Bayesian and non-Bayesian estimation for the multiple-stress model under progressively type-Ⅱ hybrid censoring with optimal design

Neama Salah Youssef Temraz; Neama Salah Youssef Temraz

doi:10.3934/math.2026267

AIMS Mathematics

2026, Volume 11, Issue 3: 6464-6484. doi: 10.3934/math.2026267

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Research article

Bayesian and non-Bayesian estimation for the multiple-stress model under progressively type-Ⅱ hybrid censoring with optimal design

Neama Salah Youssef Temraz ^{1,2
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1.
Prince Sattam Bin Abdulaziz University, College of Sciences and Humanities, Saudi Arabia
2.
Mathematics Department, Faculty of Science, Tanta University, Tanta, Egypt

Received: 09 January 2026 Revised: 24 February 2026 Accepted: 02 March 2026 Published: 13 March 2026
MSC : MathematicsSubjectClassification: 62N02, 62F15, 62N05

In this paper, estimation of the accelerated life testing (ALT) for the stress model in multiple case is considered. Maximum likelihood estimation is employed to obtain the estimated parameters of the multiple-stress model by a maximum likelihood approach. A Bayesian estimation procedure is proposed to estimate the parameters of the multiple-stress model. Estimation methods are considered under a progressively type-Ⅱ hybrid censoring scheme with generalized inverted exponential distribution. Different criteria are discussed to determine the optimal design of the progressively type-Ⅱ hybrid censoring scheme for the multiple-stress model. A simulation study is conducted to obtain results for the maximum likelihood and Bayesian estimates of the parameters of the multiple-stress model. Real data application involving the breakdown voltage of insulating oil is introduced to analyze the performance of the multiple-stress model under progressively type-Ⅱ hybrid censoring scheme with generalized inverted exponential distribution.
Citation: Neama Salah Youssef Temraz. Bayesian and non-Bayesian estimation for the multiple-stress model under progressively type-Ⅱ hybrid censoring with optimal design[J]. AIMS Mathematics, 2026, 11(3): 6464-6484. doi: 10.3934/math.2026267

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Abstract

In this paper, estimation of the accelerated life testing (ALT) for the stress model in multiple case is considered. Maximum likelihood estimation is employed to obtain the estimated parameters of the multiple-stress model by a maximum likelihood approach. A Bayesian estimation procedure is proposed to estimate the parameters of the multiple-stress model. Estimation methods are considered under a progressively type-Ⅱ hybrid censoring scheme with generalized inverted exponential distribution. Different criteria are discussed to determine the optimal design of the progressively type-Ⅱ hybrid censoring scheme for the multiple-stress model. A simulation study is conducted to obtain results for the maximum likelihood and Bayesian estimates of the parameters of the multiple-stress model. Real data application involving the breakdown voltage of insulating oil is introduced to analyze the performance of the multiple-stress model under progressively type-Ⅱ hybrid censoring scheme with generalized inverted exponential distribution.

