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

研究者提出研究区间截尾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有一个分布函数f，设（Y，Z）为随机筛选区间。观测值为（L1,R1），…，（Ln,Rn），它们是来自总体（L，R）的i.d个随机向量，其中，如果X不在区间（Y，Z）内，则L=R=X，否则为（L，R）=（Y，Z）。在医学后续研究或工业寿命试验中，很自然地会出现间隔剔除数据。迄今为止，广义极大似然估计量（GMLE）还没有封闭形式的表达式。研究者与Wong共同提出了一个通过向中心再分配方法获得的估计量，称为RTCE。RTCE有一个显式表达式，在许多情况下是GMLE。该提案的资助将导致在更一般的间隔审查模型中明确表达GMLE，而不是这些特殊情况，并更好地理解GMLE和RTCE的特性。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推导自洽算法。在医学后续研究或工业寿命试验中，例如在研究化学预防剂效果的乳腺癌化学预防研究中，自然会出现间隔剔除数据。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，女性能坚持多久不服用药物。令X表示从停止使用药物到丧失其保护作用的时间间隔，符合中间生物标志物恢复到基线水平的要求。当生物标志物水平在预定的时间间隔内监测时，X的确切值通常是未知的，除非它位于一个时间间隔内。本提案的总体技术目标是对生存函数进行非参数估计，即任意给定时间t下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.