Survivorship analysis allows to statistically analyze situations that can be modeled as waiting times to an event. These waiting times are characterized by the cumulative hazard rate, which can be estimated by the Nelson-Aalen estimator or diverse confidence estimators based on asymptotic statistics. To better understand the small sample properties of these estimators, the speed of convergence of the estimate to the exact value is examined. This is done by deriving large deviation principles and their rate functions for the estimators and examining their properties. It is shown that these rate functions are asymmetric, leading to a tendency of the estimated cumulative hazard rate to overestimate the true cumulative hazard rate. This tendency is strongest in the cases of (1) small sample sizes and (2) low tail probabilities. Taking this tendency into account can improve risk assessments of rare events and of cases where only little data is available.