References

[1]	K. A. Doksum, A. Hbyland, Models for variable-stress accelerated life testing experiments based on Wiener processes and the inverse Gaussian distribution, Technometrics, 34 (1992), 74–82. https://doi.org/10.1080/00401706.1992.10485235 doi: 10.1080/00401706.1992.10485235
[2]	K. Chaloner, K. Larntz, Bayesian design for accelerated life testing, J. Stat. Plan. Infer., 33 (1992), 245–259. https://doi.org/10.1016/0378-3758(92)90071-Y doi: 10.1016/0378-3758(92)90071-Y
[3]	V. Bagdonavičius, O. Cheminade, M. Nikulin, Statistical planning and inference in accelerated life testing using the CHSS model, J. Stat. Plan. Infer., 126 (2004), 535–551. https://doi.org/10.1016/j.jspi.2003.09.011 doi: 10.1016/j.jspi.2003.09.011
[4]	J. R. V. Dorp, T. A. Mazzuchi, A general Bayes Weibull inference model for accelerated life testing, Reliab. Eng. Syst. Saf., 90 (2005), 140–147. https://doi.org/10.1016/j.ress.2004.10.012 doi: 10.1016/j.ress.2004.10.012
[5]	Y. Zhu, E. A. Elsayed, Design of equivalent accelerated life testing plans under different stress applications, Qual. Technol. Quant. Manag., 8 (2011), 463–478. https://doi.org/10.1080/16843703.2011.11673271 doi: 10.1080/16843703.2011.11673271
[6]	D. Han, T. Bai, Design optimization of a simple step-stress accelerated life test–Contrast between continuous and interval inspections with non-uniform step durations, Reliab. Eng. Syst. Saf., 199 (2020), 106875. https://doi.org/10.1016/j.ress.2020.106875 doi: 10.1016/j.ress.2020.106875
[7]	M. Abd El-Raheem, M. H. Abu-Moussa, M. M. Mohie El-Din, E. H. Hafez, Accelerated life tests under Pareto-Ⅳ lifetime distribution: Real data application and simulation study, Mathematics, 8 (2020), 1786. https://doi.org/10.3390/math8101786 doi: 10.3390/math8101786
[8]	A. Rahman, M. Kamal, S. Khan, M. F. Khan, M. S. Mustafa, E. Hussam, et al., Statistical inferences under step stress partially accelerated life testing based on multiple censoring approaches using simulated and real-life engineering data, Sci. Rep., 13 (2023), 12452. https://doi.org/10.1038/s41598-023-39170-x doi: 10.1038/s41598-023-39170-x
[9]	W. B. Nelson, Advances in accelerated life tests with step and varying stress, IEEE Trans. Reliab., 73 (2024), 21–30. https://doi.org/10.1109/TR.2024.3358233 doi: 10.1109/TR.2024.3358233
[10]	B. Pradhan, D. Kundu, On progressively censored generalized exponential distribution, Test, 18 (2009), 497–515. https://doi.org/10.1007/s11749-008-0110-1 doi: 10.1007/s11749-008-0110-1
[11]	M. Salah, Parameter estimation of the Marshall-Olkin exponential distribution under type-Ⅱ hybrid censoring schemes and its applications, J. Stat. Appl. Probab., 5 (2016), 1–8. https://doi.org/10.18576/jsap/050301 doi: 10.18576/jsap/050301
[12]	T. Sen, S. Singh, Y. M. Tripathi, Statistical inference for lognormal distribution with type-Ⅰ progressive hybrid censored data, Am. J. Math. Manag. Sci., 38 (2019), 70–95. https://doi.org/10.1080/01966324.2018.1484826 doi: 10.1080/01966324.2018.1484826
[13]	R. B. Arabi, M. A. Noori, Estimation based on progressively type-Ⅰ hybrid censored data from the Burr Ⅻ distribution, Stat. Papers, 60 (2019), 761–803. https://doi.org/10.1007/s00362-016-0849-5 doi: 10.1007/s00362-016-0849-5
[14]	E. S. A El-Sherpieny, E. M. Almetwally, H. Z. Muhammed, Progressive type-Ⅱ hybrid censored schemes based on maximum product spacing with application to power Lomax distribution, Phys. A: Stat. Mech. Appl., 553 (2020), 124–251. https://doi.org/10.1016/j.physa.2020.124251 doi: 10.1016/j.physa.2020.124251
