Death-only rate correction under exponential censoring (${\hat{\lambda }}_{\text{plug-in}}=\hat{\pi }\hat{s}$)

Under independent exponential death and censoring, a rate fit using only observed-death times estimates $s=\lambda +\gamma$. Multiplying by the observed death fraction consistently estimates $\lambda$. This plug-in correction generally differs in a finite sample from the full maximum-likelihood estimate, which uses observed exposure from both deaths and censored subjects.