  In a neutron lifetime experiment conducted at the National Institute of
Standards and Technology, protons produced by neutron decay events are confined
in a Penning trap. In each run of the experiment, there is a trapping stage of
duration $\tau$. 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
$\delta$ 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 $\lambda$ 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
$\lambda$. I present asymptotically valid formulas for the bias,
root-mean-square prediction error, and standard deviation of the maximum
likelihood estimate of $\lambda$ for this subsample. Based on nominal values of
$\lambda$ and the dead time $\delta$, I determine the optimal duration of the
trapping stage $\tau$ by minimizing the root-mean-square prediction error of
the estimate.
