In reliability theory, the reliability inference for s-out-of-k systems holds significant importance. In this paper, we explore the estimation of reliability for s-out-of-k systems based on partially constant stress accelerated life tests. Assume that the latent failure times of the components follow the Kumaraswamy distribution. Maximum likelihood estimates for the unknown parameters are established, and their uniqueness is demonstrated. In addition, confidence intervals for the unknown parameters are constructed using the covariance matrix. Confidence intervals for the reliability functions are determined by the Delta method, while Bootstrap intervals are provided for comparison purposes. Subsequently, Bayesian point and interval estimates based on MCMC techniques considering different loss functions are discussed. Lastly, we conduct an extensive simulation study and analyse one real data set, which reveals that the Bayesian approach yields the best results.

References

• Aban, I., Meerschaert, M., & Panorska, A. (2006). Parameter estimation for the truncated Pareto distribution. Journal of the American Statistical Association, 101(473), 270–277. https://doi.org/10.1198/016214505000000411
• Aljohani, H. M., & Alfar, N. M. (2020). Estimations with step-stress partially accelerated life tests for competing risks Burr XII lifetime model under type-II censored data. Alexandria Engineering Journal, 59(3), 1171–1180. https://doi.org/10.1016/j.aej.2020.01.022
• Asa, A., Mhe, A., Hma, B., & Ae, C. (2022). Estimations of accelerated Lomax lifetime distribution with a dependent competing risks model under type-I generalized hybrid censoring scheme. Alexandria Engineering Journal, 61(8), 6489–6499. https://doi.org/10.1016/j.aej.2021.12.006
• Barlow, R., & Proschan, F. (1975). Statistical theory of reliability and life testing: Probability models. Holt, Rinehart and Winston.
• Bhattacharyya, G. K., & Johnson, R. A. (1974). Estimation of reliability in multicomponent stress-strength model. Journal of the American Statistical Association, 69(348), 966–970. https://doi.org/10.1080/01621459.1974.10480238
• Cai, J., Shi, Y., & Liu, B. (2020). Bayesian analysis for dependent competing risks model with masked causes of failure in step-stress accelerated life test under progressive hybrid censoring. Communications in Statistics – Simulation and Computation, 49(9), 2302–2320. https://doi.org/10.1080/03610918.2018.1517212
• Dey, S., Wang, L., & Nassar, M. (2022). Inference on Nadarajah-Haghighi distribution with constant stress partially accelerated life tests under progressive type-II censoring. Journal of Applied Statistics, 49(11), 2891–2912. https://doi.org/10.1080/02664763.2021.1928014
• Efron, B. (1982). The Jackknife, the Bootstrap and other resampling plans. Society for industrial and applied mathematics.
• Fan, T. H., & Hsu, T. M. (2014). Constant stress accelerated life test on a multiple-component series system under Weibull lifetime distributions. Communications in Statistics-Theory and Methods, 43(11), 2370–2383. https://doi.org/10.1080/03610926.2013.809115
• Fu, Y., Han, G., Chen, J., & Ma, J. (2009). Research of accelerated life test of complex system based on pneumatic system (in Chinese version). Journal of Mechanical Strength, 31(4), 578–583.
• Genc, A. (2013). Estimation of P(X>Y) with Topp-Leone distribution. Journal of Statistical Computation and Simulation, 83(2), 326–339. https://doi.org/10.1080/00949655.2011.607821
• Gholizadeh, R. A., Shirazi, M. A., & Mosalmanza, S. (2011). Classical and Bayesian estimations on the Kumaraswamy distribution using grouped and un-grouped data under difference loss functions. Journal of Applied Sciences, 11(12), 2154–2162. https://doi.org/10.3923/jas.2011.2154.2162
• Ghosh, I., & Nadarajah, S. (2017). On the Bayesian inference of Kumaraswamy distributions based on censored samples. Communication in Statistics-Theory and Methods, 46(17), 8760–8777. https://doi.org/10.1080/03610926.2016.1193197
• Greene, W. H. (2000). Econometric analysis (4th ed.). Wiley.
