Preliminary and Shrinkage-Type Estimation for the Parameters of the Birnbaum–Saunders Distribution Based on Modified Moments

The two-parameter Birnbaum–Saunders (B-S) distribution is widely applied across various fields due to its favorable statistical properties. This study aims to enhance the efficiency of modified moment estimators for the B-S distribution by systematically incorporating auxiliary non-sample information. To this end, we developed and analyzed a suite of estimation strategies, including restricted estimators, preliminary test estimators, and Stein-type shrinkage estimators. A pretest procedure was formulated to guide the decision on whether to integrate the non-sample information. The relative performance of these estimators was rigorously evaluated through an asymptotic distributional analysis, comparing their asymptotic distributional bias and risk under a sequence of local alternatives. The finite-sample properties were assessed via Monte Carlo simulation studies. The practical utility of the proposed methods is demonstrated through applications to two real-world datasets: failure times for mechanical valves and bone mineral density measurements. Both numerical results and theoretical analysis confirm that the proposed shrinkage-based techniques deliver substantial efficiency gains over conventional estimators.