Finite-sample bias of Nelson–Aalen estimation

Even with a common event hazard, the exact expectation of the Nelson–Aalen estimator is ${\int }_0^t\mathbb{P}(Y(u)>0)h(u)du$, because no increments can be observed after the risk set is empty. Replacing ratios by ratios of expectations gives an approximate target and does not prove finite-sample unbiasedness. For one uncensored exponential event time, the estimator has mean $1-e^{-\lambda t}$ rather than $\lambda t$.