Reliability inference of the quadratic hazard rate model under step-stress partially accelerated life testing with progressive Type-Ⅱ censoring

Hanan Haj Ahmad; Moustafa N. Mousa; M. E. Moshref; N. Youns; Mahmoud M. M. Mansour; Hanan Haj Ahmad; Moustafa N. Mousa; M. E. Moshref; N. Youns; Mahmoud M. M. Mansour

doi:10.3934/math.2026253

AIMS Mathematics

2026, Volume 11, Issue 3: 6106-6140. doi: 10.3934/math.2026253

Previous Article Next Article

Research article Special Issues

Reliability inference of the quadratic hazard rate model under step-stress partially accelerated life testing with progressive Type-Ⅱ censoring

1.
Department of Mathematics and Statistics, College of Science, King Faisal University, Hofuf 31982, Al-Ahsa, Saudi Arabia
2.
Department of Mathematics, Faculty of Science, Kafrelsheikh University, Kafrelsheikh 33516, Egypt
3.
Department of Mathematics, Faculty of Science, Al-Azhar University, Nasr City 11884, Cairo, Egypt
4.
Department of Basic Science, Faculty of Engineering, The British University in Egypt, El Sherouk City, Cairo, Egypt

Received: 30 January 2026 Revised: 03 March 2026 Accepted: 04 March 2026 Published: 11 March 2026
MSC : 60G35, 62F10, 62F15, 62N01, 62N02, 62N05

Step-stress partially accelerated life testing (SSPALT) provides an effective framework for obtaining timely reliability information when products operate under varying stress levels. In this paper, statistical inference and applications of the quadratic hazard rate distribution are developed under SSPALT with progressive Type-Ⅱ censoring and binomial removals. This censoring scheme offers flexibility in balancing the experimental cost and statistical efficiency while accommodating practical testing constraints. Maximum likelihood and Bayesian estimation are derived for the model parameters and the acceleration factor. Computational implementations are carried out using the Markov chain Monte Carlo method. Reliability analyses, including survival and hazard rate functions, are carried out to support practical reliability assessment and decision-making. Asymptotic confidence intervals and credible intervals are constructed to quantify the estimation uncertainty. The performance of the proposed estimators is evaluated through Monte Carlo simulations under different experimental settings. The methodology is illustrated using three real data sets from solar lighting equipment, nanocrystalline devices, and micro-unmanned aerial vehicles. The results emphasize the flexibility and accurate inference for accelerated reliability testing, making it suitable for a wide range of applications.
- accelerated life testing,
- quadratic hazard rate,
- progressive censoring,
- maximum likelihood,
- Bayesian inference,
- reliability analysis
Citation: Hanan Haj Ahmad, Moustafa N. Mousa, M. E. Moshref, N. Youns, Mahmoud M. M. Mansour. Reliability inference of the quadratic hazard rate model under step-stress partially accelerated life testing with progressive Type-Ⅱ censoring[J]. AIMS Mathematics, 2026, 11(3): 6106-6140. doi: 10.3934/math.2026253

Related Papers:

Abstract

Step-stress partially accelerated life testing (SSPALT) provides an effective framework for obtaining timely reliability information when products operate under varying stress levels. In this paper, statistical inference and applications of the quadratic hazard rate distribution are developed under SSPALT with progressive Type-Ⅱ censoring and binomial removals. This censoring scheme offers flexibility in balancing the experimental cost and statistical efficiency while accommodating practical testing constraints. Maximum likelihood and Bayesian estimation are derived for the model parameters and the acceleration factor. Computational implementations are carried out using the Markov chain Monte Carlo method. Reliability analyses, including survival and hazard rate functions, are carried out to support practical reliability assessment and decision-making. Asymptotic confidence intervals and credible intervals are constructed to quantify the estimation uncertainty. The performance of the proposed estimators is evaluated through Monte Carlo simulations under different experimental settings. The methodology is illustrated using three real data sets from solar lighting equipment, nanocrystalline devices, and micro-unmanned aerial vehicles. The results emphasize the flexibility and accurate inference for accelerated reliability testing, making it suitable for a wide range of applications.

