Bayesian Shrinkage Estimation of the Shape Parameter of the Power Distribution under Various Priors and Loss Functions

This study develops and evaluates a suite of Bayesian shrinkage estimators for the shape parameter ($\alpha$) of the two-parameter Power distribution under a complete sampling scheme. The primary objective is to identify a robust estimation strategy that improves upon classical and standard Bayesian methods, particularly in scenarios with limited data. The methodological framework incorporates three distinct prior distributions (non-informative Uniform and Jeffreys; informative Exponential), five symmetric and asymmetric loss functions (Squared Error Loss Function (SELF), Weighted Squared Error Loss Function (WSELF), Kullback-Leibler Loss Function (KLF), Modified Quadratic Squared Error Loss Function (M/Q SELF), and Precautionary Loss Function (PLF)), and four shrinkage strategies (one with a constant factor and three modified Thompson-type adaptive estimators). Performance is rigorously assessed via a comprehensive Monte Carlo simulation study, comparing the proposed estimators against standard Maximum Likelihood and Bayes estimators in terms of absolute bias and mean squared error (MSE). The results consistently demonstrate the superiority of Bayesian methods over the Maximum Likelihood Estimator (MLE), especially for small sample sizes. Crucially, a modified shrinkage estimator based on the work of Mehta \& Srinivasan (1971) exhibits the lowest bias and MSE across nearly all considered scenarios, establishing its exceptional robustness and efficiency. The principal implication of this study is that the proposed adaptive shrinkage estimator offers a significant improvement in estimation accuracy for the Power distribution's shape parameter, presenting a valuable tool for practitioners in reliability engineering and lifetime data analysis.