Fuzzy Bayesian, E-Bayesian, and Hierarchical Bayesian Estimations of R = P(X>Y ) in Weibull Distribution under Type II censored Data
Kazem Fayyaz Heydari
1
(
Department of Statistics, Payame Noor University, Tehran, Iran.
)
Fereshteh Momeni
2
(
Department of Mathematics and Statistics, Behshahr Branch, Islamic Azad University, Behshahr, Iran.
)
Shahram Yaghoobzadeh Sharastani
3
(
Department of Statistics, Payame Noor University, Tehran, Iran.
)
الکلمات المفتاحية: E-Bayesian estimator, Hierarchical Bayesian estimator, Fuzzy Data, Stress-strength parameter, Type II censoring.,
ملخص المقالة :
This study examines Bayesian, E-Bayesian (E-B), and hierarchical Bayesian (H-B) estimation methods for the stress-strength reliability parameter (SSRP) R = P(X>Y ), within the Weibull distribution framework under Type II censoring and fuzzy data conditions. Stress and strength random variables are modeled as Weibull distributions with distinct scale parameters but a common shape parameter. Estimations are conducted using the squared error (SE) loss function and Lindleys approximation. Furthermore, a comprehensive simulation study, complemented by real-world data analysis, has been carried out to assess and compare the performance of the proposed estimators. The results from both simulation and empirical analyses demonstrate that the H-B estimator consistently outperforms both the Bayesian and E-B estimators under the squared error (SE) loss function.
[1] Awad AM, Azzam MM, Hamdan MA. Some inference results on P (Y
[2] Kundu D, Gupta RD. Estimation of P (Y
[3] Gupta RD, Gupta RC. Estimation of pr(ax >> by) in the multivariate normal case. Statistics. 1990; 21(1): 91-97. DOI: https://doi.org/10.1080/02331889008802229
[4] Raqab MZ, Kundu D. Comparison of different estimators of P(Y < X) for a scaled Burr Type X distribution. Communications in StatisticsSimulation and Computation. 2005; 34(2): 465-483. DOI: https://doi.org/10.1081/SAC-200055741
[5] Lio YL, Tsai TR. Estimation of δ = P (X
[6] Kundu D, Gupta RD. Estimation of P (Y
[7] Raqab MZ, Madi MT, Kundu D. Estimation of P(Y < X) for the three-parameter generalized exponential distribution. Communications in StatisticsTheory and Methods. 2008; 37(18): 2854-2864. DOI: https://doi.org/10.1080/03610920802162664
[8] Baklizi A. Likelihood and Bayesian estimation of P r(X < Y ) using lower record values from the generalized exponential distribution. Computational Statistics and Data Analysis. 2008; 52(7): 3468-3473. DOI: https://doi.org/10.1016/j.csda.2007.11.002
[9] Asgharzadeh A, Valiollahi R, Raqab MZ. Stress-strength reliability of Weibull distribution based on progressively censored samples. SORT-Statistics and Operations Research Transactions. 2011; 35(2): 103-124.
[10] Saracoglu B, Kinaci I, Kundu D. On estimation of R = P(Y < X) for exponential distribution under progressive Type-II censoring. Journal of Statistical Computation and Simulation. 2012; 82(5): 729-744. DOI: https://doi.org/10.1080/00949655.2010.551772
[11] Al-Mutairi DK, Ghitany ME, Kundu D. Inferences on stress-strength reliability from Lindley distributions. Communications in Statistics-Theory and Methods. 2013; 42(8): 1443-1463. DOI: https://doi.org/10.1080/03610926.2011.563011
[12] Ghitany ME, Al-Mutairi DK, Aboukhamseen SM. Estimation of the reliability of a stress-strength system from power Lindley distributions. Communications in Statistics- Simulation and Computation. 2015; 44(1): 118-136. DOI: https://doi.org/10.1080/03610918.2013.767910
[13] Lindley DV, Smith AF. Bayes estimation for the linear model. Journal of the Royal Statistical Society. Series B (Methodological). 1972; 34(1): 1-41.
[14] Han M. The Structure of hierarchical prior distribution and its applications. Chinese Operations Research and Management Science. 1997; 6(3): 31-40.
