= Greenwood formula
{c}
{title2=$\widehat{\operatorname{Var}}(\widehat S(t))$}
For the <Kaplan–Meier estimator>, the estimated <variance> is
$$
\widehat{\operatorname{Var}}(\widehat S(t))=\widehat S(t)^2\sum_{t_j\le t}\frac{d_j}{r_j(r_j-d_j)}.
$$
The formula follows by summing approximate variances of the logarithms of the conditional survival factors and applying the <delta method> to their product. Its square root is the estimated <standard error>. The same expression uses the actual delayed-entry <risk sets> for the <Kaplan–Meier estimator with delayed entry>. When $0<\widehat S(t)<1$ and every contributing $r_j>d_j$, an approximate log-scale <confidence interval> is $\exp\{\log\widehat S(t)\pm z_{1-\alpha/2}\widehat{\operatorname{se}}(\widehat S(t))/\widehat S(t)\}$, with the upper endpoint capped at one. This interval can be inaccurate with very small <risk sets>.
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