Analysis of competing risks model using the generalized progressive hybrid censored data from the generalized Lomax distribution

Amal Hassan; Sudhansu Maiti; Rana Mousa; Najwan Alsadat; Mahmoued Abu-Moussa; Amal Hassan; Sudhansu Maiti; Rana Mousa; Najwan Alsadat; Mahmoued Abu-Moussa

doi:10.3934/math.20241611

AIMS Mathematics

2024, Volume 9, Issue 12: 33756-33799. doi: 10.3934/math.20241611

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Research article

Analysis of competing risks model using the generalized progressive hybrid censored data from the generalized Lomax distribution

1.
Department of Mathematical Statistics, Faculty of Graduate Studies for Statistical Research, Cairo University, Giza, Egypt
2.
Department of Statistics, Visvs-Bharati University, Santiniketan, India
3.
Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt
4.
Department of Quantitative Analysis, College of Business Administration, King Saud University, Riyadh, Saudi Arabia
5.
Department of Mathematics, Faculty of Science, Galala University, Galala, Suez 43511, Egypt

Received: 19 September 2024 Revised: 04 November 2024 Accepted: 11 November 2024 Published: 28 November 2024
MSC : 62F15, 62N02

The competing risk (CR) model is crucial for studying various areas, such as biology, econometrics, and engineering. When multiple factors could cause a product to fail, these factors often work against each other, resulting in the product's failure. This scenario is known as the CR problem. This study focused on parameter estimation of the generalized Lomax distribution under a generalized progressive hybrid censoring scheme in the presence of CR when the cause of failure for each item was known and independent. Both maximum likelihood (ML) and Bayesian approaches were used to estimate the unknown parameters, reliability characteristics, and relative risks due to two causes. Bayesian estimators under gamma priors with different loss functions were generated using Markov chain Monte Carlo, and confidence intervals (CIs) were generated using the ML estimation method. Additionally, two bootstrap CIs for the unknown parameters were presented. According to the conditional posterior distribution, credible intervals and the highest posterior density intervals were further generated. The performance of different estimators was compared using Monte Carlo simulation, and real-data applications were used to verify the proposed estimates.
- competing risks,
- generalized Lomax distribution,
- generalized progressive hybrid censoring,
- bootstrap confidence intervals
Citation: Amal Hassan, Sudhansu Maiti, Rana Mousa, Najwan Alsadat, Mahmoued Abu-Moussa. Analysis of competing risks model using the generalized progressive hybrid censored data from the generalized Lomax distribution[J]. AIMS Mathematics, 2024, 9(12): 33756-33799. doi: 10.3934/math.20241611

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Abstract

The competing risk (CR) model is crucial for studying various areas, such as biology, econometrics, and engineering. When multiple factors could cause a product to fail, these factors often work against each other, resulting in the product's failure. This scenario is known as the CR problem. This study focused on parameter estimation of the generalized Lomax distribution under a generalized progressive hybrid censoring scheme in the presence of CR when the cause of failure for each item was known and independent. Both maximum likelihood (ML) and Bayesian approaches were used to estimate the unknown parameters, reliability characteristics, and relative risks due to two causes. Bayesian estimators under gamma priors with different loss functions were generated using Markov chain Monte Carlo, and confidence intervals (CIs) were generated using the ML estimation method. Additionally, two bootstrap CIs for the unknown parameters were presented. According to the conditional posterior distribution, credible intervals and the highest posterior density intervals were further generated. The performance of different estimators was compared using Monte Carlo simulation, and real-data applications were used to verify the proposed estimates.

