Cure-rates and shape-restricted hazard functions under censoring

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In this project I plan to address two important issues in survival analysis. The first is estimation of the proportion, or testing for presence, of immune individuals (long-term survivors) in a sample where we observe some time-to-event data for each individual (for instance, time to pain-relief following a drug administration, or time to relapse following a remission; the individuals who do not exhibit relief or relapse are termed 'immune'). I propose to study the problem under different types of 'censoring' where instead of the exact time we can often only observe whether the latter is bigger or smaller than some censoring time (for instance, under 'Case-1 interval censoring' we only have the inspection times and whether or not the event in question ocurred before it (the 'current status')). Interestingly, even if we do not have the actual number of immune individuals, censoring information such as above often helps us draw valid inferences. The problem precisely is to figure out how to calculate the required proportion and its sampling variation approximately.