Inference on stress-strength reliability from censored data using the asymmetric generalized Poisson Lomax model

Rashad M. EL-Sagheer; Mohamed F. Abouelenein; Mohamed S. Eliwa; Mahmoud El-Morshedy; Noura Roushdy; Mahmoud M. Ramadan; Rashad M. EL-Sagheer; Mohamed F. Abouelenein; Mohamed S. Eliwa; Mahmoud El-Morshedy; Noura Roushdy; Mahmoud M. Ramadan

doi:10.3934/math.2025888

AIMS Mathematics

2025, Volume 10, Issue 8: 19896-19921. doi: 10.3934/math.2025888

Previous Article Next Article

Research article Special Issues

Inference on stress-strength reliability from censored data using the asymmetric generalized Poisson Lomax model

1.
High Institute of Computers and Management Information Systems, First Statement, New Cairo 11865, Cairo, Egypt
2.
Department of Insurance and Risk Management, College of Business, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11432, Riyadh, Saudi Arabia
3.
Department of Statistics and Operations Research, College of Science, Qassim University, Saudi Arabia
4.
Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia
5.
Department of Mathematics, Faculty of Education, Ain Shams University, Roxy 11341, Cairo, Egypt

Received: 29 May 2025 Revised: 12 August 2025 Accepted: 13 August 2025 Published: 28 August 2025
MSC : 62F10, 62F12

This paper investigated the estimation of the stress-strength reliability parameter $ R = P(X > Y) $, where the random variables $ X $ (strength) and $ Y $ (stress) were independently modeled by the generalized Poisson Lomax distribution (GPLD). The analysis was conducted under a progressive Type II censoring scheme, which provided greater flexibility in practical life-testing experiments by allowing for staggered removal of surviving units. Assuming shared scale and shape parameters, both maximum likelihood estimators (MLEs) and Bayesian estimators of $ R $ were derived. Due to the absence of closed-form solutions, numerical optimization techniques were applied for MLEs, while Bayesian estimates were obtained under squared error and LINEX loss functions using Markov chain Monte Carlo methods with importance sampling. Asymptotic confidence intervals and highest posterior density credible intervals were constructed to assess uncertainty. A comprehensive simulation study was performed to evaluate the efficiency and robustness of the proposed estimators under varying sample sizes and censoring schemes. Furthermore, two real data applications were analyzed that show survival times of melanoma patients and failure times of high-voltage insulating fluids to illustrate the practical utility of the methodology. The results demonstrated that Bayesian approaches, particularly under asymmetric loss functions, yielded superior performance compared to their frequentist counterparts.
- statistical model,
- generalized Poisson Lomax distribution,
- stress-strength reliability model,
- Bayes theorem,
- highest posterior density,
- computer simulation,
- data analysis
Citation: Rashad M. EL-Sagheer, Mohamed F. Abouelenein, Mohamed S. Eliwa, Mahmoud El-Morshedy, Noura Roushdy, Mahmoud M. Ramadan. Inference on stress-strength reliability from censored data using the asymmetric generalized Poisson Lomax model[J]. AIMS Mathematics, 2025, 10(8): 19896-19921. doi: 10.3934/math.2025888

Related Papers:

Abstract

This paper investigated the estimation of the stress-strength reliability parameter $ R = P(X > Y) $, where the random variables $ X $ (strength) and $ Y $ (stress) were independently modeled by the generalized Poisson Lomax distribution (GPLD). The analysis was conducted under a progressive Type II censoring scheme, which provided greater flexibility in practical life-testing experiments by allowing for staggered removal of surviving units. Assuming shared scale and shape parameters, both maximum likelihood estimators (MLEs) and Bayesian estimators of $ R $ were derived. Due to the absence of closed-form solutions, numerical optimization techniques were applied for MLEs, while Bayesian estimates were obtained under squared error and LINEX loss functions using Markov chain Monte Carlo methods with importance sampling. Asymptotic confidence intervals and highest posterior density credible intervals were constructed to assess uncertainty. A comprehensive simulation study was performed to evaluate the efficiency and robustness of the proposed estimators under varying sample sizes and censoring schemes. Furthermore, two real data applications were analyzed that show survival times of melanoma patients and failure times of high-voltage insulating fluids to illustrate the practical utility of the methodology. The results demonstrated that Bayesian approaches, particularly under asymmetric loss functions, yielded superior performance compared to their frequentist counterparts.