• Guo, Z., Zhang, J., & Yan, R. (2020). The residual lifetime of surviving components of coherent system under periodical inspections. Mathematics, 8(12), 2227–7390. https://doi.org/10.3390/math8122181
• Guo, Z., Zhang, J., & Yan, R. (2022). On inactivity times of failed components of coherent system under double monitoring. Probability in the Engineering and Informational Sciences, 36(3), 923–940. https://doi.org/10.1017/S0269964821000152
• Guo, M., Zhang, J., Zhang, Y., & Zhao, P. (2024). Optimal redundancy allocations for series systems under hierarchical dependence structures. Quality and Reliability Engineering International, 40(4), 1540–1565. https://doi.org/10.1002/qre.3508
• Hua, R., & Gui, W. (2022). Inference for copula-based dependent competing risks model with step-stress accelerated life test under generalized progressive hybrid censoring. Computational Statistics, 37(5), 2399–2436. https://doi.org/10.1007/s00180-022-01203-w
• Ismail, A. A. (2014). Inference for a step-stress partially accelerated life test model with an adaptive type-II progressively hybrid censored data from Weibull distribution. Journal of Computational and Applied Mathematics, 260, 533–542. https://doi.org/10.1016/j.cam.2013.10.014
• Ismail, A. A. (2016). Statistical inference for a step-stress partially-accelerated life test model with an adaptive type-I progressively hybrid censored data from Weibull distribution. Statistical Papers, 57(2), 271–301. https://doi.org/10.1007/s00362-014-0639-x
• Kizilaslan, F., & Nadar, M. (2016). Estimation and prediction of the Kumaraswamy distribution based on record values and inter-record times. Journal of Statistical Computation & Simulation, 86(12), 2471–2493. https://doi.org/10.1080/00949655.2015.1119832
• Kohansal, A. (2019). On estimation of reliability in a multicomponent stress-strength model for a Kumaraswamy distribution based on progressively censored sample. Statistical Papers, 60(6), 2185–2224. https://doi.org/10.1007/s00362-017-0916-6
• Kumaraswamy, P. (1980). A generalized probability density function for double-bounded random processes. Journal of Hydrology, 46(1-2), 79–88. https://doi.org/10.1016/0022-1694(80)90036-0
• Liu, B., Shi, Y., Zhang, F., & Bai, X. (2017). Reliability nonparametric Bayesian estimation for the masked data of parallel systems in step-stress accelerated life tests. Journal of Computational & Applied Mathematics, 311, 375–386. https://doi.org/10.1016/j.cam.2016.07.029
• Lu, B., Zhang, J., & Yan, R. (2023). Optimal allocation of a coherent system with statistical dependent subsystems. Probability in the Engineering and Informational Sciences, 37(1), 29–48. https://doi.org/10.1017/S0269964821000437
• Nelson, W. (2008). Accelerated testing: Statistical models, test plans, and data analysis. Wiley.
• Rastogi, M. K., Saad, J., Farghal, A. A., & G. A. Elmougod (2022). Partially constant-stress accelerated life tests model for parameters estimation of Kumaraswamy distribution under adaptive type-II progressive censoring. Alexandria Engineering Journal, 61(7), 5133–5143. https://doi.org/10.1016/j.aej.2021.10.035
• Roy, S. (2018). Bayesian accelerated life test plans for series systems with Weibull component lifetimes. Applied Mathematical Modelling, 62, 383–403. https://doi.org/10.1016/j.apm.2018.06.007
• Samanta, D., Gupta, A., & Kundu, D. (2019). Analysis of Weibull step-stress model in presence of competing risk. IEEE Transactions on Reliability, 68(2), 420–438. https://doi.org/10.1109/TR.24
• Wang, L. (2017). Inference of constant-stress accelerated life test for a truncated distribution under progressive censoring. Applied Mathematical Modelling, 44, 743–757. https://doi.org/10.1016/j.apm.2017.02.011
• Wang, L., Dey, S., Tripathi, Y. M., & Wu, S. J. (2020). Reliability inference for a multicomponent stress-strength model based on Kumaraswamy distribution. Journal of Computational and Applied Mathematics, 376, Article 112823. https://doi.org/10.1016/j.cam.2020.112823