[15]	T. Zhu, Statistical inference of Weibull distribution based on generalized progressively hybrid censored data, J. Comput. Appl. Math., 371 (2020), 112705. https://doi.org/10.1016/j.cam.2019.112705 doi: 10.1016/j.cam.2019.112705
[16]	M. Berzborn, E. Cramer, Inference for type-Ⅰ and type-Ⅱ hybrid censored minimal repair and record data, J. Stat. Theory Pract., 17 (2023), 1–38. https://doi.org/10.1007/s42519-023-00351-6 doi: 10.1007/s42519-023-00351-6
[17]	F. H. Riad, E. H. Hafez, A. M. Mubarak, Study on step-stress accelerated life testing for the Burr-Ⅻ distribution using cumulative exposure model under progressive type-Ⅱ censoring with real data example, J. Stat. Appl. Probab., 10 (2021), 35–44. https://doi.org/10.18576/jsap/100104 doi: 10.18576/jsap/100104
[18]	T. A. Abushal, A. H. Abdel-Hamid, Inference on a new distribution under progressive-stress accelerated life tests and progressive type-Ⅱ censoring based on a series-parallel system, AIMS Math., 7 (2022), 425–454. https://doi.org/10.3934/math.2022028
[19]	M. M. Yousef, R. Alsultan, S. G. Nassr, Parameter inference on partially accelerated life testing for the inversed Kumaraswamy distribution based on type-Ⅱ progressive censoring data, Math. Biosci. Eng., 20 (2023), 1674–1694. https://doi.org/10.3934/mbe.2023076 doi: 10.3934/mbe.2023076
[20]	A. A. Ismail, Progressive stress accelerated life test for inverse Weibull failure model: A parametric inference, J. King Saud Univ. Sci., 34 (2022), 101994. https://doi.org/10.1016/j.jksus.2022.101994 doi: 10.1016/j.jksus.2022.101994
[21]	R. Alotaibi, F. S. Alamri, E. M. Almetwally, M. Wang, H. Rezk, Classical and Bayesian inference of a progressive-stress model for the Nadarajah-Haghighi distribution with type Ⅱ progressive censoring and different loss functions, Mathematics, 10 (2022), 1602. https://doi.org/10.3390/math10091602 doi: 10.3390/math10091602
[22]	F. A. Moala, K. D. Chagas, Bayesian analysis for multiple step-stress accelerated life test model under gamma lifetime distribution and type-Ⅱ censoring, Int. J. Qual. Reliab. Manag., 40 (2023), 1068–1091. https://doi.org/10.1108/IJQRM-09-2021-0336 doi: 10.1108/IJQRM-09-2021-0336
[23]	A. K. Mahto, Y. M. Tripathi, S. Dey, B. S. Alsaedi, M. H. Alhelali, F. M. Alghamdi, et al., Bayesian estimation and prediction under progressive-stress accelerated life test for a log-logistic model, Alex. Eng. J., 101 (2024), 330–342. https://doi.org/10.1016/j.aej.2024.05.045 doi: 10.1016/j.aej.2024.05.045
[24]	R. R. Barton, Optimal accelerated life-time plans that minimize the maximum test-stress, IEEE Trans. Reliab., 40 (1991), 166–172. https://doi.org/10.1109/24.87122 doi: 10.1109/24.87122
[25]	D. S. Bai, S. W. Chung, Optimal design of partially accelerated life tests for the exponential distribution under Type-Ⅰ censoring, IEEE Trans. Reliab., 41 (1992), 400–406. https://doi.org/10.1109/24.159807 doi: 10.1109/24.159807
[26]	D. S. Bai, M. S. Cha, S. W. Chung, Optimum simple ramp-tests for the Weibull distribution and Type-Ⅰ censoring, IEEE Trans. Reliab., 41 (1992), 407–413. https://doi.org/10.1109/24.159808 doi: 10.1109/24.159808
[27]	F. Haghighi, Optimal design of accelerated life tests for an extension of the exponential distribution, Reliab. Eng. Syst. Saf., 131 (2014), 251–256. https://doi.org/10.1016/j.ress.2014.04.017 doi: 10.1016/j.ress.2014.04.017
[28]	Q. Guan, Y. Tang, J. Fu, A. Xu, Optimal multiple constant-stress accelerated life tests for generalized exponential distribution, Commun. Stat. B: Simul. Comput., 43 (2014), 1852–1865. https://doi.org/10.1080/03610918.2013.810257 doi: 10.1080/03610918.2013.810257
