In this paper, we mainly consider reliability inference and remaining useful life prediction for the unit Gompertz distribution and the related stress-strength model. The exact confidence interval for the model parameter $ \beta $ is derived. The generalized confidence intervals for model parameter $ \alpha $ and some commonly used reliability metrics such as the failure rate function, the quantile, and the reliability function are explored. Based on the observed failure data set, the prediction intervals for the remaining useful life and the future failure times are developed. In addition, when the stress and strength variables follow the unit Gompertz distributions with different parameters, the generalized confidence interval for the reliability of the stress-strength model is also proposed. A Monte Carlo simulation study is implemented to evaluate the accuracy of the proposed inferential procedures and compared with the Wald and bootstrap methods. Finally, two examples are provided to illustrate the applicability of the proposed methods.
Citation: Xiaofei Wang, Peihua Jiang. Reliability assessment and remaining useful life prediction based on the unit Gompertz distribution[J]. AIMS Mathematics, 2025, 10(10): 22911-22928. doi: 10.3934/math.20251018
In this paper, we mainly consider reliability inference and remaining useful life prediction for the unit Gompertz distribution and the related stress-strength model. The exact confidence interval for the model parameter $ \beta $ is derived. The generalized confidence intervals for model parameter $ \alpha $ and some commonly used reliability metrics such as the failure rate function, the quantile, and the reliability function are explored. Based on the observed failure data set, the prediction intervals for the remaining useful life and the future failure times are developed. In addition, when the stress and strength variables follow the unit Gompertz distributions with different parameters, the generalized confidence interval for the reliability of the stress-strength model is also proposed. A Monte Carlo simulation study is implemented to evaluate the accuracy of the proposed inferential procedures and compared with the Wald and bootstrap methods. Finally, two examples are provided to illustrate the applicability of the proposed methods.
| [1] |
M. Arshad, Q. J. Azhad, N. Gupta, A. K. Pathak, Bayesian inference of Unit Gompertz distribution based on dual generalized order statistics, Commun. Stat.-Simul. Comput., 52 (2023), 3657–3675. https://doi.org/10.1080/03610918.2021.1943441 doi: 10.1080/03610918.2021.1943441
|
| [2] |
J. Mazucheli, A. F. Menezes, S. Dey, Unit-Gompertz distribution with applications, Statistica, 79 (2019), 25–43. https://doi.org/10.6092/issn.1973-2201/8497 doi: 10.6092/issn.1973-2201/8497
|
| [3] |
M. Z. Anis, D. De, An expository note on unit-Gompertz distribution with applications, Statistica, 80 (2020), 469–490. https://doi.org/10.6092/issn.1973-2201/11135 doi: 10.6092/issn.1973-2201/11135
|
| [4] |
M. K. Jha, S. Dey, R. M. Alotaibi, Y. M. Tripathi, Reliability estimation of a multicomponent stress-strength model for unit Gompertz distribution under progressive Type II censoring, Qual. Reliab. Eng. Int., 36 (2020), 965–987. https://doi.org/10.1002/qre.2610 doi: 10.1002/qre.2610
|
| [5] |
D. Kumar, S. Dey, E. Ormoz, S. MirMostafaee, Inference for the unit-Gompertz model based on record values and inter-record times with an application, Rend. Circ. Mat. Palermo, Ser. 2, 69 (2020), 1295–1319. https://doi.org/10.1007/s12215-019-00471-8 doi: 10.1007/s12215-019-00471-8
|
| [6] |
M. K. Jha, S. Dey, Y. M. Tripathi, Reliability estimation in a multicomponent stress–strength based on unit-Gompertz distribution, Int. J. Qual. Reliab. Manage., 37 (2000), 428–450. https://doi.org/10.1108/IJQRM-04-2019-0136 doi: 10.1108/IJQRM-04-2019-0136
|
| [7] |
S. Dey, R. Al-Mosawi, Classical and Bayesian inference of unit Gompertz distribution based on progressively Type II censored data, Am. J. Math. Manage. Sci., 43 (2024), 61–89. https://doi.org/10.1080/01966324.2024.2311286 doi: 10.1080/01966324.2024.2311286
|
| [8] | Z. W. Birnbaum, On a use of the Mann-Whitney statistic, Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, Volume 1: Contributions to the Theory of Statistics, 13 (1956), 13–17. |
| [9] |