• Wang, L., Wu, S. J., Zhang, C., Dey, S., & Tripathi, Y. M. (2022). Analysis for constant-stress model on multicomponent system from generalized inverted exponential distribution with stress dependent parameters. Mathematics and Computers in Simulation, 193(C), 301–316. https://doi.org/10.1016/j.matcom.2021.10.017
• Wang, L., Zhang, C., Tripathi, Y. M., Dey, S., & Wu, S. (2021). Reliability analysis of Weibull multicomponent system with stress-dependent parameters from accelerated life data. Quality and Reliability Engineering International, 37(6), 2603–2621. https://doi.org/10.1002/qre.v37.6
• Xu, A., Fu, J., Tang, Y., & Guan, Q. (2015). Bayesian analysis of constant-stress accelerated life test for the Weibull distribution using noninformative priors. Applied Mathematical Modelling, 39(20), 6183–6195. https://doi.org/10.1016/j.apm.2015.01.066
• Xu, A., & Tang, Y. (2011). Objective Bayesian analysis of accelerated competing failure models under type-I censoring. Computational Statistics & Data Analysis, 55(10), 2830–2839. https://doi.org/10.1016/j.csda.2011.04.009
• Yan, R., & Niu, J. (2022). Stochastic comparisons of second-order statistics from dependent and heterogenous modified proportional hazard rate observations. Statistics, 57(2), 328–353. https://doi.org/10.1080/02331888.2023.2177999
• Yan, R., Wang, J., & Lu, B. (2021). Orderings of component level versus system level at active redundancies for coherent systems with dependent components. Probability in the Engineering and Informational Sciences, 36(4), 1275–1297. https://doi.org/10.1017/S0269964821000401
• Yan, R., Zhang, J., & Wang, J. (2022). Optimal allocation of relevations in coherent systems. Journal of Applied Probability, 58(4), 293–298.
• Zhang, J., Guo, Z., Niu, J., & Yan, R. (2024). Increasing convex order of capital allocation with dependent assets under threshold model. Statistical Theory and Related Fields, 1–12. https://doi.org/10.1080/24754269.2023.2301630
• Zhang, C., Shi, Y., & Wu, M. (2016). Statistical inference for competing risks model in step-stress partially accelerated life tests with progressively type-I hybrid censored Weibull life data. Journal of Computational & Applied Mathematics, 297, 65–74. https://doi.org/10.1016/j.cam.2015.11.002
• Zhang, T., & Xie, M. (2011). On the upper truncated Weibull distribution and its reliability implications. Reliability Engineering & System Safety, 96(1), 194–200. https://doi.org/10.1016/j.ress.2010.09.004
• Zhang, J., Yan, R., & Wang, J. (2022). Reliability optimization of parallel-series and series-parallel systems with statistically dependent components. Applied Mathematical Modelling, 102, 618–639. https://doi.org/10.1016/j.apm.2021.10.003
• Zhang, J., Yan, R., & Zhang, Y. (2022). Reliability analysis of fail-safe systems with heterogeneous and dependent components subject to random shocks. Proceedings of the Institution of Mechanical Engineers, Part O: Journal of Risk and Reliability, 237(6), 1073–1087.
• Zhang, J., Yan, R., & Zhang, Y. (2023). Stochastic comparisons of largest claim amount from heterogeneous and dependent insurance portfolios. Journal of Computational and Applied Mathematics, 431, Article 115265. https://doi.org/10.1016/j.cam.2023.115265
• Zhang, J., & Zhang, Y. (2022). A copula-based approach on optimal allocation of hot standbys in series systems. Naval Research Logistics (NRL), 69(6), 902–913. https://doi.org/10.1002/nav.v69.6
• Zhang, J., & Zhang, Y. (2023). Stochastic comparisons of relevation allocation policies in coherent systems. Test, 32(3), 865–907. https://doi.org/10.1007/s11749-023-00855-0
• Zuo, M. J. (2003). Optimal reliability modeling, principles and applications. Wiley.

To cite this article: Dongzhu Lamu & Rongfang Yan (2024) Reliability estimation of s-out-of-k system with Kumaraswamy distribution based on partially constant stress accelerated life tests, Statistical Theory and Related Fields, 8:4, 295-314, DOI: 10.1080/24754269.2024.2359826

To link to this article: https://doi.org/10.1080/24754269.2024.2359826