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

研究者提出研究区间删失数据下未知生存函数S（=1-F）的非参数估计。 假设生存时间X具有分布函数F。 设（Y，Z）为随机截尾区间。 观测值为（L1，R1），...，(Ln，Rn），其是i.i.d.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（L，R）=（Y，Z）。 区间删失数据在医学随访研究或工业寿命测试中很自然地出现。 广义极大似然估计（GMLE）至今没有一个封闭的表达式。 研究人员与Wong共同提出了一种通过再分布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。 这一建议的资金将导致一个更一般的间隔审查模型，而不是这些特殊的情况下，一个明确的表达GMLE，并更好地理解的属性GMLE和RTCE。 这些结果，然后导致一个更方便和有用的估计比目前的程序来自自洽算法。 间隔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，在乳腺癌化学预防研究中，研究了化学预防剂的效果。 在这样的研究中，一个重要的问题是，在药物的保护作用消失之前，一个女人可以在多长时间内不服用药物。 令X表示从停止使用药剂到其保护作用丧失的时间间隔，其保护作用被限定为恢复到中间生物标志物的基线水平。 当在预定的间隔中监测生物标志物水平时，X的确切值通常是未知的，除非它位于间隔中。 本提案的总体技术目标是对生存函数进行非参数估计，即，的 在任何给定时间t的概率Xt。

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.