References

[1]	D. Kundu, R. Gupta, Estimation of $P\left(Y < X\right) $ for generalized exponential distribution, Metrika, 61 (2005), 291–308. https://doi.org/10.1007/s001840400345 doi: 10.1007/s001840400345
[2]	J. Q. Surles, W. J. Padgett, Inference for $P\left(Y < X\right) $ in the Burr type X model, J. Appl. Stat. Sci., 7 (1998), 225–238.
[3]	M. Z. Raqab, D. Kundu, Comparison of different estimators of $P\left(Y < X\right) $ for a scaled Burr type X distribution, Commun. Stat. Simul. C., 34 (2005), 465–483. https://doi.org/10.1081/sac-200055741 doi: 10.1081/sac-200055741
[4]	D. Kundu, R. D. Gupta, Estimation of $P\left(Y < X\right) $ for Weibull distribution, IEEE T. Reliab., 55 (2006), 270–280. https://doi.org/10.1109/TR.2006.874918 doi: 10.1109/TR.2006.874918
[5]	M. Nadar, F. Kizilaslan, A. Papadopoulos, Classical and Bayesian estimation of $P\left(Y < X\right) $ for Kumaraswamy distribution, J. Stat. Comput. Simul., 84 (2014), 1505–1529. https://doi.org/10.1080/00949655.2012.750658 doi: 10.1080/00949655.2012.750658
[6]	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
[7]	V. K. Sharma, S. K. Singh, U. Singh, V. Agiwal, The inverse Lindley distribution: A stress-strength reliability model with application to head and neck cancer data, J. Ind. Prod. Eng., 32 (2015), 162–173. https://doi.org/10.1080/21681015.2015.1025901 doi: 10.1080/21681015.2015.1025901
[8]	A. Baklizi, Estimation of $P\left(Y < X\right) $ using record values in the one and two parameter exponential distributions, Commun. Stat. Theor. M., 37 (2008), 692–698. https://doi.org/10.1080/03610920701501921 doi: 10.1080/03610920701501921
[9]	A. Baklizi, Inference on $P\left(Y < X\right) $ in the two-parameter Weibull model based on records, Int. Scholarly Res. Notices, 2012 (2012), 1–11. https://doi.org/10.5402/2012/263612 doi: 10.5402/2012/263612
[10]	A. S. Hasan, M. Abd-Allah, H. F. Nagy, Estimation of $P\left(Y < X\right) $ using record values from the generalized inverted exponential distribution, Pak. J. Stat. Oper. Res., 14 (2018), 645–660. https://doi.org/10.18187/pjsor.v14i3.2201 doi: 10.18187/pjsor.v14i3.2201
[11]	M. Nadar, F. Kizilaslan, Classical and Bayesian estimation of $P\left(X < Y\right) $ using upper record values from Kumaraswamy distribution, Stat. Papers, 55 (2014), 751–783. https://doi.org/10.1007/s00362-013-0526-x doi: 10.1007/s00362-013-0526-x
[12]	A. W. M. Mahmoud, R. M. EL-Sagheer, A. A. Soliman, A. H. Abd Ellah, Bayesian estimation of $P\left(Y < X\right) $ based on record values from the Lomax distribution and MCMC technique, J. Mod. Appl. Stat. Meth., 15 (2016), 488–510. https://doi.org/10.22237/jmasm/1462076640 doi: 10.22237/jmasm/1462076640
[13]	N. Balakrishnan, R. A. Sandhu, A simple simulation algorithm for generating progressive type-II censored samples, Am. Stat., 49 (1995), 229–230. https://doi.org/10.1080/00031305.1995.10476150 doi: 10.1080/00031305.1995.10476150
[14]	B. Saraçoğlu, I. Kinaci, D. Kundu, On estimation of $P\left(Y < X\right) $ for exponential distribution under progressive type-II censoring, J. Stat. Comput. Simul., 82 (2012), 729–744. https://doi.org/10.1080/00949655.2010.551772 doi: 10.1080/00949655.2010.551772
[15]	K. Kumar, H. Krishna, R. Garg, Estimation of $P\left(Y < X\right) $ in Lindley distribution using progressively first failure censoring, Int. J. Syst. Assur. Eng., 6 (2014), 330–341. https://doi.org/10.1007/s13198-014-0267-9 doi: 10.1007/s13198-014-0267-9
[16]	H. Krishna, M. Dube, R. Garg, Estimation of $P\left(Y < X\right) $ for progressively first failure censored generalized inverted exponential distribution, J. Stat. Comput. Simul., 87 (2017), 2274–2289. https://doi.org/10.1080/00949655.2017.1326119 doi: 10.1080/00949655.2017.1326119
[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. Simul., 91 (2021), 1366–1393. https://doi.org/10.1080/00949655.2020.1856846 doi: 10.1080/00949655.2020.1856846
[18]	R. De la Cruz, H. S. Salinas, C. Meza, Reliability estimation for stress-strength model based on Unit-Half-Normal distribution, Symmetry, 14 (2022), 837. https://doi.org/10.3390/sym14040837 doi: 10.3390/sym14040837
