Reliability analysis of independent Burr-X competing risks model based on improved adaptive progressively Type-Ⅱ censored samples with applications

Refah Alotaibi; Mazen Nassar; Ahmed Elshahhat; Refah Alotaibi; Mazen Nassar; Ahmed Elshahhat

doi:10.3934/math.2025686

AIMS Mathematics

2025, Volume 10, Issue 7: 15302-15332. doi: 10.3934/math.2025686

Previous Article Next Article

Research article Special Issues

Reliability analysis of independent Burr-X competing risks model based on improved adaptive progressively Type-Ⅱ 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.
Department of Statistics, Faculty of Commerce, Zagazig University, Egypt
4.
Faculty of Technology and Development, Zagazig University, Zagazig 44519, Egypt

Received: 28 May 2025 Revised: 16 June 2025 Accepted: 24 June 2025 Published: 02 July 2025
MSC : 62F10, 62F15, 62N01, 62N05

In the analysis of failure time data, researchers often encounter situations in which events arise from multiple mutually exclusive causes. This framework is commonly referred to as competing risks. Traditional methods are inadequate in such settings, as they typically assume the presence of a single type of failure and do not account for the influence of other competing events. This study investigates the competing risks model under an improved adaptive progressive Type-Ⅱ censoring scheme, which is particularly beneficial in contexts where the duration of testing is critical. The lifetimes associated with competing risks are assumed to follow independent Burr-X distributions, a flexible model capable of accommodating a variety of data types. Both classical and Bayesian estimation methods are utilized to estimate the model parameters and the reliability function, a key metric in reliability assessment. Maximum likelihood estimates are computed numerically, and approximate confidence intervals are derived. For Bayesian inference, squared error and linear-exponential loss functions are employed. Given the complexity of the posterior distribution, Markov chain Monte Carlo techniques are utilized to obtain Bayesian estimates and construct Bayesian credible intervals. A comprehensive numerical analysis, which includes a simulation study and the examination of real competing risk datasets, is conducted to evaluate and compare the performance of the proposed estimation methods.
- Burr-X competing risk model,
- reliability function,
- improved adaptive progressive Type-Ⅱ censoring,
- Bayesian estimation,
- MCMC technique
Citation: Refah Alotaibi, Mazen Nassar, Ahmed Elshahhat. Reliability analysis of independent Burr-X competing risks model based on improved adaptive progressively Type-Ⅱ censored samples with applications[J]. AIMS Mathematics, 2025, 10(7): 15302-15332. doi: 10.3934/math.2025686

Related Papers:

Abstract

In the analysis of failure time data, researchers often encounter situations in which events arise from multiple mutually exclusive causes. This framework is commonly referred to as competing risks. Traditional methods are inadequate in such settings, as they typically assume the presence of a single type of failure and do not account for the influence of other competing events. This study investigates the competing risks model under an improved adaptive progressive Type-Ⅱ censoring scheme, which is particularly beneficial in contexts where the duration of testing is critical. The lifetimes associated with competing risks are assumed to follow independent Burr-X distributions, a flexible model capable of accommodating a variety of data types. Both classical and Bayesian estimation methods are utilized to estimate the model parameters and the reliability function, a key metric in reliability assessment. Maximum likelihood estimates are computed numerically, and approximate confidence intervals are derived. For Bayesian inference, squared error and linear-exponential loss functions are employed. Given the complexity of the posterior distribution, Markov chain Monte Carlo techniques are utilized to obtain Bayesian estimates and construct Bayesian credible intervals. A comprehensive numerical analysis, which includes a simulation study and the examination of real competing risk datasets, is conducted to evaluate and compare the performance of the proposed estimation methods.

