It is well known that the nonparametric maximum likelihood estimator (NPMLE) can severely underestimate the survival probabilities at early times for left-truncated and interval-censored (LT-IC) data. For arbitrarily truncated and censored data, Pan and Chappel (JAMA Stat Probab Lett 38:49–57, 1998a, Biometrics 54:1053–1060, 1998b) proposed a nonparametric estimator of the survival function, called the iterative Nelson estimator (INE). Their simulation study showed that the INE performed well in overcoming the under-estimation of the survival function from the NPMLE for LT-IC data. In this article, we revisit the problem of inconsistency of the NPMLE. We point out that the inconsistency is caused by the likelihood function of the left-censored observations, where the left-truncated variables are used as the left endpoints of censoring intervals. This can lead to severe underestimation of the survival function if the NPMLE is obtained using Turnbull’s (JAMA 38:290–295, 1976) EM algorithm. To overcome this problem, we propose a modified maximum likelihood estimator (MMLE) based on a modified likelihood function, where the left endpoints of censoring intervals for left-censored observations are the maximum of left-truncated variables and the estimated left endpoint of the support of the left-censored times. Simulation studies show that the MMLE performs well for finite sample and outperforms both the INE and NPMLE.

中文翻译：

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带左截断的混合情况区间删失模型下生存函数的非参数估计量。

众所周知，非参数最大似然估计量 (NPMLE) 可能会严重低估早期左截断和区间删失 (LT-IC) 数据的生存概率。对于任意截断和删失的数据，Pan 和 Chappel (JAMA Stat Probab Lett 38:49–57, 1998a, Biometrics 54:1053–1060, 1998b) 提出了生存函数的非参数估计器，称为迭代 Nelson 估计器 (INE)。他们的模拟研究表明，INE 在克服 NPMLE 对 LT-IC 数据的生存函数的低估方面表现良好。在本文中，我们重新审视了 NPMLE 的不一致问题。我们指出不一致是由左删失观测的似然函数引起的，其中左截断变量用作删失区间的左端点。如果 NPMLE 是使用 Turnbull 的 (JAMA 38:290–295, 1976) EM 算法获得的，这可能会导致对生存函数的严重低估。为了克服这个问题，我们提出了一种基于修改似然函数的修改最大似然估计器（MMLE），其中左删失观测的删失区间的左端点是左截断变量的最大值和支持的估计左端点左删失的时代。模拟研究表明，MMLE 在有限样本上表现良好，并且优于 INE 和 NPMLE。其中左删失观察的删失区间的左端点是左截断变量的最大值和左删失时间支持的估计左端点。模拟研究表明，MMLE 在有限样本上表现良好，并且优于 INE 和 NPMLE。其中左删失观察的删失区间的左端点是左截断变量的最大值和左删失时间支持的估计左端点。模拟研究表明，MMLE 在有限样本上表现良好，并且优于 INE 和 NPMLE。