Analysis of reliability index $ \mathfrak{R} = P(Y &lt; X) $ for newly extended xgamma progressively first-failure censored samples with applications

Refah Alotaibi; Mazen Nassar; Zareen A. Khan; Ahmed Elshahhat; Refah Alotaibi; Mazen Nassar; Zareen A. Khan; Ahmed Elshahhat

doi:10.3934/math.20241546

AIMS Mathematics

2024, Volume 9, Issue 11: 32200-32231. doi: 10.3934/math.20241546

Previous Article Next Article

Research article Special Issues

Analysis of reliability index $ \mathfrak{R} = P(Y < X) $ for newly extended xgamma progressively first-failure censored samples with applications

1.
Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia
2.
Department of Statistics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia
3.
Faculty of Technology and Development, Zagazig University, Zagazig 44519, Egypt

Received: 06 September 2024 Revised: 27 October 2024 Accepted: 05 November 2024 Published: 14 November 2024
MSC : 62F10, 62F15, 62N01, 62N02, 62N05

The stress-strength index measures the likelihood that a system's strength exceeds its stress. This study focuses on deducting the stress-strength index, denoted as $ \mathfrak{R} = P(Y < X) $, where the strength $ (X) $ and stress $ (Y) $ are independent random variables following new extended xgamma distributions. Inferences are made based on progressively first-failure censored samples. Both maximum likelihood and Bayesian estimation approaches, including point and interval estimations, are considered. The estimations take into account the model parameters as well as the reliability index. The Bayes estimates are obtained using the Markov chain Monte Carlo sampling procedure with the squared error loss function. Additionally, the approximate confidence intervals and Bayes credible intervals are developed. A simulation experiment is conducted to assess the different estimates presented in this paper. Precision metrics such as root mean square error, mean relative absolute bias, and interval length are used to evaluate the efficiency of various point and interval estimates. Two insulating fluid data sets are analyzed to demonstrate the relevance and applicability of the proposed estimation methods.
- extended xgamma distribution,
- stress strength index,
- classical estimation,
- squared error loss function,
- Bayesian estimation
Citation: Refah Alotaibi, Mazen Nassar, Zareen A. Khan, Ahmed Elshahhat. Analysis of reliability index $ \mathfrak{R} = P(Y < X) $ for newly extended xgamma progressively first-failure censored samples with applications[J]. AIMS Mathematics, 2024, 9(11): 32200-32231. doi: 10.3934/math.20241546

Related Papers:

Abstract

The stress-strength index measures the likelihood that a system's strength exceeds its stress. This study focuses on deducting the stress-strength index, denoted as $ \mathfrak{R} = P(Y < X) $, where the strength $ (X) $ and stress $ (Y) $ are independent random variables following new extended xgamma distributions. Inferences are made based on progressively first-failure censored samples. Both maximum likelihood and Bayesian estimation approaches, including point and interval estimations, are considered. The estimations take into account the model parameters as well as the reliability index. The Bayes estimates are obtained using the Markov chain Monte Carlo sampling procedure with the squared error loss function. Additionally, the approximate confidence intervals and Bayes credible intervals are developed. A simulation experiment is conducted to assess the different estimates presented in this paper. Precision metrics such as root mean square error, mean relative absolute bias, and interval length are used to evaluate the efficiency of various point and interval estimates. Two insulating fluid data sets are analyzed to demonstrate the relevance and applicability of the proposed estimation methods.

