Likelihood models for two-stage neutron lifetime experiments

At the NIST Cold Neutron Research Facility, a multiple run neutron lifetime experiment is underway. Each run of the two-stage experiment consists of a neutron production stage and a neutron decay stage. Salient features of the experimental data are that the number N of neutrons which decay is an unobservable random variable. Also, the neutron decay events are contaminated by background events. Under the assumption that the number of trapped neutrons is a realization of a Markovian birth and death process, we deduce the distribution of N and the decay times subject to a variety of experimental constraints. These distributions serve as likelihood models for estimation of the mean neutron lifetime. The method of maximum likelihood and the method of minimum chi-square are employed and compared for estimation. For fairly large sample sizes, the minimum chi-square estimates suitable for binned decay data are shown, by simulation, to be comparable to the maximum likelihood estimates based on the data of exact decay times. But for moderate sample sizes, the former method gives smaller variances than the latter. The loss of efficiency due to using binned data rather than the complete decay time data is minimal provided the bin widths are reasonably small and the observation period for observing decay is not too short. Our systematic and unified approach facilitates the application of the developed statistical methods to similar two-stage lifetime experiments in which radioactive decay processes are studied. It also clarifies some of the ambiguities in the literature in using the conditional distributions for statistical estimation.