References

[1]	J. Boag, Maximum likelihood estimates of the proportion of patients cured by cancer therapy, J. R. Stat. Soc., 11 (1949), 15–44. https://doi.org/10.1111/j.2517-6161.1949.tb00020.x doi: 10.1111/j.2517-6161.1949.tb00020.x
[2]	M. Pintilie, Competing risks: a practical perspective, John Wiley & Sons, Inc., 2006.
[3]	T. Chen, S. Zheng, H. Luo, X. Liu, J. Feng, Reliability analysis of multiple causes of failure in presence of independent competing risks, Qual. Reliab. Eng. Int., 32 (2016), 363–372. https://doi.org/10.1002/qre.1755 doi: 10.1002/qre.1755
[4]	L. Liu, X. Liu, X. Wang, Y. Wang, C. Li, Reliability analysis and evaluation of a brake system based on competing risk, J. Eng. Res., 5 (2017), 150–161.
[5]	H. Li, M. Yazdi, Stochastic game theory approach to solve system safety and reliability decision-making problem under uncertainty, In: H. Li, M. Yazdi, Advanced decision-making methods and applications in system safety and reliability problems, Springer, 2022. https://doi.org/10.1007/978-3-031-07430-1-8
[6]	M. Yazdi, E. Zarei, S. Adumene, R. Abbassi, P. Rahnamayiezekavat, Uncertainty modeling in risk assessment of digitalized process systems, Methods Chem. Process Saf., 6 (2022), 389–416. https://doi.org/10.1016/bs.mcps.2022.04.005 doi: 10.1016/bs.mcps.2022.04.005
[7]	E. Zarei, M. Yazdi, R. Moradi, A. B. Toroody, Expert judgment and uncertainty in sociotechnical systems analysis, In: E. Zarei, Safety causation analysis in sociotechnical systems: advanced models and techniques, Springer, 2024. https://doi.org/10.1007/978-3-031-62470-4_18
[8]	N. Balakrishnan, Progressive censoring methodology: an appraisal, Test, 16 (2007), 211–259. https://doi.org/10.1007/s11749-007-0061-y doi: 10.1007/s11749-007-0061-y
[9]	H. K. Ng, P. S. Chan, Comments on: progressive censoring methodology: an appraisal, Test, 16 (2007), 287–289. https://doi.org/10.1007/s11749-007-0071-9 doi: 10.1007/s11749-007-0071-9
[10]	Y. Cho, H. Sun, K. Lee, Exact likelihood inference for an exponential parameter under generalized progressive hybrid censoring scheme, Stat. Methodol., 23 (2015), 18–34. https://doi.org/10.1016/j.stamet.2014.09.002 doi: 10.1016/j.stamet.2014.09.002
[11]	D. Kundu, A. Joarder, Analysis of type-II progressively hybrid censored competing risks data, J. Mod. Appl. Stat. Methods, 5 (2006), 152–170. https://doi.org/10.22237/JMASM/1146456780 doi: 10.22237/JMASM/1146456780
[12]	H. Liao, W. Gui, Statistical inference of the Rayleigh distribution based on progressively type II censored competing risks data, Symmetry, 11 (2019), 898. https://doi.org/10.3390/sym11070898 doi: 10.3390/sym11070898
[13]	L. Wang, Y. M. Tripathi, C. Lodhi, Inference for Weibull competing risks model with partially observed failure causes under generalized progressive hybrid censoring, J. Comput. Appl. Math., 368 (2020), 112537. https://doi.org/10.1016/j.cam.2019.112537 doi: 10.1016/j.cam.2019.112537
[14]	A. M. Almarashi, A. Algarni, G. A. Abd-Elmougod, Statistical analysis of competing risks lifetime data from Nadarajaha and Haghighi distribution under type-II censoring, J. Intell. Fuzzy Syst., 38 (2020), 2591–2601. https://doi.org/10.3233/JIFS-179546 doi: 10.3233/JIFS-179546
