Optimal proton trapping strategy for a neutron lifetime experiment

In a neutron lifetime experiment conducted at the National Institute of Standards and Technology, protons produced by neutron decay events are confined in a proton trap. In each run of the experiment, there is a trapping stage of duration τ. After the trapping stage, protons are purged from the trap. A proton detector provides incomplete information because it goes dead after detecting the first of any purged protons. Further, there is a dead time δ between the end of the trapping stage in one run and the beginning of the next trapping stage in the next run. Based on the fraction of runs where a proton is detected, I estimate the trapping rate λ by the method of maximum likelihood. I show that the expected value of the maximum likelihood estimate is infinite. To obtain a maximum likelihood estimate with a finite expected value and a well-defined and finite variance, I restrict attention to a subsample of all realizations of the data. This subsample excludes an exceedingly rare realization that yields an infinite-valued estimate of λ. I present asymptotically valid formulas for the bias, root-mean-square prediction error, and standard deviation of the maximum likelihood estimate of λ for this subsample. Based on nominal values of λ and the dead time δ, I determine the optimal duration of the trapping stage τ by minimizing the root-mean-square prediction error of the estimate.