References

[1]	D. G. Hoel, A representation of mortality data by competing risks, Biometrics, 28 (1972), 475–488. https://doi.org/10.2307/2556161 doi: 10.2307/2556161
[2]	M. J. Crowder, Classical competing risks, New York: Chapman and Hall/CRC, 2001. https://doi.org/10.1201/9781420035902
[3]	J. R. King, Probability charts for decision making, Industrial Press, 1971.
[4]	D. Kundu, N. Kannan, N. Balakrishnan, Analysis of progressively censored competing risks data, Handbook of statistics, 23 (2004), 331–348. https://doi.org/10.1016/S0169-7161(03)23018-2 doi: 10.1016/S0169-7161(03)23018-2
[5]	M. Pintilie, Analysing and interpreting competing risk data, Stat. Med., 26 (2007), 1360–1367. https://doi.org/10.1002/sim.2655 doi: 10.1002/sim.2655
[6]	Z. Zhang, Survival analysis in the presence of competing risks, Ann. Transl. Med., 5 (2017), 47. https://doi.org/10.21037/atm.2016.08.62 doi: 10.21037/atm.2016.08.62
[7]	S. O. A. El-Azeem, M. H. Abu-Moussa, M. M. M. El-Din, L. S. Diab, On step-stress partially accelerated life testing with competing risks under progressive type-Ⅱ censoring, Ann. Data Sci., 11 (2024), 909–930. https://doi.org/10.1007/s40745-022-00454-0 doi: 10.1007/s40745-022-00454-0
[8]	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
[9]	N. Balakrishnan, R. A. Sandhu, A simple simulational 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
[10]	N. Balakrishnan, E. Cramer, U. Kamps, N. Schenk, Progressive type Ⅱ censored order statistics from exponential distributions, Statistics, 35 (2001), 537–556. https://doi.org/10.1080/02331880108802753 doi: 10.1080/02331880108802753
[11]	M. K. Rastogi, Y. M. Tripathi, Estimating the parameters of a Burr distribution under progressive type Ⅱ censoring, Stat. Methodol., 9 (2012), 381–391. https://doi.org/10.1016/j.stamet.2011.10.002 doi: 10.1016/j.stamet.2011.10.002
[12]	S. Dey, L. Wang, M. Nassar, Inference on Nadarajah-Haghighi distribution with constant stress partially accelerated life tests under progressive type-Ⅱ censoring, J. Appl. Stat., 49 (2022), 2891–2912. https://doi.org/10.1080/02664763.2021.1928014 doi: 10.1080/02664763.2021.1928014
[13]	H. Krishna, R. Goel, Inferences for two Lindley populations based on joint progressive type-Ⅱ censored data, Commun. Stat. Simul. C., 51 (2022), 4919–4936. https://doi.org/10.1080/03610918.2020.1751851 doi: 10.1080/03610918.2020.1751851
[14]	Y. E. Jeon, S. B. Kang, J. I. Seo, Novel estimation based on a minimum distance under the progressive Type-Ⅱ censoring scheme, Commun. Stat. Appl. Met., 30 (2023), 411–421. https://doi.org/10.29220/CSAM.2023.30.4.411 doi: 10.29220/CSAM.2023.30.4.411
[15]	R. Kumari, Y. M. Tripathi, L. Wang, R. K. Sinha, Reliability estimation for Kumaraswamy distribution under block progressive type-Ⅱ censoring, Statistics, 58 (2024), 142–175. https://doi.org/10.1080/02331888.2024.2301736 doi: 10.1080/02331888.2024.2301736
[16]	H. K. T. Ng, D. Kundu, P. S. Chan, Type-Ⅱ progress Statistical analysis of exponential lifetimes under an adaptive censoring scheme, Nav. Res. Log., 56 (2009), 687–698. https://doi.org/10.1002/nav.20371 doi: 10.1002/nav.20371
[17]	H. Panahi, S. Asadi, On adaptive progressive hybrid censored Burr type Ⅲ distribution: Application to the nano droplet dispersion data, Qual. Technol. Quant. M., 18 (2021), 179–201. https://doi.org/10.1080/16843703.2020.1806431 doi: 10.1080/16843703.2020.1806431
[18]	H. Haj Ahmad, M. M. Salah, M. S. Eliwa, Z. Ali Alhussain, E. M. Almetwally, E. A. Ahmed, Bayesian and non-Bayesian inference under adaptive type-Ⅱ progressive censored sample with exponentiated power Lindley distribution, J. Appl. Stat., 49 (2022), 2981–3001. https://doi.org/10.1080/02664763.2021.1931819 doi: 10.1080/02664763.2021.1931819
[19]	A. Elshahhat, M. Nassar, Analysis of adaptive Type-Ⅱ progressively hybrid censoring with binomial removals, J. Stat. Comput. Sim., 93 (2023), 1077–1103. https://doi.org/10.1080/00949655.2022.2127149 doi: 10.1080/00949655.2022.2127149
