Pitman closeness of maximum likelihood estimators under type-II hybrid censoring with exponential lifetimes

Assuming exponential lifetimes, exact expressions of the Pitman closeness (PC) probabilities were derived for comparing two maximum likelihood estimators (MLEs) of θ under different termination times from a Type-II hybrid censoring scheme (HCS). Assuming that the lifetime data follow an exponential distribution with scale parameter θ, prior work had computed the Pitman closeness (PC) probabilities for estimators based on Type-I right-censoring, Type-II right-censoring and Type-I hybrid censoring schemes (HCS). However, the derivation of the PC under a Type-II HCS has not yet been addressed in the literature. This paper examines two comparisons of MLEs for θ, the scale parameter, for exponentially distributed lifetimes arising from the Type-II HCS: (1) between estimators corresponding to different numbers of observed failures and (2) between estimators with different censoring times. Closed-form expressions for the PC probabilities are derived, and numerical results are reported for various sample sizes, censoring times and study durations. The results show that increasing the pre-fixed termination time or the number of failures led to an estimator that was Pitman closer to the true parameter, which confirm the intuition that increasing the termination time or the number of observed failures will lead to an estimator that is Pitman closer. However, the PC probability also accounts for the cases where the estimators are equal. Conditional on the estimators being different, there were a surprising couple of instances where there is a higher probability that the estimator based on a shorter termination time is closer to θ compared to the estimator based on the longer termination time.