Control and Optimization in Applied Mathematics

In collaboration with Payame Noor University and the Iranian Society of Instrumentation and Control Engineers

Current Issue

By Issue

By Author

By Subject

Author Index

Keyword Index

About Journal

Aims and Scope

Editorial Board

Publication Ethics

Indexing and Abstracting

Related Links

FAQ

Peer Review Process

Journal Metrics

News

Special Issues

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

Articles in Press

Document Type : Research Article

Authors

Department of Mathematics, College of Education for Pure Sciences, University of Thi-Qar, Nasiriyah, Iraq

10.30473/coam.2026.77066.1388

Abstract

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.

Highlights

  • 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.

Keywords

Main Subjects

References
[1] Abd Ali, M.H., Jiheel, A.K., Al-Hemyari, Z. (2022). “Two-stage shrinkage Bayesian estimators for the shape parameter of Pareto distribution dependent on Katti’s regions”. Iraqi Journal for Computer Science and Mathematics, 3(2), 42–54. DOI: https://doi.org/10.52866/ijcsm.2022.02.01.005
[2] Aghamohammadi, S. Z. (2026). “On double-stage Bayesian Shrinkage estimator for two-parameter exponential distribution under progressively type-II censored data”. Journal of Statistical Computation and Simulation, 1–19. DOI: https://doi.org/10.1080/ 00949655.2025.2612260
[3] Al-Bayyati, H. A., Arnold, J.C. (1972). “On double-stage estimation in simple linear regression using prior knowledge”. Technometrics, 14(2), 405–414. DOI: https://doi. org/10.1080/00401706.1972.10488925
[4] Arnold, B.C. (1983). Pareto Distributions. International Cooperative Publishing House, Fairland, MD.
[5] Atewi, I.J., Jiheel, A.K., Shanubhogue, A.R.R. (2023). “Double-stage shrinkage estimation of reliability function for Burr XII distribution”. Iraqi Journal for Computer Science and Mathematics, 4(1), 35–52. DOI: https://doi.org/10.52866/ijcsm.2023.01. 01.005
[6] Balakrishnan, N., Aggarwala, R. (2000). Progressive censoring: Theory, methods, and applications. Birkhäuser Boston, MA. DOI: https://doi.org/10.1007/ 978-1-4612-1334-5
[7] Balakrishnan, N., Cramer, E. (2014). The art of progressive censoring: Applications to reliability and quality. Birkhäuser New York, NY. DOI: https://doi.org/10.1007/ 978-0-8176-4807-7
[8] Chiou, P. (2007). “Shrinkage estimation of reliability in the exponential distribution”. Communications in Statistics – Theory and Methods, 21(6), 1745–1758. DOI: https://doi.org/10.1080/03610929208830876
[9] Gupta, R., Bhougal, S., Joorel, J.P.S. (2006). “A two-stage procedure for estimation of the common mean of several normal populations having unequal and unknown variances”. Model Assisted Statistics and Applications. DOI: https://doi.org/10.3233/ MAS-2006-1401
[10] Jiheel, A.K., Al-Qatifi, A.B.J. (2020). “Bayes pre-test shrinkage estimation of Rayleigh distribution under different loss functions”. International Journal of Advances in Applied Mathematics and Mechanics, 7(4), 72–90. https://www.ijaamm.com/uploads/2/1/ 4/8/21481830/v7n4p7_72-90.pdf
[11] Kumar, I., Kumar, K. (2022). “On estimation of P(V < U) for inverse Pareto distribution under progressively censored data”. International Journal of System Assurance Engineering and Management, 13, 189–202. DOI: https://doi.org/10.1007/ s13198-021-01193-w
[12] Norström, J.G. (1996). “The use of the precautionary loss function in risk analysis”. IEEE Transactions on Reliability, 45(3), 400–403. DOI: https://doi.org/10.1109/ 24.536992
[13] Sabr, M.R., Jiheel, A.K. (2024). “Pre-test shrinkage estimation for reliability function of Burr XII distribution using progressive Type II censored sample under precautionary loss function (PLF)”. Journal of Education for Pure Science, 14(3). DOI: https://doi.org/ 10.32792/jeps.v14i3.549
[14] Thompson, J.R. (1968). “Some shrinkage techniques for estimating the mean”. Journal of the American Statistical Association, 63(321), 113–122. DOI: https://doi.org/10. 1080/01621459.1968.11009226
[15] Ye, Z., Gui, W. (2024). “Estimation and Bayesian prediction of the generalized Pareto distribution based on progressively Type-II censored samples”. Applied Sciences, 14(18), 8433. DOI: https://doi.org/10.3390/app14188433
[16] Public Health Agency of Canada (PHAC). (2020–2023). Public Health Infobase – Data on COVID-19 in Canada. Government of Canada Open Data Portal. Dataset ID: 261c32ab- 4cfd-4f81-9dea-7b64065690dc. (Accessed: 2024). https://open.canada.ca/data/en/dataset/261c32ab-4cfd-4f81-9dea-7b64065690dc .
 
Control and Optimization in Applied Mathematics

Articles in Press, Accepted Manuscript
Available Online from 03 April 2026
Files
Share
How to cite
Statistics