The mean lifetime is an important characteristic of particles to be identified in nuclear physics. State-of-the-art particle detectors can identify the arrivals of single radioactive nuclei as well as their subsequent radioactive decays (departures). Challenges arise when the arrivals and departures are unmatched and the departures are only partially observed. An inefficient solution is to run experiments where the arrival rate is set very low to allow for the matching of arrivals and departures. We propose an estimation method that works for a wide range of arrival rates. The method combines an initial estimator and a numerical bias correction technique. Simulations and examples based on data on the alpha decays of Lutetium isotope 155 demonstrate that the method produces unbiased estimates regardless of the arrival rate. As a practical benefit, the estimation method enables the use of all data collected in the particle detector, which will lead to more accurate estimates and, in some cases, to shorter experiments.