[20]	W. Yan, P. Li, Y. Yu, Statistical inference for the reliability of Burr-Ⅻ distribution under improved adaptive Type-Ⅱ progressive censoring, Appl. Math. Model., 95 (2021), 38–52. https://doi.org/10.1016/j.apm.2021.01.050 doi: 10.1016/j.apm.2021.01.050
[21]	I. Elbatal, M. Nassar, A. Ben Ghorbal, L. S. G. Diab, A. Elshahhat, Reliability analysis and its applications for a newly improved type-Ⅱ adaptive progressive alpha power exponential censored sample, Symmetry, 15 (2023), 2137. https://doi.org/10.3390/sym15122137 doi: 10.3390/sym15122137
[22]	S. Dutta, S. Kayal, Inference of a competing risks model with partially observed failure causes under improved adaptive type-Ⅱ progressive censoring, P. I. Mech. Eng. O-J. Ris., 237 (2023), 765–780. https://doi.org/10.1177/1748006X221104555 doi: 10.1177/1748006X221104555
[23]	M. Irfan, S. Dutta, A. K. Sharma, Statistical inference and optimal plans for improved adaptive type-Ⅱ progressive censored data following Kumaraswamy-G family of distributions, Phys. Scr., 100 (2025), 025213. https://doi.org/10.1088/1402-4896/ada216 doi: 10.1088/1402-4896/ada216
[24]	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
[25]	W. I. Burr, Cumulative frequency distribution, Ann. Math. Statist., 13 (1942), 215–232. https://doi.org/10.1214/aoms/1177731607 doi: 10.1214/aoms/1177731607
[26]	B. Tarvirdizade, H. K. G. H. K. Gharehchobogh, Inference on $Pr(X> Y)$ based on record values from the Burr type Ⅹ distribution, Hacet. J. Math. Stat., 45 (2016), 267–278.
[27]	A. Rabie, J. Li, E-Bayesian estimation for Burr-X distribution based on generalized type-Ⅰ hybrid censoring scheme, Am. J. Math. Manag. Sci., 39 (2020), 41–55. https://doi.org/10.1080/01966324.2019.1579123 doi: 10.1080/01966324.2019.1579123
[28]	M. Kayid, V. B. Nagarjuna, M. Elgarhy, Statistical inference for heavy‐tailed Burr X distribution with applications, J. Math., 2024 (2024), 9552629. https://doi.org/10.1155/2024/9552629 doi: 10.1155/2024/9552629
[29]	Y. Lio, D. G. Chen, T. R. Tsai, L. Wang, The reliability inference for multicomponent stress–strength model under the Burr X distribution, AppliedMath, 4 (2024), 394–426. https://doi.org/10.3390/appliedmath4010021 doi: 10.3390/appliedmath4010021
[30]	S. Dutta, S. Kayal, Analysis of the improved adaptive type-Ⅱ progressive censoring based on competing risk data, 2021.
[31]	L. Wang, H. Li, Inference for exponential competing risks data under generalized progressive hybrid censoring, Commun. Stat. Simul. C., 51 (2022), 1255–1271. https://doi.org/10.1080/03610918.2019.1667388 doi: 10.1080/03610918.2019.1667388
[32]	Y. Du, W. Gui, Statistical inference of Burr-XII distribution under adaptive type ⅱ progressive censored schemes with competing risks, Results Math., 77 (2022), 81. https://doi.org/10.1007/s00025-022-01617-4 doi: 10.1007/s00025-022-01617-4
[33]	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
[34]	M. Plummer, N. Best, K. Cowles, K. Vines, CODA: convergence diagnosis and output analysis for MCMC, R News, 6 (2006), 7–11.
[35]	D. Kundu, N. Kannan, N. Balakrishnan, Analysis of progressively censored competing risks data, Handbook of statistics, 23 (2003), 331–348. https://doi.org/10.1016/S0169-7161(03)23018-2 doi: 10.1016/S0169-7161(03)23018-2
[36]	R. Alotaibi, M. Nassar, Z. A. Khan, W. A. Alajlan, A. Elshahhat, Analysis and data modelling of electrical appliances and radiation dose from an adaptive progressive censored XGamma competing risk model, J. Radiat. Res. Appl. Sci., 18 (2025), 101188. https://doi.org/10.1016/j.jrras.2024.101188 doi: 10.1016/j.jrras.2024.101188
[37]	O. A. Alqasem, M. Nassar, M. E. Abd Elwahab, A. Elshahhat, Analyzing Burr-X competing risk model using adaptive progressive Type-Ⅱ censored binomial removal data with application to electrodes and electronics, J. Radiat. Res. Appl. Sci., 17 (2024), 101107. https://doi.org/10.1016/j.jrras.2024.101107 doi: 10.1016/j.jrras.2024.101107

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)