth, 2014. "Empirical likelihood bivariate nonparametric maximum likelihood estimator with right censored data," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 66(5), pages 913-930, October.
  • Handle: RePEc:spr:aistmt:v:66:y:2014:i:5:p:913-930
    DOI: 10.1007/s10463-013-0433-x
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    References listed on IDEAS

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    1. Dabrowska, Dorota M., 1989. "Kaplan-Meier estimate on the plane: Weak convergence, LIL, and the bootstrap," Journal of Multivariate Analysis, Elsevier, vol. 29(2), pages 308-325, May.
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    Cited by:

    1. Jian-Jian Ren & Yiming Lyu, 2024. "Weighted Empirical Likelihood for Accelerated Life Model with Various Types of Censored Data," Stats, MDPI, vol. 7(3), pages 1-11, September.
    2. Jian-Jian Ren & Yuyin Shi, 2024. "Empirical likelihood MLE for joint modeling right censored survival data with longitudinal covariates," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 76(4), pages 617-648, August.

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