M. E. Ghitany, D. K. Al-Mutairi, S. M. Aboukhamseen, Estimation of the reliability of a stress-strength system from power Lindley distributions, Commun. Stat.-Simul. Comput., 44 (2015), 118–136. https://doi.org/10.1080/03610918.2013.767910 doi: 10.1080/03610918.2013.767910
|
| [10] |
A. Pak, A. Kumar Gupta, N. Bagheri Khoolenjani, On reliability in a multicomponent stress-strength model with power Lindley distribution, Rev. Colomb. Estad., 41 (2018), 251–267. https://doi.org/10.15446/rce.v41n2.69621 doi: 10.15446/rce.v41n2.69621
|
| [11] | A. Asgharzadeh, R. Valiollahi, M. Z. Raqab, Stress-strength reliability of Weibull distribution based on progressively censored samples, SORT, 35 (2011), 103–124. |
| [12] |
A. M. Almarashi, A. Algarni, M. Nassar, On estimation procedures of stress-strength reliability for Weibull distribution with application, Plos One, 15 (2020), e0237997. https://doi.org/10.1371/journal.pone.0237997 doi: 10.1371/journal.pone.0237997
|
| [13] | D. Kundu, R. D. Gupta, Estimation of P [Y < X] for Weibull distributions, IEEE Trans. Reliab., 55 (2006), 270–280. |
| [14] |
Y. L. Lio, T. R. Tsai, Estimation of $\delta$ = P (X < Y) for Burr XII distribution based on the progressively first failure-censored samples, J. Appl. Stat., 39 (2012), 309–322. https://doi.org/10.1080/02664763.2011.586684 doi: 10.1080/02664763.2011.586684
|
| [15] |
A. S. Hassan, A. Al-Omari, H. F. Nagy, Stress–strength reliability for the generalized inverted exponential distribution using MRSS, Iran. J. Sci. Technol., Trans. A: Sci., 45 (2021), 641–659. https://doi.org/10.1007/s40995-020-01033-9 doi: 10.1007/s40995-020-01033-9
|
| [16] |
B. X. Wang, Y. Geng, J. X. Zhou, Inference for the generalized exponential stress-strength model, Appl. Math. Model., 53 (2018), 267–275. https://doi.org/10.1016/j.apm.2017.09.012 doi: 10.1016/j.apm.2017.09.012
|
| [17] |
S. Rostamian, N. Nematollahi, Estimation of stress–strength reliability in the inverse Gaussian distribution under progressively Type II censored data, Math. Sci., 13 (2019), 175–191. https://doi.org/10.1007/s40096-019-0289-1 doi: 10.1007/s40096-019-0289-1
|
| [18] |
X. Wang, B. X. Wang, X. Pan, Y. Hu, Y. Chen, J. Zhou, Interval estimation for inverse Gaussian distribution, Qual. Reliab. Eng. Int., 37 (2021), 2263–2275. https://doi.org/10.1002/qre.2856 doi: 10.1002/qre.2856
|
| [19] | S. Kotz, Y. Lumelskii, M. Pensky, The stress–strength model and its generalizations: theory and applications, World Scientific, 2003. https://doi.org/10.1142/5015 |
| [20] | S. Weerahandi, Generalized inference in repeated measures: exact methods in MANOVA and mixed models, New York: John Wiley & Sons, 2004. |
| [21] |
H. K. Iyer, C. M. J. Wang, T. Mathew, Models and confidence intervals for true values in interlaboratory trials, J. Am. Stat. Assoc., 99 (2004), 1060–1071. https://doi.org/10.1198/016214504000001682 doi: 10.1198/016214504000001682
|
| [22] |
J. Hannig, H. Iyer, P. Patterson, Fiducial generalized confidence intervals, J. Am. Stat. Assoc., 101 (2006), 254–269. https://doi.org/10.1198/016214505000000736 doi: 10.1198/016214505000000736
|
| [23] |
R. Viveros, N. Balakrishnan, Interval estimation of parameters of life from progressively censored data, Technometrics, 36 (1994), 84–91. https://doi.org/10.2307/1269201 doi: 10.2307/1269201
|
| [24] |
B. X. Wang, K. Yu, M. C. Jones, Inference under progressively Type II right-censored sampling for certain lifetime distributions, Technometrics, 52 (2010), 453–460. https://doi.org/10.1198/Tech.2010.08210 doi: 10.1198/Tech.2010.08210
|
| [25] |
S. Qin, B. X. Wang, T. R. Tsai, X. Wang, The prediction of remaining useful lifetime for the Weibull k-out-of-n load-sharing system, Reliab. Eng. Syst. Safety, 233 (2023), 109091. https://doi.org/10.1016/j.ress.2023.109091 doi: 10.1016/j.ress.2023.109091
|
| [26] |
R. Dumonceaux, C. E. Antle, Discrimination between the log-normal and the Weibull distributions, Technometrics, 15 (1973), 923–926. https://doi.org/10.2307/1267401 doi: 10.2307/1267401
|
| [27] |
N. Alsadat, A. S. Hassan, M. Elgarhy, C. Chesneau, R. E. Mohamed, An efficient stress–strength reliability estimate of the unit Gompertz distribution using ranked set sampling, Symmetry, 15 (2023), 1121. https://doi.org/10.3390/sym15051121 doi: 10.3390/sym15051121
|
Xiaofei Wang, Peihua Jiang. Reliability assessment and remaining useful life prediction based on the unit Gompertz distribution[J]. AIMS Mathematics, 2025, 10(10): 22911-22928. doi: 10.3934/math.20251018
DownLoad: