Estimations for the Kumaraswamy distribution with a partially observed competing-risks model under progressive censoring data

Manal H. Alloqmani; Faten S. Alamri; Gamal M. Ismail; Samah M. Ahmed; Raga Hassan Ali Shiekh; Al-Wageh A. Farghal; Manal H. Alloqmani; Faten S. Alamri; Gamal M. Ismail; Samah M. Ahmed; Raga Hassan Ali Shiekh; Al-Wageh A. Farghal

doi:10.3932/math.2026886

AIMS Mathematics

2026, Volume 11, Issue 7: 21928-21960. doi: 10.3932/math.2026886

Previous Article Next Article

Research article Special Issues

Estimations for the Kumaraswamy distribution with a partially observed competing-risks model under progressive censoring data

1.
Department of Mathematics, Faculty of Sciences and Arts, King Abdulaziz University, Rabigh 21911, Saudi Arabia
2.
Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia
3.
Department of Mathematics, Faculty of Science, Islamic University of Madinah, Madinah 42351, Saudi Arabia
4.
Mathematics Department, Faculty of Science, Sohag University, Sohag 82524, Egypt
5.
Department of Statistics, Faculty of Science, University of Tabuk, Tabuk 71491, Saudi Arabia

Received: 24 March 2026 Revised: 20 June 2026 Accepted: 22 June 2026 Published: 23 July 2026
MSC : 62N01, 62F10, 62F15

This research examined the application of a two-competing-risks framework using the Kumaraswamy distribution within the context of progressive Type-Ⅱ censoring. In this study, we assumed that the model follows the Kumaraswamy distribution and that the failure modes are independent and partially observed. The Type-Ⅱ progressive censoring scheme, a cornerstone of modern reliability studies, was employed here as the primary data collection framework, particularly when evaluating highly reliable products where testing time and cost are significant factors. Maximum likelihood estimators (MLEs) were established. Also, the corresponding confidence intervals for the unknown parameters were computed, and the existence and uniqueness of these estimates were rigorously proven. Furthermore, approximate confidence intervals are constructed using the observed Fisher information matrix and the percentile bootstrap interval. In the Bayesian framework, Bayes estimates and their associated credible intervals were obtained utilizing Markov chain Monte Carlo (MCMC) sampling methods. Finally, a real-world application and simulation studies were presented to illustrate the practical utility of the proposed methodology.
Citation: Manal H. Alloqmani, Faten S. Alamri, Gamal M. Ismail, Samah M. Ahmed, Raga Hassan Ali Shiekh, Al-Wageh A. Farghal. Estimations for the Kumaraswamy distribution with a partially observed competing-risks model under progressive censoring data[J]. AIMS Mathematics, 2026, 11(7): 21928-21960. doi: 10.3932/math.2026886

Related Papers:

Abstract

This research examined the application of a two-competing-risks framework using the Kumaraswamy distribution within the context of progressive Type-Ⅱ censoring. In this study, we assumed that the model follows the Kumaraswamy distribution and that the failure modes are independent and partially observed. The Type-Ⅱ progressive censoring scheme, a cornerstone of modern reliability studies, was employed here as the primary data collection framework, particularly when evaluating highly reliable products where testing time and cost are significant factors. Maximum likelihood estimators (MLEs) were established. Also, the corresponding confidence intervals for the unknown parameters were computed, and the existence and uniqueness of these estimates were rigorously proven. Furthermore, approximate confidence intervals are constructed using the observed Fisher information matrix and the percentile bootstrap interval. In the Bayesian framework, Bayes estimates and their associated credible intervals were obtained utilizing Markov chain Monte Carlo (MCMC) sampling methods. Finally, a real-world application and simulation studies were presented to illustrate the practical utility of the proposed methodology.

