Estimating the stress-strength reliability parameter of the inverse power Lomax distribution

Abdelfattah Mustafa; M. I. Khan; Samah M. Ahmed; Abdelfattah Mustafa; M. I. Khan; Samah M. Ahmed

doi:10.3934/math.2025700

AIMS Mathematics

2025, Volume 10, Issue 7: 15632-15652. doi: 10.3934/math.2025700

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Estimating the stress-strength reliability parameter of the inverse power Lomax distribution

1.
Mathematics Department, Faculty of Science, Islamic University of Madinah, Madinah 42351, Saudi Arabia
2.
Mathematics Department, Faculty of Science, Mansoura University, Mansoura 35516, Egypt
3.
Mathematics Department, Faculty of Science, Sohag University, Sohag 82524, Egypt

Received: 29 January 2025 Revised: 28 June 2025 Accepted: 03 July 2025 Published: 09 July 2025
MSC : 60E05, 62F10, 62F15

This research focused on estimating the stress-strength parameter by considering stress and strength as distinct random variables, both characterized by the inverse power Lomax (IPL) distribution. The maximum likelihood estimate (MLE) for stress-strength reliability was then calculated using the Newton-Raphson method. Using the asymptotic normality of MLEs, this study developed approximate confidence intervals. Bootstrap confidence intervals for the stress-strength reliability parameter ($ R $) were investigated. The Bayes estimator of $ R $ was considered. Furthermore, we utilized the Markov chain Monte Carlo (MCMC) method to create both symmetric and asymmetric loss functions, allowing for a more comprehensive analysis. The highest posterior density (HPD) credible intervals under a gamma prior distribution were calculated. The different approaches were assessed using a Monte Carlo simulation. Finally, a numerical example was given to show the effectiveness of the proposed methods.
- maximum likelihood estimator,
- the inverse power Lomax model,
- stress-strength reliability,
- symmetric and asymmetric loss functions,
- bootstrap resampling,
- Bayes estimator
Citation: Abdelfattah Mustafa, M. I. Khan, Samah M. Ahmed. Estimating the stress-strength reliability parameter of the inverse power Lomax distribution[J]. AIMS Mathematics, 2025, 10(7): 15632-15652. doi: 10.3934/math.2025700

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Abstract

This research focused on estimating the stress-strength parameter by considering stress and strength as distinct random variables, both characterized by the inverse power Lomax (IPL) distribution. The maximum likelihood estimate (MLE) for stress-strength reliability was then calculated using the Newton-Raphson method. Using the asymptotic normality of MLEs, this study developed approximate confidence intervals. Bootstrap confidence intervals for the stress-strength reliability parameter ($ R $) were investigated. The Bayes estimator of $ R $ was considered. Furthermore, we utilized the Markov chain Monte Carlo (MCMC) method to create both symmetric and asymmetric loss functions, allowing for a more comprehensive analysis. The highest posterior density (HPD) credible intervals under a gamma prior distribution were calculated. The different approaches were assessed using a Monte Carlo simulation. Finally, a numerical example was given to show the effectiveness of the proposed methods.

