Kaplan-Meier Estimator

Question

In the absence of censoring, show that the Kaplan-Meier estimator for the survivor function is S(t) = 1 - (1/n)(Sum of dj) where j is the time of event. I'm not sure how to solve this, do I use the original formula and go from there or start from the beginning? Please help.

Answer 1

the idea behind the KM estimation is fairly simple ... the probability of surviving past the $i^{th}$ event is equal to the probability of surviving til just before that event multiplied by the probability of surviving during that event.

$$S(T>(t+\Delta t))=S(T>t)s(\Delta t \vert T>t)$$

where small $s$ (my notation?) is the probability of surviving the event that takes place during the interval $(t,t+\Delta t)$ ... the probability of surviving this time interval is equal to one minus the probability of mortality during this time interval, namely $1-d/n$ where $d$ is the number of failures during that interval and $n$ is the number surviving just prior