References

[1]	S. Bessler, H. Chernoff, A. W. Marshall, An optimal sequential accelerated life test, Technometrics, 4 (1962), 367–379. http://dx.doi.org/10.1080/00401706.1962.10490019 doi: 10.1080/00401706.1962.10490019
[2]	H. Chernoff, Optimal accelerated life designs for estimation, Technometrics, 4 (1962), 381–408. http://dx.doi.org/10.1080/00401706.1962.10490020 doi: 10.1080/00401706.1962.10490020
[3]	W. Nelson, Accelerated testing: Statistical models, test plans and data analysis, New York: Wiley, 1990. http://dx.doi.org/10.1002/9780470316795
[4]	W. Q. Meeker, L. A. Escobar, C. J. Lu, Accelerated degradation tests: Modeling and analysis, Technometrics, 40 (1998), 89–99. http://dx.doi.org/10.1080/00401706.1998.10485191 doi: 10.1080/00401706.1998.10485191
[5]	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. Safe., 199 (2020), 106875. http://dx.doi.org/10.1016/j.ress.2020.106875 doi: 10.1016/j.ress.2020.106875
[6]	A. Rahman, T. N. Sindhu, S. A. Lone, M. Kamal, Statistical inference for Burr type X distribution using geometric process in accelerated life testing design for time censored data, Pak. J. Stat. Oper. Res., 16 (2020), 577–586. http://dx.doi.org/10.18187/pjsor.v16i3.2252 doi: 10.18187/pjsor.v16i3.2252
[7]	S. J. Wu, Estimation of the parameters of the Weibull distribution with progressively censored data, J. Japan Statist. Soc., 32 (2002), 155–163. http://dx.doi.org/10.14490/jjss.32.155 doi: 10.14490/jjss.32.155
[8]	E. A. Ahmed, Estimation of some lifetime parameters of generalized Gompertz distribution under progressively type-Ⅱ censored data, Appl. Math. Model., 39 (2015), 5567–5578. http://dx.doi.org/10.1016/j.apm.2015.01.023 doi: 10.1016/j.apm.2015.01.023
[9]	M. A. W. Mahmoud, D. A. Ramadan, M. M. M. Mansour, Estimation of lifetime parameters of the modified extended exponential distribution with application to a mechanical model, Commun. Stat. Simul. C., 51 (2022), 7005–7018. http://dx.doi.org/10.1080/03610918.2020.1821887 doi: 10.1080/03610918.2020.1821887
[10]	L. Wang, K. Wu, X. Zuo, Inference and prediction of progressive Type-Ⅱ censored data from Unit-Generalized Rayleigh distribution, Hacet. J. Math. Stat., 51 (2022), 1752–1767. http://dx.doi.org/10.15672/hujms.988054 doi: 10.15672/hujms.988054
[11]	D. A. Ramadan, Y. A. Tashkandy, M. E. Bakr, O. S. Balogun, M. M. Hasaballah, Analysis of Marshall-Olkin extended Gumbel type-Ⅱ distribution under progressive type-Ⅱ censoring with applications, AIP Adv., 14 (2024), 055137. http://dx.doi.org/10.1063/5.0210905 doi: 10.1063/5.0210905
[12]	Y. Tashkandy, E. Almetwally, R. Ragab, A. Gemeay, M. A. El-Raouf, S. Khosa, et al., Statistical inferences for the extended inverse Weibull distribution under progressive type-Ⅱ censored sample with applications, Alex. Eng. J., 65 (2023), 493–502. http://dx.doi.org/10.1016/j.aej.2022.09.023 doi: 10.1016/j.aej.2022.09.023
[13]	H. Haj Ahmad, Best prediction method for progressive Type-Ⅱ censored samples under new Pareto model with applications, J. Math., 2021 (2021), 1355990. http://dx.doi.org/10.1155/2021/1355990 doi: 10.1155/2021/1355990
[14]	O. E. Abo-Kasem, A. R. El Saeed, A. I. El Sayed, Optimal sampling and statistical inferences for Kumaraswamy distribution under progressive type-Ⅱ censoring schemes, Sci. Rep., 13 (2023), 12063. http://dx.doi.org/10.1038/s41598-023-38594-9 doi: 10.1038/s41598-023-38594-9
[15]	D. S. Bai, S. W. Chung, Y. R. Chun, Optimal design of partially accelerated life tests for the lognormal distribution under type I censoring, Reliab. Eng. Syst. Saf., 40 (1993), 85–92. http://dx.doi.org/10.1016/0951-8320(93)90122-F doi: 10.1016/0951-8320(93)90122-F
[16]	A. Rahman, S. Lone, A. Islam, Analysis of exponentiated exponential model under step stress partially accelerated life testing plan using progressive type-Ⅱ censored data, Investigacion Operacional, 39 (2018), 551–559.
