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

研究了区间删失数据下未知生存函数S(=1-F)的非参数估计问题。设生存时间X为分布函数F，设(Y，Z)为随机截尾区间。观测结果是(L1，r1)，…，(Ln，Rn)，它们是I.I.D.来自总体(L，R)的随机向量，其中L=R=X，如果X不在区间(Y，Z)内，则(L，R)=(Y，Z)。间隔删失数据在医学跟踪研究或工业寿命测试中非常自然地出现。到目前为止，广义最大似然估计(GMLE)还没有一个封闭的表达式。这位研究人员与Wong共同提出了一种通过重分布到中心的方法获得的估计值，称为RTCE。RTCE有一个明确的表达式，并且在许多情况下是GMLE。对这项提议的资助将导致以更一般的间隔审查模式而不是这些特殊情况明确地表达全球MLE，并更好地理解GMLE和RTCE的性质。然后，这些结果导致了比当前由自洽算法导出的过程更方便和有用的估计器。间隔删失数据自然出现在医学随访研究或工业生命测试中，例如，在研究化学预防药物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，女性在不服用该制剂的情况下可以坚持多久，直到该制剂的保护作用消失。设X表示从停止使用该试剂到失去其保护效果的时间间隔，该时间间隔被鉴定为恢复到中间生物标志物的基线水平。当以预定的间隔监测生物标记物水平时，X的确切值通常是未知的，除非它位于某个间隔内。该方案的总体技术目标是执行生存函数的非参数估计，即任意给定时间t的XT的概率。

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.