[15]	M. R. Mahmoud, H. Z. Muhammed, A. R. El-Saeed, Analysis of progressively type-I censored data in competing risks models with generalized inverted exponential distribution, J. Stat. Appl. Probab., 9 (2020), 109–117. https://doi.org/10.18576/jsap/090110 doi: 10.18576/jsap/090110
[16]	A. M. Abd El-Raheem, M. Hosny, M. H. Abu-Moussa, On progressive censored competing risks data: real data application and simulation study, Mathematics, 9 (2021), 1805. https://doi.org/10.3390/math9151805 doi: 10.3390/math9151805
[17]	T. A. Abushal, A. A. Soliman, G. A. Abd-Elmougod, Inference of partially observed causes for failure of Lomax competing risks model under type-II generalized hybrid censoring scheme, Alex. Eng. J., 61 (2022), 5427–5439. https://doi.org/10.1016/j.aej.2021.10.058 doi: 10.1016/j.aej.2021.10.058
[18]	X. Qin, W. Gui, Statistical inference of Lomax distribution based on adaptive progressive type-II hybrid censored competing risks data, Commun. Stat., 52 (2023), 8114–8135. https://doi.org/10.1080/03610926.2022.2056750 doi: 10.1080/03610926.2022.2056750
[19]	A. S. Hassan, R. M. Mousa, M. H. Abu-Moussa, Analysis of progressive type-II competing risks data, with applications, Lobachevskii J. Math., 43 (2022), 2279–2292. https://doi.org/10.1134/S1995080222120149 doi: 10.1134/S1995080222120149
[20]	A. Elshahhat, M. Nassar, Inference of improved adaptive progressively censored competing risks data for Weibull lifetime models, Stat. Papers, 65 (2024), 1163–1196. https://doi.org/10.1007/s00362-023-01417-0 doi: 10.1007/s00362-023-01417-0
[21]	A. S. Hassan, R. M. Mousa, M. H. Abu-Moussa, Bayesian analysis of generalized inverted exponential distribution based on generalized progressive hybrid censoring competing risks data, Ann. Data Sci., 11 (2024), 1225–1264. https://doi.org/10.1007/s40745-023-00488-y doi: 10.1007/s40745-023-00488-y
[22]	Y. Tian, Y. Liang, W. Gui, Inference and optimal censoring scheme for a competing-risks model with type-II progressive censoring, Math. Popul. Stud., 31 (2024), 1–39. https://doi.org/10.1080/08898480.2023.2225349 doi: 10.1080/08898480.2023.2225349
[23]	Q. Lv, R. Hua, W. Gui, Statistical inference of Gompertz distribution under general progressive type II censored competing risks sample, Commun. Stat., 53 (2024), 682–701. https://doi.org/10.1080/03610918.2022.2028834 doi: 10.1080/03610918.2022.2028834
[24]	S. A. Salem, O. Abo-Kasem, A. A. Khairy, Inference for generalized progressive hybrid type-II censored Weibull lifetimes under competing risk data, Comput. J. Math. Stat. Sci., 3 (2024), 177–202. https://doi.org/10.21608/cjmss.2024.256760.1035 doi: 10.21608/cjmss.2024.256760.1035
[25]	R. Maurya, Y. Tripathi, C. Lodhi, M. Rastogi, On a generalized Lomax distribution, Int. J. Syst. Assur. Eng. Manag., 10 (2019), 1091–1104. https://doi.org/10.1007/s13198-019-00839-0 doi: 10.1007/s13198-019-00839-0
[26]	A. Hassan, S. M. Assar, A. Shelbaia, Optimum step stress accelerated life test plan for Lomax distribution with an adaptive type-II progressive hybrid censoring, Br. J. Math. Comput. Sci., 13 (2016), 1–9. https://doi.org/10.9734/BJMCS/2016/21964 doi: 10.9734/BJMCS/2016/21964
[27]	A. Alghamdi, Study of generalized Lomax distribution and change point problem, Ph. D. thesis, Bowling Green State University, 2018.