[17]	R. M. El-Sagheer, M. A. W. Mahmoud, M. M. M. Mansour, Inferences for Weibull-Gamma distribution in presence of partially accelerated life test, J. Stat. Appl. Probab., 9 (2020), 93–107. http://dx.doi.org/10.18576/jsap/090109 doi: 10.18576/jsap/090109
[18]	I. Alam, S. Anwar, L. K. Sharma, Inference on adaptive Type-Ⅱ progressive hybrid censoring under partially accelerated life test for Gompertz distribution, Int. J. Syst. Assur. Eng. Manag., 14 (2023), 2661–2673. http://dx.doi.org/10.1007/s13198-023-02129-2 doi: 10.1007/s13198-023-02129-2
[19]	L. A. Al-Essa, A. H. Abdel-Hamid, T. Alballa, A. Hashem, Reliability analysis of the triple modular redundancy system under step-partially accelerated life tests using Lomax distribution, Sci. Rep., 13 (2023), 14719. http://dx.doi.org/10.1038/s41598-023-41363-3 doi: 10.1038/s41598-023-41363-3
[20]	M. M. Yousef, R. Alsultan, S. G. Nassr, Parametric inference on partially accelerated life testing for the inverted Kumaraswamy distribution based on Type-Ⅱ progressive censoring data, Math. Biosci. Eng., 20 (2023), 1674–1694. http://dx.doi.org/10.3934/mbe.2023076 doi: 10.3934/mbe.2023076
[21]	I. Alam, A. Ahmed, Parametric and interval estimation under step-stress partially accelerated life tests using adaptive Type-Ⅱ progressive hybrid censoring, Ann. Data. Sci., 10 (2023), 441–453. https://doi.org/10.1007/s40745-020-00249-1 doi: 10.1007/s40745-020-00249-1
[22]	M. M. Yousef, A. Fayomi, E. M. Almetwally, Simulation techniques for strength component partially accelerated to analyze stress-strength model, Symmetry, 15 (2023), 1183. http://dx.doi.org/10.3390/sym15061183 doi: 10.3390/sym15061183
[23]	Y. Tian, W. Gui, Inference and expected total test time for step-stress life test in the presence of complementary risks and incomplete data, Comput. Stat., 39 (2024), 1023–1060. http://dx.doi.org/10.1007/s00180-023-01343-7 doi: 10.1007/s00180-023-01343-7
[24]	H. H. Ahmad, D. A. Ramadan, E. M. Almetwally, Tampered random variable analysis in step-stress testing: Modeling, inference, and applications, Mathematics, 12 (2024), 1248. http://dx.doi.org/10.3390/math12081248 doi: 10.3390/math12081248
[25]	L. J. Bain, Analysis for the linear failure-rate life-testing distribution, Technometrics, 16 (1974), 551–559. http://dx.doi.org/10.1080/00401706.1974.10489237 doi: 10.1080/00401706.1974.10489237
[26]	I. Elbatal, N. Butt, A new generalization of quadratic hazard rate distribution, Pak. J. Stat. Oper. Res., 9 (2013), 4.
[27]	A. Mustafa, A new bivariate distribution with generalized quadratic hazard rate marginals, J. Math. Stat., 12 (2016), 255–270.
[28]	F. Bhatti, G. Hamedani, W. Sheng, M. Ahmad, On extended quadratic hazard rate distribution: Development, properties, characterizations and applications, Stoch. Qual. Contr., 33 (2018), 45–60. http://dx.doi.org/10.1515/eqc-2018-0002 doi: 10.1515/eqc-2018-0002
[29]	F. Merovci, I. Elbatal, The beta quadratic hazard rate distribution, Pak. J. Statist., 31 (2015), 427–446.
[30]	H. Okasha, M. Kayid, A. Abouammoh, I. Elbatal, A new family of quadratic hazard rate-geometric distributions with reliability applications, J. Test. Eval., 44 (2016), 1397–1948. http://dx.doi.org/10.1520/JTE20150116 doi: 10.1520/JTE20150116
[31]	A. Al-Hossain, J. Dar, Some results on moment of order statistics for the quadratic hazard rate distribution, J. Stat. Appl. Pro., 5 (2016), 371–376. http://dx.doi.org/10.18576/jsap/050218 doi: 10.18576/jsap/050218
[32]	M. Nagy, A. H. Mansi, M. N. Mousa, M. E. Moshref, N. Youns, M. M. M. Mansour, On estimation of the quadratic hazard rate model parameters: Simulation and application, AIP Adv., 15 (2025), 055126. http://dx.doi.org/10.1063/5.0257356 doi: 10.1063/5.0257356
[33]	M. N. Mousa, M. E. Moshref, N. Youns, M. M. M. Mansour, Inference under hybrid censoring for the quadratic hazard rate model: Simulation and applications to COVID-19 mortality, Mod. J. Stat., 2 (2025), 1–31. http://dx.doi.org/10.64389/mjs.2026.02113 doi: 10.64389/mjs.2026.02113