References

[1]	Z. W. Birnbaum, On a use of the Mann-Whitney statistic, In: Proceedings of the third Berkeley symposium on mathematical statistics and probability, volume 1: contributions to the theory of statistics, 1956, 13–17.
[2]	R. A. Johnson, Stress strength models for reliability, Handbook of statistics, 7 (1988), 27–54. https://doi.org/10.1016/S0169-7161(88)07005-1
[3]	A. M. Awad, M. K. Charraf, Estimation of $P(Y < X)$ in the Burr case: a comparative study, Commun. Stat.-Simul. Comput., 15 (1986), 389–403. http://doi.org/10.1080/03610918608812514 doi: 10.1080/03610918608812514
[4]	K. E. Ahmed, M. E. Fakhry, Z. F. Jaheen, Empirical Bayes estimation of $P(Y < X)$ and characterizations of Burr-Type X model, J. Stat. Plan. Infer., 64 (1997), 297–308. https://doi.org/10.1016/S0378-3758(97)00038-4 doi: 10.1016/S0378-3758(97)00038-4
[5]	A. H. Khan, T. R. Jan, Estimation of stress-strength reliability model using finite mixture of two parameter Lindley distributions, J. Stat. Appl. Pro., 4 (2015), 147–159.
[6]	B. Saraçoğlu, I. Kinaci, D. Kundu, On estimation of $R = P(Y X)$ for exponential distribution under progressive Type-II censoring, J. Stat. Comput. Sim., 82 (2012), 729–744. https://doi.org/10.1080/00949655.2010.551772 doi: 10.1080/00949655.2010.551772
[7]	D. Kundu, R. D. Gupta, Estimation of $P(X < Y)$ for generalized exponential distribution, Metrika, 61 (2005), 291–308. https://doi.org/10.1007/s001840400345 doi: 10.1007/s001840400345
[8]	S. Nadarajah, Reliability for Laplace distributions, Math. Probl. Eng., 2004 (2004), 169–183. https://doi.org/10.1155/S1024123X0431104X doi: 10.1155/S1024123X0431104X
[9]	J. G. Surles, W. J. Padgett, Inference for reliability and stress-strength for a scaled Burr-type X distribution, Lifetime Data Anal., 7 (2001), 187–200. https://doi.org/10.1023/A:1011352923990 doi: 10.1023/A:1011352923990
[10]	S. Nadarajah, Reliability for lifetime distributions, Math. Comput. Model., 37 (2003), 683–688. https://doi.org/10.1016/S0895-7177(03)00074-8 doi: 10.1016/S0895-7177(03)00074-8
[11]	D. Kundu, M. Z. Raqab, Estimation of $R = P[Y X]$ for three-parameter Weibull distribution, Stat. Probab. Lett., 79 (2009), 1839–1846. https://doi.org/10.1016/j.spl.2009.05.026 doi: 10.1016/j.spl.2009.05.026
[12]	M. Nassar, R. Alotaibi, C. Zhang, Product of spacing estimation of stress–strength reliability for alpha power exponential progressively Type-II censored data, Axioms, 12 (2023), 752. http://doi.org/10.3390/axioms12080752 doi: 10.3390/axioms12080752
[13]	K. Krishnamoorthy, Y. Lin, Confidence limits for stress-strength reliability involving Weibull models, J. Stat. Plan. Infer., 140 (2010), 1754–1764. https://doi.org/10.1016/j.jspi.2009.12.028 doi: 10.1016/j.jspi.2009.12.028
[14]	R. C. Gupta, C. Peng, Estimating reliability in proportional odds ratio models, Comput. Stat. Data Anal., 53 (2009), 1495–1510. https://doi.org/10.1016/j.csda.2008.10.014 doi: 10.1016/j.csda.2008.10.014
[15]	D. K. Al-Mutairi, M. E. Ghitany, R. C. Gupta, Estimation of reliability in a series system with random sample size, Comput. Stat. Data Anal., 55 (2011), 964–972. https://doi.org/10.1016/j.csda.2010.07.027 doi: 10.1016/j.csda.2010.07.027
[16]	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
[17]	S. Ghanbari, A. Rezaei Roknabadi, M. Salehi, Estimation of stress–strength reliability for Marshall–Olkin distributions based on progressively Type-II censored samples, J. Appl. Stat., 49 (2022), 1913–1934. https://doi.org/10.1080/02664763.2021.1884207 doi: 10.1080/02664763.2021.1884207
[18]	S. Asadi, H. Panahi, Estimation of stress–strength reliability based on censored data and its evaluation for coating processes, Qual. Technol. Quant. Manag., 19 (2022), 379–401. https://doi.org/10.1080/16843703.2021.2001129 doi: 10.1080/16843703.2021.2001129
[19]	X. Hu, H. Ren, Statistical inference of the stress-strength reliability for inverse Weibull distribution under an adaptive progressive type-II censored sample, AIMS Math., 8 (2023), 28465–28487. https://doi.org/10.3934/math.20231457 doi: 10.3934/math.20231457
