So, let $\hat{S}(t)$ be the Kaplan–Meier estimator of an unknown survival function $S(t)$. Consider the logistic transform of this estimator:
$$\hat{S}_{\text{logistic}}(t) = \log \left(\frac{\hat{S}(t)}{1-\hat{S}(t)}\right)$$
Using the delta method and Greenwood’s formula for the variance of $\hat{S}(t)$ , I need to derive an approximate expression for the variance $\operatorname{Var}(\hat{S}_{\text{logistic}}(t))$.
Can anyone give me some pointers about how to start this question? Been struggling on it for a while now. Thanks in advance.