References

[1]	N. Balakrishnan, R. Aggarwala, Progressive censoring: theory, methods, and applications, Boston, MA: Birkhäuser, 2000.
[2]	H. K. T. Ng, P. S. Chan, N. Balakrishnan, Optimal progressive censoring plans for the Weibull distribution, Technometrics, 46 (2004), 470–481. https://doi.org/10.1198/004017004000000482 doi: 10.1198/004017004000000482
[3]	A. Fayomi, E. M. Almetwally, M. E. Qura, Bivariate length-biased exponential distribution under progressive type-Ⅱ censoring: incorporating random removal and applications to industrial and computer science data, Axioms, 13 (2024), 664. https://doi.org/10.3390/axioms13100664 doi: 10.3390/axioms13100664
[4]	X. Gao, W. Gui, Statistical inference of Burr-Hatke exponential distribution with partially accelerated life test under progressively type-Ⅱ censoring, Mathematics, 11 (2023), 2939. https://doi.org/10.3390/math11132939 doi: 10.3390/math11132939
[5]	K. Lee, Inference for parameters of exponential distribution under combined type Ⅱ progressive hybrid censoring scheme, Mathematics, 12 (2024), 820. https://doi.org/10.3390/math12060820 doi: 10.3390/math12060820
[6]	S. Mondal, D. Kundu, Inference on Weibull parameters under a balanced two-sample type Ⅱ progressive censoring scheme, Qual. Reliab. Eng. Int., 36 (2020), 1–17. https://doi.org/10.1002/qre.2553 doi: 10.1002/qre.2553
[7]	S. Sel, M. Jung, Y. Chung, Bayesian and maximum likelihood estimations from parameters of McDonald extended Weibull model based on progressive type-Ⅱ censoring, J. Stat. Theory Pract., 12 (2018), 231–254. https://doi.org/10.1080/15598608.2017.1343693 doi: 10.1080/15598608.2017.1343693
[8]	R. Goel, H. Krishna, Progressive type-Ⅱ random censoring scheme with Lindley failure and censoring time distributions, Int. J. Agric. Stat. Sci., 16 (2020), 23–34.
[9]	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
[10]	R. Goel, K. Kumar, H. K. T. Ng, I. Kumar, Statistical inference in Burr type Ⅻ lifetime model based on progressive randomly censored data, Qual. Eng., 36 (2024), 150–165. https://doi.org/10.1080/08982112.2023.2276771 doi: 10.1080/08982112.2023.2276771
[11]	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
[12]	D. R. Cox, The analysis of exponentially distributed life-times with two types of failure, J. R. Stat. Soc. Ser. B, 21 (1959), 411–421. https://doi.org/10.1111/J.2517-6161.1959.TB00349.X doi: 10.1111/J.2517-6161.1959.TB00349.X
[13]	M. J. Crowder, Classical competing risks, Boca Raton, FL: Chapman & Hall/CRC, 2001.
[14]	N. Balakrishnan, D. Han, Exact inference for a simple step-stress model with competing risks for failure from exponential distribution under Type-Ⅱ censoring, J. Stat. Plann. Inference, 138 (2008), 4172–4186. https://doi.org/10.1016/j.jspi.2008.03.036 doi: 10.1016/j.jspi.2008.03.036
[15]	S. H. Feizjavadian, R. Hashemi, Analysis of dependent competing risks in the presence of progressive hybrid censoring using Marshall-Olkin bivariate Weibull distribution, Comput. Stat. Data Anal., 82 (2015), 19–34. https://doi.org/10.1016/j.csda.2014.08.002 doi: 10.1016/j.csda.2014.08.002
[16]	S. M. Ahmed, M. Abdalla, A. A. Farghal, Statistical inference for competing risks model with Type-Ⅱ generalized hybrid censored of inverse Lomax distribution with applications, Alex. Eng. J., 118 (2025), 132–146. https://doi.org/10.1016/j.aej.2025.01.004 doi: 10.1016/j.aej.2025.01.004
[17]	S. Dutta, S. Kayal, Bayesian and non-Bayesian inference of Weibull lifetime model based on partially observed competing risks data under unified hybrid censoring scheme, Qual. Reliab. Eng. Int., 38 (2022), 3867–3891. https://doi.org/10.1002/qre.3180 doi: 10.1002/qre.3180
[18]	S. Dutta, S. Kayal, Inference of a competing risks model with partially observed failure causes under improved adaptive type-Ⅱ progressive censoring, Proc. Inst. Mech. Eng. Part O, 237 (2023), 765–780. https://doi.org/10.1177/1748006X221104555 doi: 10.1177/1748006X221104555