[15] Han M. E-Bayesian estimation and hierarchical Bayesian estimation of failure rate. Applied Mathematical Modelling. 2009; 33(4): 1915-1922. DOI: https://doi.org/10.1016/j.apm.2008.03.019
[16] Han M. The E-Bayesian and hierarchical Bayesian estimations for the system reliability parameter. Communications in Statistics-Theory and Methods. 2017; 46(4): 1606-162. DOI: https://doi.org/10.1080/03610926.2015.1024861
[17] Wang J, Li D, Chen D. E Bayesian estimation and hierarchical Bayesian estimation of the system reliability parameter. Systems Engineering Procedia. 2012; 3: 282-289. DOI:
https://doi.org/10.1016/j.sepro.2011.11.031
[18] Wu SJ. Estimations of the parameters of the Weibull distribution with progressively censored data. Journal of the Japan Statistical Society. 2002; 32(2): 155-163. DOI: https://doi.org/10.14490/jjss.32.155
[19] Okasha HM. E-Bayesian estimation of system reliability with Weibull distribution of components based on Type-2 censoring. Journal of Advanced Research in Scientific Computing. 2012; 4(4): 34-45.
[20] Micheas AC, Wikle CK. A Bayesian hierarchical nonoverlapping random disc growth model. Journal of the American Statistical Association. 2009; 104(485): 274-283. DOI:
https://doi.org/10.1198/jasa.2009.0124
[21] Richard DM. A Bayesian hierarchical model for the measurement of working memory capacity. Journal of Mathematical Psychology. 2011; 55(1): 8-24. DOI: https://doi.org/10.1016/j.jmp.2010.08.008
[22] Khan MJ, Khatoon B. Statistical inferences of R = P(X < Y ) for exponential distribution based on generalized order statistics. Annals of Data Science. 2020; 7(3): 525-545. DOI:
https://doi.org/10.1007/s40745-019-00207-6
[23] Marwa MM, Ali Sharawy, Mahmoud HA. E-Bayesian estimation for the parameters and hazard function of Gompertz distribution based on progressively Type-II right censoring with application. Quality and Reliability Engineering International. 2023; 39(4): 1299-1317. DOI: https://doi.org/10.1002/qre.3292
[24] Yadav AS, Singh SK, Singh U. Bayesian estimation of stressstrength reliability for Lomax distribution under Type-II hybrid censored data using asymmetric loss function. Life Cycle Reliability and Safety Engineering. 2019; 8(3): 257-267. DOI: https://doi.org/10.1007/s41872-019-00086-z
[25] Rao GS, Mbwambo S, Josephat PK. Estimation of stressstrength reliability from exponentiated inverse Rayleigh distribution. International Journal of Reliability, Quality and Safety Engineering. 2019; 26(1): 1950005. DOI: https://doi.org/10.1142/S0218539319500050
[26] Alamri OA, El-Raouf M MA, Ismail EA, Almaspoor Z, Alsaedi BSO, Khosa SK, Yusuf M. Estimate stressstrength reliability model using Rayleigh and Half-normal distribution. Computational Intelligence and Neuroscience. 2021; 2021: 7653581. DOI: https://doi.org/10.1155/2021/7653581
[27] Hemati A, Khodadadi Z, Zare K, Jafarpour H. Bayesian and classical estimation of strength-stress reliability for Gompertz distribution based on upper record values. Journal of Mathematical Extension. 2022; 16(7): 1-27.
