Mean accumulated hazard under independent censoring ($\mathbb{E}H(\min (T,C))=\mathbb{P}(T\le C)$)

Let $X=\min (T,C)$, $V=1_{\{T\le C\}}$, and suppose $T$ and $C$ are independent, with continuous failure density $f=hS$. Then the Tonelli theorem givesThus $V-H(X)$ has mean zero. The identity also holds conditionally on covariates when censoring is independent conditionally on them. For an estimated hazard, the mean error is controlled by $\mathbb{E}|\hat{H}(X)-H(X)|$, not by pointwise fit alone.

// Target: survival-analysis.bigb