[19]	M. Nassar, R. Alotaibi, S. Dey, Estimation based on adaptive progressively censored under competing risks model with engineering applications, Math. Probl. Eng., 2022 (2022), 6731230. https://doi.org/10.1155/2022/6731230 doi: 10.1155/2022/6731230
[20]	A. A. Farghal, S. K. Badr, H. Zinadah, G. A. Abd-Elmougod, Analysis of generalized inverted exponential competing risks model in presence of partially observed failure modes, Alex. Eng. J., 78 (2023), 74–87. https://doi.org/10.1016/j.aej.2023.07.021 doi: 10.1016/j.aej.2023.07.021
[21]	P. Kumaraswamy, A generalized probability density function for double-bounded random processes, J. Hydrol., 46 (1980), 79–88. https://doi.org/10.1016/0022-1694(80)90036-0 doi: 10.1016/0022-1694(80)90036-0
[22]	S. Nadarajah, On the distribution of Kumaraswamy, J. Hydrol., 348 (2008), 568–569. https://doi.org/10.1016/j.jhydrol.2007.09.008 doi: 10.1016/j.jhydrol.2007.09.008
[23]	N. A. Abou-Elheggag, A. W. A. Farghal, H. A. Ramadan, Estimation of R = P(Y $ < $ X) using k-upper record values from kumaraswamy distribution, J. Stat. Appl. Probab., 7 (2018), 263–273.
[24]	M. C. Jones, Kumaraswamy's distribution: a beta-type distribution with some tractability advantages, Stat. Methodol., 6 (2009), 70–81. https://doi.org/10.1016/j.stamet.2008.04.001 doi: 10.1016/j.stamet.2008.04.001
[25]	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
[26]	S. Dey, T. Dey, On progressively censored generalized inverted exponential distribution, J. Appl. Stat., 41 (2014), 2557–2576. https://doi.org/10.1080/02664763.2014.922165 doi: 10.1080/02664763.2014.922165
[27]	B. Efron, R. J. Tibshirani, An introduction to the bootstrap, New York: Chapman & Hall, 1994. https://doi.org/10.1201/9780429246593
[28]	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
[29]	A. Fathi, A. W. A. Farghal, A. A. Soliman, Inference on Weibull inverted exponential distribution under progressive first-failure censoring with constant-stress partially accelerated life test, Stat. Pap., 65 (2024), 5021–5053. https://doi.org/10.1007/s00362-024-01583-9 doi: 10.1007/s00362-024-01583-9
[30]	A. Mustafa, M. I. Khan, S. M. Ahmed, Estimating the stress-strength reliability parameter of the inverse power Lomax distribution, AIMS Math., 10 (2025), 15632–15652. https://doi.org/10.3934/math.2025700 doi: 10.3934/math.2025700
[31]	S. M. Ahmed, M. I. Khan, A. Mustafa, Bayesian estimation and prediction of inverse power Lomax model under censored data with applications, J. Math., 2025 (2025), 7285331. https://doi.org/10.1155/jom/7285331 doi: 10.1155/jom/7285331
[32]	M. H. Chen, Q. M. Shao, Monte Carlo estimation of Bayesian credible and HPD intervals, J. Comput. Graph. Stat., 8 (1999), 69–92. https://doi.org/10.1080/10618600.1999.10474802 doi: 10.1080/10618600.1999.10474802
[33]	A. A. Soliman, E. A. Ahmed, N. A. Abou-Elheggag, S. M. Ahmed, Step-stress partially accelerated life tests model in estimation of inverse Weibull parameters under progressive type-Ⅱ censoring, Appl. Math. Inf. Sci., 11 (2017), 1369–1381. http://dx.doi.org/10.18576/amis/110514 doi: 10.18576/amis/110514
[34]	D. G. Hoel, A representation of mortality data by competing risks, Biometrics, 28 (1972), 475–488.
[35]	S. G. Meintanis, Test of fit for Marshall-Olkin distributions with applications, J. Statist. Plann. Inference, 137 (2007), 3954–3963. https://doi.org/10.1016/j.jspi.2007.04.013 doi: 10.1016/j.jspi.2007.04.013
[36]	L. Wang, Inference of progressively censored competing risks data from Kumaraswamy distributions, J. Comput. Appl. Math., 343 (2018), 719–736. https://doi.org/10.1016/j.cam.2018.05.013 doi: 10.1016/j.cam.2018.05.013

Reader Comments

Your name:*

Email:*
© 2026 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)