[34]	P. K. Goel, Some estimation problems in the study of tampered random variables, Carnegie Mellon University, 1971.
[35]	N. Balakrishnan, R. Aggarwala, Progressive censoring: Theory, methods and applications, New York: Springer, 2000. http://dx.doi.org/10.1007/978-1-4612-1334-5
[36]	N. Balakrishnan, E. Cramer, The art of progressive censoring: Applications to reliability and quality, New York: Springer, 2014. http://dx.doi.org/10.1007/978-0-8176-4807-7
[37]	N. Balakrishnan, Progressive censoring methodology: An appraisal, TEST, 16 (2007), 211–259. http://dx.doi.org/10.1007/s11749-007-0061-y doi: 10.1007/s11749-007-0061-y
[38]	S. K. Tse, C. Yang, H. K. Yuen, Statistical analysis of Weibull distributed lifetime data under type Ⅱ progressive censoring with binomial removals, J. Appl. Stat., 27 (2000), 1033–1043. http://dx.doi.org/10.1080/02664760050173355 doi: 10.1080/02664760050173355
[39]	A. Azzalini, Statistical inference based on the likelihood, London: Chapman & Hall, 1996. http://dx.doi.org/10.1201/9780203738627
[40]	S. R. Eliason, Maximum likelihood estimation: Logic and practice, SAGE Publications, 1993. http://dx.doi.org/10.4135/9781412984928
[41]	L. Held, D. Sabanés Bové, Applied statistical inference: Likelihood and bayes, Berlin: Springer, 2020. http://dx.doi.org/10.1007/978-3-642-37887-4
[42]	A. Zellner, Bayesian estimation and prediction using asymmetric loss functions, J. Am. Stat. Assoc., 81 (1986), 446–451. http://dx.doi.org/10.1080/01621459.1986.10478289 doi: 10.1080/01621459.1986.10478289
[43]	R. Calabria, G. Pulcini, An engineering approach to Bayes estimation for the Weibull distribution, Microelectron. Reliab., 34 (1994), 789–802. http://dx.doi.org/10.1016/0026-2714(94)90004-3 doi: 10.1016/0026-2714(94)90004-3
[44]	S. M. Lynch, Introduction to applied Bayesian statistics and estimation for social scientists, New York: Springer, 2007. http://dx.doi.org/10.1007/978-0-387-71265-9
[45]	L. Tierney, Markov chains for exploring posterior distributions, Ann. Statist., 22 (1994), 1701–1728. http://dx.doi.org/10.1214/aos/1176325750 doi: 10.1214/aos/1176325750
[46]	W. R. Gilks, S. Richardson, D. Spiegelhalter, Markov chain Monte Carlo in Practice, New York: Chapman & Hall/CRC, 1995. http://dx.doi.org/10.1201/b14835
[47]	Z. F. Jaheen, Empirical Bayes inference for generalized exponential distribution based on records, Commun. Stat. Theor. M., 33 (2004), 1851–1861. http://dx.doi.org/10.1081/STA-120037445 doi: 10.1081/STA-120037445
[48]	S. Dey, S. Singh, Y. M. Tripathi, A. Asgharzadeh, Estimation and prediction for a progressively censored generalized inverted exponential distribution, Stat. Methodol., 32 (2016), 185–202. http://dx.doi.org/10.1016/j.stamet.2016.05.007 doi: 10.1016/j.stamet.2016.05.007
[49]	S. Singh, Y. Tripathi, Estimating the parameters of an inverse Weibull distribution under progressive type-Ⅰ interval censoring, Stat. Papers, 59 (2018), 21–56. http://dx.doi.org/10.1007/s00362-016-0750-2 doi: 10.1007/s00362-016-0750-2
[50]	S. L. Lohr, Sampling: Design and analysis, New York: Chapman & Hall/CRC, 2021. http://dx.doi.org/10.1201/9780429298899
[51]	W. H. Greene, Econometric analysis, Stat. Papers, 52 (2011), 983–984. http://dx.doi.org/10.1007/s00362-010-0315-8
[52]	L. Al-Essa, A. Abdel-Hamid, T. Alballa, A. Hashem, Reliability analysis of the triple modular redundancy system under step-partially accelerated life tests using Lomax distribution, Sci. Rep., 13 (2023), 14719. http://dx.doi.org/10.1038/s41598-023-41363-3 doi: 10.1038/s41598-023-41363-3
[53]	M. A. Amleh, M. Z. Raqab, Inference for step-stress plan with Khamis-Higgins model under type-Ⅱ censored Weibull data, Qual. Reliab. Eng. Int., 39 (2023), 982–1000. http://dx.doi.org/10.1002/qre.3275 doi: 10.1002/qre.3275

Reader Comments

Your name:*

Email:*
© 2026 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)