Sharp minimax distribution estimation for current status censoring with or without missing

IF 4.3 3区 材料科学 Q1 ENGINEERING, ELECTRICAL & ELECTRONIC
S. Efromovich
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引用次数: 1

Abstract

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.
有或没有缺失的当前状态审查的尖锐极小最大值分布估计
通过当前状态截尾(CSC)修改的寿命X的累积分布函数和概率密度的非参数估计,包括左右缺失数据的情况,是一个具有偏差数据的经典不适定问题。CSC数据的偏差性质可能会使我们无法进行一致的估计,除非偏差函数是已知的或可以估计的,并且其不适定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,其中连续监测时间$Z$独立于$X$,$\Delta:=I(X\leq Z)$是状态,并且观测的偏差由$Z$的密度产生,这是可估计的。在存在右或左缺失的情况下,我们观察到来自$(\Delta Z,\Delta)$或$((1-\Delta(Z,\Deleta))$的相应样本;数据再次有偏差,但现在不能根据数据估计$Z$的密度。因此,为了解决估计问题,必须知道$Z$的密度(就像在对照研究中一样),或者进行额外的$Z$横截面抽样,这通常比基础CSC研究更简单。本文的主要目的是针对这一有偏和不适定性问题发展有效(尖锐极小极大)估计理论,该理论受到直接观测$X$情况下已知结果的启发。在所发展的理论的有趣方面中:(i)虽然缺失CSC的尖锐极小极大分析可能遵循经典的Pinsker方法,但CSC的分析需要基于频域和时域中的特殊平滑的更复杂的估计过程;(ii)有效的估计需要解决近似非周期Sobolev函数的老问题;(iii)如果$X$的cdf的光滑性是已知的,则即使$Z$的密度更粗糙,其速率最小最大估计也是可能的。给出了真实和模拟的例子,以及将核心模型扩展到依赖$X$和Z以及病例对照CSC。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
CiteScore
7.20
自引率
4.30%
发文量
567
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