This book presents in-depth research on positive parameters of hierarchical models under Stein’s loss function and proposes a novel empirical Bayesian estimation method. By integrating Stein’s loss function with empirical Bayesian estimation, the book tackles key challenges in estimating positive parameters that traditional methods struggle to address. It provides numerical simulations for each hierarchical model from at least four perspectives and analyzes extensive real-world data to empirically validate the effectiveness of the proposed method. The findings demonstrate that the MLE method outperforms the moment method in terms of consistency, goodness-of-fit, Bayes estimators, and PESLs.

The book is intended for graduate students, teachers, and researchers in statistics, particularly those interested in empirical Bayes analysis, positive parameters, hierarchical models and mixture distributions, Stein’s loss function, and other loss functions.