[20]	I. Elbatal, A. S. Hassan, L. S. Diab, A. Ben Ghorbal, M. Elgarhy, A. R. El-Saeed, Stress–strength reliability analysis for different distributions using progressive Type-II censoring with binomial removal, Axioms, 12 (2023), 1054. https://doi.org/10.3390/axioms12111054 doi: 10.3390/axioms12111054
[21]	N. S. Y. Temraz, Inference on the stress strength reliability with exponentiated generalized Marshall Olkin-G distribution, PLOS ONE, 18 (2023), e0280183. https://doi.org/10.1371/journal.pone.0280183 doi: 10.1371/journal.pone.0280183
[22]	T. Xavier, J. K. Jose, S. C. Bagui, Stress-strength reliability estimation of a series system with cold standby redundancy based on Kumaraswamy half-Logistic distribution, American Journal of Mathematical and Management Sciences, 42 (2023), 183–201. https://doi.org/10.1080/01966324.2023.2213835 doi: 10.1080/01966324.2023.2213835
[23]	A. S. Hassan, N. Alsadat, M. Elgarhy, H. Ahmad, H. F. Nagy, On estimating multi stress strength reliability for inverted Kumaraswamy under ranked set sampling with application in engineering, J. Nonlinear Math. Phys., 31 (2024), 30. https://doi.org/10.1007/s44198-024-00196-y doi: 10.1007/s44198-024-00196-y
[24]	A. Xu, R. Wang, X. Weng, Q. Wu, L. Zhuang, Strategic integration of adaptive sampling and ensemble techniques in federated learning for aircraft engine remaining useful life prediction, Appl. Soft Comput., 175 (2025), 113067. https://doi.org/10.1016/j.asoc.2025.113067 doi: 10.1016/j.asoc.2025.113067
[25]	A. Xu, G. Fang, L. Zhuang, C. Gu, A multivariate student-$t$ process model for dependent tail-weighted degradation data, IISE Trans., 57 (2025), 1071–1087. https://doi.org/10.1080/24725854.2024.2389538 doi: 10.1080/24725854.2024.2389538
[26]	A. S. Hassan, M. Abd-Allah, On the inverse power Lomax distribution, Ann. Data Sci., 6 (2019), 259–278. https://doi.org/10.1007/s40745-018-0183-y doi: 10.1007/s40745-018-0183-y
[27]	P. Kumar, H. Sharma, Prior preferences for the inverse power Lomax distribution: Bayesian method, Journal of Statistics Applications & Probability Letters, 10 (2023), 49–62. https://doi.org/10.18576/jsapl/100104 doi: 10.18576/jsapl/100104
[28]	X. Shi, Y. Shi, Inference for inverse power Lomax distribution with progressive first-failure censoring, Entropy, 23 (2021), 1099. https://doi.org/10.3390/e23091099 doi: 10.3390/e23091099
[29]	S. M. Ahmed, A. Mustafa, Estimation of the coefficients of variation for inverse power Lomax distribution, AIMS Math., 9 (2024), 33423–33441. https://doi.org/10.3934/math.20241595 doi: 10.3934/math.20241595
[30]	A. P. Basu, N. Ebrahimi, Bayesian approach to life testing and reliability estimation using asymmetric loss function, J. Stat. Plan. Infer., 29 (1991), 21–31. https://doi.org/10.1016/0378-3758(92)90118-C doi: 10.1016/0378-3758(92)90118-C
[31]	H. R. Varian, A Bayesian approach to real estate assessment, In: Studies in Bayesian econometrics and statistics: in honor of L. J. Savage, North Holland, Amsterdam, 1975,195–208.
[32]	U. Balasooriya, N. Balakrishnan, Reliability sampling plans for log-normal distribution based on progressively censored samples, IEEE Trans. Reliab., 49 (2000), 199–203. https://doi.org/10.1109/24.877338 doi: 10.1109/24.877338
[33]	G. Prakash, D. C. Singh, Shrinkage estimation in exponential Type-II censored data under LINEX loss, J. Korean Stat. Soc., 37 (2008), 53–61. https://doi.org/10.1016/j.jkss.2007.07.002 doi: 10.1016/j.jkss.2007.07.002
[34]	R. Calabria, G. Pulcini, Bayes 2-sample prediction for the inverse Weibull distribution, Commun. Stat.-Theor. Method., 23 (1994), 1811–1824. https://doi.org/10.1080/03610929408831356 doi: 10.1080/03610929408831356
[35]	J. F. Lawless, Statistical models and methods for lifetime data, New York: John Wiley and Sons, 2003. https://doi.org/10.1002/9781118033005
[36]	B. Efron, R. J. Tibshirani, An introduction to the bootstrap, 1 Eds., New York: Chapman and Hall/CRC, 1994. https://doi.org/10.1201/9780429246593
[37]	M. E. Ghitany, B. Atieh, S. Nadarajah, Lindley distribution and its application, Math. Comput. Simul., 78 (2008), 493–506. https://doi.org/10.1016/j.matcom.2007.06.007 doi: 10.1016/j.matcom.2007.06.007

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