[28] Asadi S, Panahi H. Estimation of stressstrength reliability based on censored data and its evaluation for coating processes. Quality Technology and Quantitative Management. 2022; 19(3): 379-401. DOI: https://doi.org/10.1080/16843703.2021.2001129
[29] Kohansal A. Inference on stress-strength model for a Kumaraswamy distribution based on hybrid progressive censored sample. REVSTAT-Statistical Journal. 2022; 20(1): 51-83. DOI: https://doi.org/10.1007/s00362-017-0916-6
[30] Muenz LR, Green SB. Time savings in censored life testing. Journal of the Royal Statistical Society Series B: Statistical Methodology. 1977; 39(2): 269-275. DOI: https://doi.org/10.1111/j.2517- 6161.1977.tb01625.x
[31] Balakrishnan N, Aggarwala R. Progressive censoring: theory, methods, and applications. Boston: Springer Science & Business Media; 2000. DOI:
https://doi.org/10.1007/978-1-4612-1334-5
[32] Wang Y, Xiang J, Gui W. Statistical inference for two lomax populations under balanced joint progressive Type-II censoring scheme. Mathematics. 2025; 13(9): 1536. DOI: https://doi.org/10.3390/math13091536
[33] Shojaee O, Zarei H, Naruei F. E-Bayesian estimation and the corresponding E-MSE under progressive Type-II censored data for some characteristics of Weibull distribution. Statistics, Optimization & Information Computing. 2024; 12(4): 962-981. DOI: https://doi.org/10.19139/soic-2310-5070-1709
[34] shojaee O, Piriaei H, Babanezhad M. E-Bayesian Estimations and Its E-MSE For Compound Rayleigh Progressive Type-II Censored Data. Statistics, Optimization & Information Computing. 2022; 10(4): 1056-1071. DOI: https://doi.org/10.19139/soic-2310-5070-1359
[35] Pak A. Statistical inference for the parameter of Lindley distribution based on fuzzy data. Brazilian Journal of Probability and Statistics. 2017; 31(3): 502-515. DOI: https://doi.org/10.1214/16-BJPS321
[36] Pak A, Parham GA, Saraj M. Inference for the Weibull distribution based on fuzzy data. Revista Colombiana de Estadistica. 2013; 36(2): 337-356.
[37] Seham M. Bayesian analysis of the kumaraswamy distribution based on fuzzy data. Journal of Statistics Applications & Probability. 2024; 13(6): 1589-1601. DOI: https://doi.org/10.18576/jsap/130603
[38] Shafiq M, Viertl R. On the estimation of parameters, survival functions, and hazard rates based on fuzzy life time data. Communications in Statistics-Theory and Methods. 2017; 46(10): 5035-5055. DOI: https://doi.org/10.1080/03610926.2015.1099667
[39] Yaghoobzadeh Shahrastani S. Estimating E-Bayesian and hierarchical Bayesian of scalar parameter of Gompertz distribution under Type II censoring schemes based on fuzzy
data. Communications in Statistics-Theory and Methods. 2019; 48(4): 831-84. DOI: https://doi.org/10.1080/03610926.2017.1417438
[40] Fayyaz Heidari K, Deiri E, Baloui Jamkhaneh E. E-Bayesian and hierarchical Bayesian estimation of Rayleigh distribution parameter with Type-II censoring from imprecise data. Journal of the Indian Society for Probability and Statistics. 2022; 23(1): 63-76. DOI: https://doi.org/10.1007/s41096-021-00112-3
[41] Makhdoom I, Yaghoobzadeh Shahrastani S, Sharifonnasabi F. Bayesian inference for the Lindley distribution under Type-II Censoring with fuzzy data. Journal of Data Science and Modeling. 2022; 2(2): 245-265. DOI: https://doi.org/10.22054/jdsm.2025.83708.1061
[42] Zarei R, Yaghoobzadeh Shahrestani S, Fadaei F. Fuzzy E-Bayesian and H-Bayesian estimation of scalar parameter in Gompertz distribution under asymmetric loss functions. Journal of Statistical Theory and Practice. 2025; 19(1): 1-2. DOI: https://doi.org/10.1007/s42519-025-00429-3
[43] Zadeh LA. Probability measures of fuzzy events. Journal of Mathematical Analysis and Applications. 1968; 23(2): 421-427. DOI: https://doi.org/10.1016/0022-247X(68)90078-4
[44] Gelman A, Carlin JB, Stern HS, Dunson DB, Vehtari A, Rubin DB. Bayesian Data Analysis. (3rd ed). London, England: Chapman and Hall/CRC; 2013.
[45] Berger JO. Statistical decision theory and Bayesian analysis. Boston: Springer Science & Business Media; 2013.
[46] Lindley DV. Approximate Bayesian Methods. Trabajos de Estadstica e Investigacin Operativa. 1980; 31(1): 223-237. DOI: https://doi.org/10.12775/eudml.40822
[47] Bader MG, Priest AM. Statistical aspects of fibre and bundle strength in hybrid composites. Progress in Science and Engineering of Composites: Proceedings of the International Conference, ICCM-IV, 25-28 Oct 1982, Tokyo, Japan. Tokyo: Scientific Research; 1982. p.1129-1136.
