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

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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.