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Al-Qaragholi, Omar, Jiheel, Alaa. (1405). Optimization-Oriented Double Pre-Test Shrinkage Estimators for Pareto Reliability under Progressive Type-II Censoring and Precautionary Loss. فصلنامه علمی زیست فناوری گیاهان زراعی, 11(2), 83-103. doi: 10.30473/coam.2026.77066.1388
Omar Jabar AlilLal Al-Qaragholi; Alaa Khlaif Jiheel. "Optimization-Oriented Double Pre-Test Shrinkage Estimators for Pareto Reliability under Progressive Type-II Censoring and Precautionary Loss". فصلنامه علمی زیست فناوری گیاهان زراعی, 11, 2, 1405, 83-103. doi: 10.30473/coam.2026.77066.1388
Al-Qaragholi, Omar, Jiheel, Alaa. (1405). 'Optimization-Oriented Double Pre-Test Shrinkage Estimators for Pareto Reliability under Progressive Type-II Censoring and Precautionary Loss', فصلنامه علمی زیست فناوری گیاهان زراعی, 11(2), pp. 83-103. doi: 10.30473/coam.2026.77066.1388
Al-Qaragholi, Omar, Jiheel, Alaa. Optimization-Oriented Double Pre-Test Shrinkage Estimators for Pareto Reliability under Progressive Type-II Censoring and Precautionary Loss. فصلنامه علمی زیست فناوری گیاهان زراعی, 1405; 11(2): 83-103. doi: 10.30473/coam.2026.77066.1388

Optimization-Oriented Double Pre-Test Shrinkage Estimators for Pareto Reliability under Progressive Type-II Censoring and Precautionary Loss

Control and Optimization in Applied Mathematics
مقاله 5، دوره 11، شماره 2 - شماره پیاپی 22، مرداد 2026، صفحه 83-103
نوع مقاله: Research Article
شناسه دیجیتال (DOI): 10.30473/coam.2026.77066.1388
نویسندگان
Omar Jabar AlilLal Al-Qaragholi* ؛ Alaa Khlaif Jiheel
Department of Mathematics, College of Education for Pure Sciences, University of Thi-Qar, Nasiriyah, Iraq
چکیده
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.

تازه های تحقیق
  • Four double pre-test shrinkage estimators are proposed for the Pareto reliability function under progressive Type-II censoring, combining a two-stage pooling rule with shrinkage toward a prior target.
  • Closed-form analytical expressions for bias ratio, PLF-risk, and relative risk are derived in terms of the regularized incomplete gamma function and the modified Bessel function of the second kind.
  • A convexity-based theorem proves that PLF-risk reduction is theoretically guaranteed near λ = θ₀/θ ≈ 1, providing a rigorous explanation for the observed numerical improvement region.
  • All numerical results are obtained by direct analytical evaluation implemented in Python, ensuring full reproducibility without Monte Carlo simulation.
  • The estimation framework admits an optimization-under uncertainty interpretation: shrinkage weight, significance level, and stage sample sizes serve as design parameters that can be tuned to minimize PLF-risk over the feasible decision space.
کلیدواژه‌ها
Double pre-test shrinkage estimation؛ Pareto reliability function؛ Progressive Type-II censoring؛ Precautionary loss function؛ Optimization under uncertainty
مراجع

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