[29]	A. M. Abd El-Raheem, Optimal design of multiple accelerated life testing for generalized half-normal distribution under type-Ⅰ censoring, J. Comput. Appl. Math., 368 (2020), 112539. https://doi.org/10.1016/j.cam.2019.112539
[30]	A. M. Abd El-Raheem, Optimal design of multiple constant-stress accelerated life testing for the extension of the exponential distribution under type-Ⅱ censoring, J. Comput. Appl. Math., 382 (2021), 113094. https://doi.org/10.1016/j.cam.2020.113094
[31]	A. M. Abouammoh, A. M. Alshingiti, Reliability estimation of generalized inverted exponential distribution, J. Stat. Comput. Simul., 79 (2009), 1301–1315. https://doi.org/10.1080/00949650802261095 doi: 10.1080/00949650802261095
[32]	U. Singh, S. K. Singh, R. K. Singh, A comparative study of traditional estimation methods and maximum product spacings method in generalized inverted exponential distribution, J. Stat. Appl. Probab., 3 (2014), 153–169. https://doi.org/10.12785/jsap/030206 doi: 10.12785/jsap/030206
[33]	A. I. Al-Omari, Time truncated acceptance sampling plans for generalized inverted exponential distribution, Electron. J. Appl. Stat. Anal., 8 (2015), 1–12. https://doi.org/10.1285/i20705948v8n1p1
[34]	M. Dube, H. Krishna, R. Garg, Generalized inverted exponential distribution under progressive first-failure censoring, J. Stat. Comput. Simul., 86 (2015), 1095–1114. https://doi.org/10.1080/00949655.2015.1052440 doi: 10.1080/00949655.2015.1052440
[35]	R. K. Singh, S. K. Singh, U. Singh, Maximum product spacings method for the estimation of parameters of generalized inverted exponential distribution under progressive type Ⅱ censoring, JSMS, 19 (2016), 219–245.https://doi.org/10.1080/09720510.2015.1023553
[36]	S. Dey, T. Dey, D. J. Luckett, Statistical inference for the generalized inverted exponential distribution based on upper record values, Math. Comput. Simul., 120 (2016), 64–78. https://doi.org/10.1016/j.matcom.2015.06.012 doi: 10.1016/j.matcom.2015.06.012
[37]	K. G. Njeri, E. G. Njenga, Maximum likelihood estimation for a progressively type Ⅱ censored generalized inverted exponential distribution via EM algorithm, Am. J. Theor. Appl. Stat., 10 (2021), 14–21. https://doi.org/10.11648/j.ajtas.20211001.13 doi: 10.11648/j.ajtas.20211001.13
[38]	Q. Chen, W. Gui, Statistical inference of the generalized inverted exponential distribution under joint progressively type-Ⅱ censoring, Entropy, 24 (2022), 576. https://doi.org/10.3390/e24050576 doi: 10.3390/e24050576
[39]	D. Kundu, A. Joarder, Analysis of type-Ⅱ progressively hybrid censored data, Comput. Stat. Data Anal., 50 (2006), 2509–2528. https://doi.org/10.1016/j.csda.2005.05.002 doi: 10.1016/j.csda.2005.05.002
[40]	A. Xu, W. Wang, Recursive Bayesian prediction of remaining useful life for gamma degradation process under conjugate priors, Scand. J. Stat., 53 (2025), 175–206. https://doi.org/10.1111/sjos.70031 doi: 10.1111/sjos.70031
[41]	D. Zhu, A. Xu, Z. Chen, S. Ding, G. Fang, An online Bayesian framework for identifying latent system degradation states, IEEE Trans. Reliab., 75 (2026), 542–554. https://doi.org/10.1109/TR.2025.3647489 doi: 10.1109/TR.2025.3647489
[42]	W. B. Nelson, Accelerated Testing: Statistical Models, Test Plans, and Data Analysis, Hoboken: John Wiley & Sons, 2009.
[43]	L. Zhuang, A. Xu, B. Wang, Y. Xue, S. Zhang, Data analysis of progressive‐stress accelerated life tests with group effects, Qual. Technol. Quant. Manag., 20 (2023), 763–783. https://doi.org/10.1080/16843703.2022.2147690 doi: 10.1080/16843703.2022.2147690

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