Mathematical Sciences: The Generalized MLE and Redistribution-to-the-Center Estimator of a Survival Function

研究者建议使用区间删失数据研究未知生存函数 S (=1-F) 的非参数估计。 假设生存时间 X 具有分布函数 F。令 (Y,Z) 为随机审查区间。 观测值是 (L1,R1),...,(Ln,Rn)，它们是独立同分布的。来自总体 (L,R) 的随机向量，其中如果 X 不在区间 (Y,Z) 内，则 L=R=X，否则为 (L,R)=(Y,Z)。 区间删失数据在医学后续研究或工业寿命测试中很自然地出现。 迄今为止，广义最大似然估计（GMLE）还没有封闭形式的表达式。 研究人员与 Wong 一起提出了一种通过重新分配到中心方法获得的估计量，称为 RTCE。 RTCE 具有显式表达式，并且在许多情况下是 GMLE。 该提案的资助将导致 GMLE 在更一般的区间审查模型（而不是这些特殊情况）中得到明确的表达，并更好地理解 GMLE 和 RTCE 的属性。 这些结果导致比当前从自洽算法导出的过程更方便、更有用的估计器。 区间删失数据自然出现在医学随访研究或工业寿命测试中，例如，在研究化学预防剂效果的乳腺癌化学预防研究中。 这项研究中的一个重要问题是，在药物的保护作用消失之前，女性可以不服用药物多长时间。 让 X 表示从停止使用药剂到丧失其保护作用（相当于中间生物标志物恢复到基线水平）的时间间隔。 当按计划的时间间隔监测生物标志物水平时，X 的确切值通常是未知的，只是它位于一个时间间隔内。 该提案的总体技术目标是对生存函数进行非参数估计，即任何给定时间 t 的 Xt 概率。

The investigator proposes to study nonparametric estimation of an unknown survival function S (=1-F) with interval-censored data. Suppose the survival time X has a distribution function F. Let (Y,Z) be the random censoring interval. The observations are (L1,R1),...,(Ln,Rn), which are i.i.d. random vectors from a population (L,R), where L=R=X if X is not inside the interval (Y,Z) and (L,R)=(Y,Z) otherwise. Interval-censored data arise quite naturally in medical follow-up studies or in industrial life-testing. To date, there is no closed form expression for the generalized maximum likelihood estimator (GMLE). The investigator, jointly with Wong, is proposing an estimator obtained by a redistribution-to-the-center method, called the RTCE. The RTCE has an explicit expression and is a GMLE in many cases. The funding of this proposal would lead to an explicit expression of a GMLE in a more general interval censorship model rather than these special cases, and a better understanding of the properties of the GMLE and the RTCE. These results then lead to a more convenient and useful estimator than the current procedure derived from a self-consistent algorithm. Interval-censored data arise naturally in medical follow-up studies or in industrial life-testing, for example, in a breast cancer chemoprevention study in which the effect of a chemopreventive agent is investigated. An important question in such a study is how long a woman can go without taking the agent before the protective effect of the agent wears off. Let X denote the time interval from cessation of use of the agent to the loss of its protective effect qualified as a return to baseline level of an intermediate biomarker. When the biomarker levels are monitored in scheduled intervals, the exact value of X is usually not known except that it lies in an interval. The overall technical objective in this proposal is to carry out nonparametric estimation of the survival function, i.e., the probability of Xt for any given time t.