Singer and Willett (2003) write the following about estimating the standard errors of estimated survival probabilities within the context of discrete time event history models (e.g. logit hazard models):
Estimating the standard error of a survival probability is a more difficult task than estimating the standard error of its associated hazard probability. This is because, unlike hazard, which is estimated as the fraction of the risk set that experience the target event in any given period, the survival probability is estimated as a product of (1 – hazard) for this and all previous time periods. Estimating the standard error of an estimate that is itself the product of several estimates is a difficult statistical task. Indeed, it is so difficult that statisticians rarely recommend that you estimate the standard error of the survival probabilities directly..." [Emphasis added]
I am interested in how I can better understand why they made the bolded assertion (specifically, because they are correct, I and some collaborating statisticians are having a very difficult time producing a reliable estimate of the variance of the survivor function). Why is this kind of estimate especially difficult?
Singer, J. D. and Willett, J. B. (2003) Applied Longitudinal Data Analysis: Modeling Change and Event Occurrence. Oxford University Press.