[28]	G. Box, D. Cox, An analysis of transformations, J. R. Stat. Soc., 26 (1964), 211–243. https://doi.org/10.1111/j.2517-6161.1964.tb00553.x doi: 10.1111/j.2517-6161.1964.tb00553.x
[29]	S. Varde, Life testing and reliability estimtation for the two parameter exponential distribution, J. Amer. Statist. Assoc., 64 (1969), 621–631. https://doi.org/10.1080/01621459.1969.10501000 doi: 10.1080/01621459.1969.10501000
[30]	H. Varian, A Bayesian approach to real estate assessment, In: L. J.Savage, S. E. Feinberg, A. Zellner, Studies in Bayesian econometric and statistics, North-Holland Publication, 1975.
[31]	A. Basu, N. Ibrahimi, Bayesian approach to life testing and reliability estimating using asymmetric loss function, J. Stat. Plan. Inference, 29 (1991), 21–31. https://doi.org/10.1016/0378-3758(92)90118-C doi: 10.1016/0378-3758(92)90118-C
[32]	R. Calabria, G. Pulcini, Point estimation under asymmetric loss functions for left-truncated exponential samples, Commun. Stat., 25 (1996), 585–600. https://doi.org/10.1080/0361092960883171583 doi: 10.1080/0361092960883171583
[33]	N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, E. Teller, Equation of state calculations by fast computing machine, J. Chem. Phys., 21 (1953), 1087–1092. https://doi.org/10.1063/1.1699114 doi: 10.1063/1.1699114
[34]	A. Cohen, Maximum likelihood estimation in the Weibull distribution based on complete and on censored samples, Technometrics, 7 (1965), 579–588. https://doi.org/10.1080/00401706.1965.10490300 doi: 10.1080/00401706.1965.10490300
[35]	W. Greene, Econometric analysis, 5 Eds., Pearson Education, Inc., 2003.
[36]	B. Efron, The Jackknife, the Bootstrap and other resampling plans, CBMS-NSF Regional Conference Series in Applied Mathematics, 1982.
[37]	P. Hall, Theoretical comparison of bootstrap confidence intervals, Ann. Stat., 16 (1988), 927–953. https://doi.org/10.1214/aos/1176350933 doi: 10.1214/aos/1176350933
[38]	D. Kundu, N. Kannan, N. Balakrishnan, Analysis of progressively censored competing risks data, Handb. Stat., 23 (2003), 331–348. https://doi.org/10.1016/S0169-7161(03)23018-2 doi: 10.1016/S0169-7161(03)23018-2
[39]	N. Balakrishnan, R. Sandhu, A simple simulational algorithm for generating progressive type-II censored samples, Amer. Stat., 49 (1995), 229–230. https://doi.org/10.1080/00031305.1995.10476150 doi: 10.1080/00031305.1995.10476150
[40]	M. M. El-Din, M. Nagy, M. Abu-Moussa, Estimation and prediction for Gompertz distribution under the generalized progressive hybrid censored data, Ann. Data Sci., 6 (2019), 673–705. https://doi.org/10.1007/s40745-019-00199-3 doi: 10.1007/s40745-019-00199-3
[41]	Z. Xia, J. Yu, L. Cheng, L. Liu, W. Wang. Study on the breaking strength of jute fibers using modified Weibull distribution, Compos. Part A, 40 (2009), 54–59. https://doi.org/10.1016/j.compositesa.2008.10.001 doi: 10.1016/j.compositesa.2008.10.001
[42]	D. Kundu, B. Pradhan, Bayesian analysis of progressively censored competing risks data, Sankhya B, 73 (2011), 276–296. https://doi.org/10.1007/s13571-011-0024-x doi: 10.1007/s13571-011-0024-x
[43]	M. Chen, Q. Shao, Monte Carlo estimation of Bayesian credible and HPD intervals, J. Comput. Graph. Stat., 6 (1988), 66–92. https://doi.org/10.1080/10618600.1999.10474802 doi: 10.1080/10618600.1999.10474802

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