Payame Noor University (PNU) Control and Optimization in Applied Mathematics 2383-3130 11 2 2026 05 21 Optimization-Oriented Double Pre-Test Shrinkage Estimators for Pareto Reliability under Progressive Type-II Censoring and Precautionary Loss 83 103 12797 10.30473/coam.2026.77066.1388 EN Omar Jabar Alial Al-Qaragholi Department of Mathematics, College of Education for Pure Sciences, University of Thi-Qar, Nasiriyah, Iraq Alaa Khlaif Jiheel Department of Mathematics, College of Education for Pure Sciences, University of Thi-Qar, Nasiriyah, Iraq Journal Article 2026 01 04 This paper develops and analyzes a class of double pre-test shrinkage estimators for the reliability function of the Pareto distribution based on progressively Type-II censored samples. The proposed approach combines a preliminary test of the shape parameter against a prior target value with shrinkage toward the corresponding prior reliability, yielding four reliability estimators with fixed and data-dependent shrinkage weights. Closed-form analytical expressions are derived for the bias and bias ratio of the proposed reliability estimators, as well as for their risk functions under the Precautionary Loss Function (PLF) and the associated relative risk with respect to the classical pooled estimator. Numerical results are obtained by direct numerical evaluation of the derived analytical expressions, including one- and two-dimensional integrals and special functions, implemented in Python. Across a wide range of design settings and reliability levels, the proposed estimators reduce PLF-risk and improve relative efficiency, with the most pronounced gains typically occurring when the prior ratio λ = θ₀/θ  is close to unity. In addition, the proposed framework can be viewed as an optimization problem under uncertainty, where the PLF-risk acts as the objective function and the design parameters, including the shrinkage weight, significance level, and stage sample sizes, define the feasible decision space. https://mathco.journals.pnu.ac.ir/article_12797_705abe8f1d88c9c92b65b56b488bb6a4.pdf