[19]	C. Cetinkaya, Reliability estimation of the stress–strength model with non-identical jointly type-II censored Weibull component strengths, J. Stat. Comput. Simul., 91 (2021), 2917–2936. https://doi.org/10.1080/00949655.2021.1910948 doi: 10.1080/00949655.2021.1910948
[20]	A. Sabry, E. M. Almetwally, H. M. Almongy, Monte Carlo simulation of stress-strength model and reliability estimation for extension of the exponential distribution, Thail. Statist., 20 (2022), 124–143.
[21]	Y. Yu, L. Wang, S. Dey, J. Liu, Estimation of stress-strength reliability from unit-Burr III distribution under records data, Math. Biosci. Eng., 20 (2023), 12360–12379. https://doi.org/10.3934/mbe.2023550 doi: 10.3934/mbe.2023550
[22]	R. M. EL-Sagheer, M. M. M. Mansour, The efficacy measurement of treatment methods: An application to stress-strength model, Appl. Math. Inf. Sci., 14 (2020), 487–492. https://doi.org/10.18576/amis/140316 doi: 10.18576/amis/140316
[23]	R. M. EL-Sagheer, A. H. Tolba, T. M. Jawa, N. Sayed-Ahmed, Inferences for stress-strength reliability model in the presence of partially accelerated life test to its strength variable, Comput. Intel. Neurosc., 2022 (2022), 4710536. https://doi.org/10.1155/2022/4710536 doi: 10.1155/2022/4710536
[24]	D. Poisson-Lomax, B. Al-Zahrani, H. Sagor, The Poisson-Lomax distribution, Rev. Colomb. Estad., 37 (2014), 225–245.
[25]	S. E. Abu-Youssef, A new generalized Poisson Lomax lifetime distribution and its application to censored data, 2016.
[26]	G. Casella, Statistical inference, 2002.
[27]	D. Kundu, H. Howlader, Bayesian inference and prediction of the inverse Weibull distribution for Type-II censored data, Comput. Stat. Data An., 54 (2010), 1547–1558. https://doi.org/10.1016/j.csda.2010.01.003 doi: 10.1016/j.csda.2010.01.003
[28]	S. Singh, Y. M. Tripathi, Estimating the parameters of an inverse Weibull distribution under progressive type-I interval censoring, Stat. Papers, 59 (2018), 21–56. https://doi.org/10.1007/s00362-016-0750-2 doi: 10.1007/s00362-016-0750-2
[29]	R. M. EL-Sagheer, M. S. Eliwa, K. M. Alqahtani, M. EL-Morshedy, Asymmetric randomly censored mortality distribution: Bayesian framework and parametric bootstrap with application to COVID-19 data, J. Math., 2022 (2022), 8300753. https://doi.org/10.1155/2022/8300753 doi: 10.1155/2022/8300753
[30]	A. M. Almarashi, A. Algarni, H. Okasha, M. Nassar, On reliability estimation of Nadarajah-Haghighi distribution under adaptive type-I progressive hybrid censoring scheme, Qual. Reliab. Eng. Int., 38 (2022), 817–833. https://doi.org/10.1002/qre.3016 doi: 10.1002/qre.3016
[31]	A. Zellner, Bayesian estimation and prediction using asymmetric loss functions, J. Am. Stat. Assoc., 81 (1986), 446–451. https://doi.org/10.1080/01621459.1986.10478289 doi: 10.1080/01621459.1986.10478289
[32]	E. T. Lee, D. R. Ishmael, R. H. Bottomley, J. L. Murray, An analysis of skin tests and their relationship to recurrence and survival in stage III and stage IV melanoma patients, Cancer, 49 (1982), 2336–2341. https://doi.org/10.1002/1097-0142(19820601)49:11<2336::AID-CNCR2820491121>3.0.CO;2-U doi: 10.1002/1097-0142(19820601)49:11<2336::AID-CNCR2820491121>3.0.CO;2-U
[33]	F. B. Boulkeroua, O. M. Bdair, M. Z. Raqab, Statistical analysis of joint progressive censoring data from Gompertz distribution, J. Stat. Appl. Probab., 11 (2022), 759–780. https://doi.org/10.18576/jsap/110302 doi: 10.18576/jsap/110302
[34]	W. B. Nelson, Applied life data analysis, New York: John Wiley & Sons, 1982. https://doi.org/10.1002/0471725234
[35]	O. E. Abo-Kasem, M. Nassar, S. Dey, A. Rasouli, Classical and Bayesian estimation for two exponential populations based on joint Type-I progressive hybrid censoring scheme, Am. J. Math. Manag. Sci., 38 (2019), 373–385. https://doi.org/10.1080/01966324.2019.1570407 doi: 10.1080/01966324.2019.1570407

Reader Comments

Your name:*

Email:*
© 2025 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)