References

[1]	S. kota, Y. Lumelskii, M. Pensky, The stress-strength model and its generalization: theory and application, Singapore: World scientific, 2003. https://doi.org/10.1142/5015
[2]	D. kundu, R. D. Gupta, Estimation of $P[Y<X]$ for generalised exponential distribution, Metrika, 61 (2005), 291–308. https://doi.org/10.1007/s001840400345 doi: 10.1007/s001840400345
[3]	K. Krishnamoorthy, S. Mukherjee, H. Guo, Inference on reliability in two-parameter exponential stress–strength model, Metrika, 65 (2007), 261–273. https://doi.org/10.1007/s00184-006-0074-7 doi: 10.1007/s00184-006-0074-7
[4]	A. Asgharzadeh, R. Valiollahi, M. Z. Raqab, Stress-strength reliability of Weibull distribution based on progressively censored samples, Stat. Oper. Res. Trans., 35 (2011), 103–124.
[5]	A. S. Yadav, S. K. Singh, U. Singh, Estimation of stress–strength reliability for inverse Weibull distribution under progressive type-Ⅱ censoring scheme, J. Ind. Prod. Eng., 35 (2018), 48–55. https://doi.org/10.1080/21681015.2017.1421590 doi: 10.1080/21681015.2017.1421590
[6]	X. Bai, Y. Shi, Y. Liu, B. Liu, Reliability estimation of stress–strength model using finite mixture distributions under progressively interval censoring, J. Comput. Appl. Math., 348 (2019), 509–524. https://doi.org/10.1016/j.cam.2018.09.023 doi: 10.1016/j.cam.2018.09.023
[7]	S. Ghanbari, A. R. 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
[8]	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. https://doi.org/10.3390/axioms12080752 doi: 10.3390/axioms12080752
[9]	F. Sultana, Ç. Çetinkaya, D. Kundu, Estimation of the stress-strength parameter under two-sample balanced progressive censoring scheme, J. Stat. Comput. Sim., 94 (2024), 1269–1299. https://doi.org/10.1080/00949655.2023.2282743 doi: 10.1080/00949655.2023.2282743
[10]	S. Sen, S. K. Ghosh, H. Al-Mofleh, The Mirra distribution for modeling time-to-event data sets, In: Strategic management, decision theory, and decision ccience, Singapore: Springer, 2021, 59–73. https://doi.org/10.1007/978-981-16-1368-5_5
[11]	S. Sen, S. S. Maiti, N. Chandra, The xgamma distribution: statistical properties and application, Journal of Modern Applied Statistical Methods, 15 (2016), 38. https://doi.org/10.22237/jmasm/1462077420 doi: 10.22237/jmasm/1462077420
[12]	M. V. de Oliveria Peres, F. S. dos Santos, R. P. de Olivera, Estimation of survival and hazard curves of mixture Mirra cure rate model: application to gastric and breast cancer data, Biom. Biostat. Int. J., 9 (2020), 132–137. https://doi.org/10.15406/bbij.2020.09.00310 doi: 10.15406/bbij.2020.09.00310
[13]	S.-J. Wu, C. Kuş, On estimation based on progressive first-failure-censored sampling, Comput. Stat. Data Anal., 53 (2009), 3659–3670. https://doi.org/10.1016/j.csda.2009.03.010 doi: 10.1016/j.csda.2009.03.010
[14]	A. A. Soliman, A. H. Abd-Ellah, N. A. Abou-Elheggag, G. A. Abd-Elmougod, Estimation of the parameters of life for Gompertz distribution using progressive first-failure censored data, Comput. Stat. Data Anal., 56 (2012), 2471–2485. https://doi.org/10.1016/j.csda.2012.01.025 doi: 10.1016/j.csda.2012.01.025
[15]	M. Dube, R. Garg, H. Krishna, On progressively first failure censored Lindley distribution, Comput. Stat., 31 (2016), 139–163. https://doi.org/10.1007/s00180-015-0622-6 doi: 10.1007/s00180-015-0622-6
[16]	E. A. Ahmed, Estimation and prediction for the generalized inverted exponential distribution based on progressively first-failure-censored data with application, J. Appl. Stat., 44 (2017), 1576–1608. https://doi.org/10.1080/02664763.2016.1214692 doi: 10.1080/02664763.2016.1214692
[17]	S. Saini, A. Chaturvedi, R. Garg, Estimation of stress–strength reliability for generalized Maxwell failure distribution under progressive first failure censoring, J. Stat. Comput. Sim., 91 (2021), 1366–1393. https://doi.org/10.1080/00949655.2020.1856846 doi: 10.1080/00949655.2020.1856846
[18]	S. K. Ashour, A. A. El-Sheikh, A. Elshahhat, Inferences and optimal censoring schemes for progressively first-failure censored Nadarajah-Haghighi distribution, Sankhya A, 84 (2022), 885–923. https://doi.org/10.1007/s13171-019-00175-2 doi: 10.1007/s13171-019-00175-2
[19]	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
[20]	Y. Xie, W. Gui, Statistical inference of the lifetime performance index with the log-logistic distribution based on progressive first-failure-censored data, Symmetry, 12 (2020), 937. https://doi.org/10.3390/sym12060937 doi: 10.3390/sym12060937
[21]	Y. Cai, W. Gui, Classical and Bayesian inference for a progressive first-failure censored left-truncated normal distribution, Symmetry, 13 (2021), 490. https://doi.org/10.3390/sym13030490 doi: 10.3390/sym13030490
[22]	M. Nassar, R. Alotaibi, A. Elshahhat, Statistical analysis of alpha power exponential parameters using progressive first-failure censoring with applications, Axioms, 11 (2022), 553. https://doi.org/10.3390/axioms11100553 doi: 10.3390/axioms11100553
[23]	A. Elshahhat, V. K. Sharma, H. S. Mohammed, Statistical analysis of progressively first-failure-censored data via beta-binomial removals, AIMS Math., 8 (2023), 22419–22446. https://doi.org/10.3934/math.20231144 doi: 10.3934/math.20231144
[24]	A. Henningsen, O. Toomet, maxLik: A package for maximum likelihood estimation in R, Comput. Stat., 26 (2011), 443–458. https://doi.org/10.1007/s00180-010-0217-1 doi: 10.1007/s00180-010-0217-1
[25]	M. Plummer, N. Best, K. Cowles, K. Vines, CODA: convergence diagnosis and output analysis for MCMC, R News, 6 (2006), 7–11.
[26]	W. Nelson, Accelerated testing: statistical model, test plan and data analysis, New York, NY: Wiley, 2004. https://doi.org/10.1002/9780470316795
[27]	M. Nassar, R. Alotaibi, A. Elshahhat, Reliability estimation of XLindley constant-stress partially accelerated life tests using progressively censored samples, Mathematics, 11 (2023), 1331. https://doi.org/10.3390/math11061331 doi: 10.3390/math11061331
[28]	R. Alotaibi, M. Nassar, A. Elshahhat, Reliability estimation under normal operating conditions for progressively Type-Ⅱ XLindley censored data, Axioms, 12 (2023), 352. https://doi.org/10.3390/axioms12040352 doi: 10.3390/axioms12040352

Reader Comments

Your name:*

Email:*
© 2024 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)