Pitman Closeness of Maximum Likelihood Estimators Under Type-II Hybrid Censoring With Exponential Lifetime

Abstract

The Pitman closeness (PC) criterion is a method to compare two statistical estimators. Assuming that the lifetime data follow an exponential distribution with scale parameter $\theta$, prior work had computed the PC probabilities for estimators of $\theta$ 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 thesis examines two comparisons of maximum likelihood estimators for $\theta$, 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. Numerical results show that increasing the pre-fixed termination time or the number of failures led to an estimator that was always Pitman closer to the true parameter. These findings confirm the intuition that increasing the termination time or the number of observed failures will usually lead to an estimator that is Pitman closer than one based on a shorter termination time or fewer observed failures.