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On the strong convergence of the product-limit estimator and its integrals under censoring and random truncation

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  • He, Shuyuan
  • Yang, Grace L.

Abstract

It is shown that the product limit estimator Fn of a continuous distribution function F based on the right censored and left truncated data is uniformly strong consistent over the entire observation interval of F allowed under censoring and truncation. Moreover, it is shown that the integral [integral operator] [phi](s) dFn(s) converges almost surely as n-->[infinity] for any nonnegative measurable function [phi] satisfying some mild conditions. The limits of these integrals, however, need not be [integral operator] [phi](s) dF(s). The results are important for studying convergence of sample moments and regression problems when both censoring and truncation are present. A condition of identifiability, often overlooked in the literature is discussed.

Suggested Citation

  • He, Shuyuan & Yang, Grace L., 2000. "On the strong convergence of the product-limit estimator and its integrals under censoring and random truncation," Statistics & Probability Letters, Elsevier, vol. 49(3), pages 235-244, September.
    • Handle: RePEc:eee:stapro:v:49:y:2000:i:3:p:235-244
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    References listed on IDEAS

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    1. Gijbels, I. & Wang, J. L., 1993. "Strong Representations of the Survival Function Estimator for Truncated and Censored Data with Applications," Journal of Multivariate Analysis, Elsevier, vol. 47(2), pages 210-229, November.
    2. Ying, Zhiliang, 1989. "A note on the asymptotic properties of the product-limit estimator on the whole line," Statistics & Probability Letters, Elsevier, vol. 7(4), pages 311-314, February.
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    Cited by:

    1. Sun, Liuquan, 2006. "The strong law under a semiparametric model for truncated and censored data," Statistics & Probability Letters, Elsevier, vol. 76(14), pages 1550-1558, August.
    2. Hwang, Yi-Ting & Wang, Chun-chao, 2008. "A goodness of fit test for left-truncated and right-censored data," Statistics & Probability Letters, Elsevier, vol. 78(15), pages 2420-2425, October.

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