Pretest and Shrinkage Type Parameter Estimation of the Birnbaum-Saunders Distribution

The two-parameter Birnbaum-Saunders distribution is widely applied in reliability engineering and fatigue life modeling due to its desirable statistical properties. In this paper, we propose improved maximum likelihood estimators for this distribution by incorporating auxiliary non-sample information into the estimation process. We consider several estimation strategies, including restricted estimators, linear shrinkage estimators, preliminary test estimators, shrinkage preliminary test estimators, and James-Stein and positive James-Stein estimators. To guide the inclusion of non-sample information, we develop a preliminary test statistic. We derive the asymptotic distributional properties of the proposed estimators, including expressions for asymptotic distributional bias and risk under sequences of local alternatives. We evaluate the finite-sample performance of the methods through extensive simulation studies. Moreover, we also provide a mechanism of cross-validation to choose the optimum value of the shrinkage intensity factor. Finally, we illustrate the practical utility of our approach using two real data sets of machine valve failure times and fatigue life data. Our numerical results and asymptotic analysis demonstrate that the proposed techniques offer substantial efficiency gains over conventional maximum likelihood estimators.