= Shared-fit Cox–Snell residuals reproduce the fitted survival curve
If a common <Kaplan–Meier estimator> $\widehat S$ is fitted and the same data are transformed to $u_i=-\log\widehat S(x_i)$ with censoring indicators retained, event ordering and risk sets are preserved under compatible tie conventions. At a transformed event time $u_j$, the residual product-limit estimate equals $\widehat S(x_j)=e^{-u_j}$. Consequently agreement of the pooled residual curve with the exponential target is largely built into the construction. Separate group curves or a model fitted with covariates can reveal differences concealed by the pooled fit. Residuals become infinite if the fitted survivor reaches zero.
// Target: survival-analysis.bigb
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