Nonparametric estimation of the cumulative distribution function and the probability density of a lifetime X modified by a current status censoring (CSC), including cases of right and left missing data, is a classical ill-posed problem with biased data. The biased nature of CSC data may preclude us from consistent estimation unless the biasing function is known or may be estimated, and its ill-posed nature slows down rates of convergence. Under a traditionally studied CSC, we observe a sample from $(Z,\Delta )$ where a continuous monitoring time $Z$ is independent of $X$, $\Delta :=I(X\leq Z)$ is the status, and the bias of observations is created by the density of $Z$ which is estimable. In presence of right or left missing, we observe corresponding samples from $(\Delta Z,\Delta )$ or $((1-\Delta )Z,\Delta )$; the data are again biased but now the density of $Z$ cannot be estimated from the data. As a result, to solve the estimation problem, either the density of $Z$ must be known (like in a controlled study) or an extra cross-sectional sampling of $Z$, which is typically simpler than an underlying CSC study, be conducted. The main aim of the paper is to develop for this biased and ill-posed problem the theory of efficient (sharp-minimax) estimation which is inspired by known results for the case of directly observed $X$. Among interesting aspects of the developed theory: (i) While sharp-minimax analysis of missing CSC may follow the classical Pinsker’s methodology, analysis of CSC requires a more complicated estimation procedure based on a special smoothing in both frequency and time domains; (ii) Efficient estimation requires solving an old-standing problem of approximating aperiodic Sobolev functions; (iii) If smoothness of the cdf of $X$ is known, then its rate-minimax estimation is possible even if the density of $Z$ is rougher. Real and simulated examples, as well as extensions of the core models to dependent $X$ and Z and case-control CSC, are presented.

中文翻译：

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锐利的极小极大分布估计，用于当前状态检查，有无丢失

由当前状态检查（CSC）修改的累积分布函数和寿命X的概率密度的非参数估计（包括左右遗失数据的情况）是带有偏差数据的经典不适定问题。CSC数据的偏差性质可能使我们无法进行一致的估计，除非偏差函数已知或可以估计，并且其不适定的性质会减慢收敛速度。在传统研究的CSC下，我们观察到来自$（Z，\ Delta）$的样本，其中连续监视时间$ Z $与$ X $无关，$ \ Delta：= I（X \ leq Z）$是状态，而观测值的偏差是由可估计的$ Z $密度造成的。在存在左右缺失的情况下，我们观察到来自$（\ Delta Z，\ Delta）$或$（（1- \ Delta ZZ，\ Delta）$的样本；数据再次出现偏差，但现在无法根据数据估算$ Z $的密度。结果，要解决估计问题，必须知道$ Z $的密度（如在对照研究中），或者要额外进行$ Z $的横截面采样，这通常比基础CSC研究更简单。进行。本文的主要目的是针对这个有偏见和不适定的问题发展有效（锐利-极小极大值）估计的理论，该理论受到直接观察到的$ X $情况下已知结果的启发。在已发展的理论的有趣方面中：（i）缺少CSC的敏锐极小极大分析可能遵循经典的Pinsker方法，而CSC的分析则需要基于频域和时域的特殊平滑化的更复杂的估计程序；（ii）有效的估计需要解决一个古老的近似非周期性Sobolev函数的问题；（iii）如果知道$ X $的cdf的平滑度，则即使$ Z $的密度更粗糙，也可以估计其最小速率。给出了真实的和模拟的示例，以及将核心模型扩展到相关的$ X $和Z以及案例控制CSC。