Classical and Bayesian inference for progressively censored competing risks data under the Gompertz-Lindley model

Mahmoud H. Abu-Moussa; Ehab M. Almetwally; Abd El-Raheem M. Abd El-Raheem; Mahmoud H. Abu-Moussa; Ehab M. Almetwally; Abd El-Raheem M. Abd El-Raheem

doi:10.3934/math.2026495

AIMS Mathematics

2026, Volume 11, Issue 4: 12064-12093. doi: 10.3934/math.2026495

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Classical and Bayesian inference for progressively censored competing risks data under the Gompertz-Lindley model

1.
Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt
2.
Faculty of Education and Arts, Sohar University, Sohar, Oman
3.
Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11432, Saudi Arabia
4.
Department of Mathematics, Faculty of Education, Ain Shams University, Cairo, Egypt

Received: 05 March 2026 Revised: 07 April 2026 Accepted: 10 April 2026 Published: 29 April 2026
MSC : 62F15, 62N05

This paper develops classical and Bayesian inferential procedures for progressively Type-Ⅱ censored competing-risks data when the latent failure times follow the Gompertz-Lindley distribution. Maximum likelihood estimators are derived for the model parameters, and asymptotic confidence intervals are constructed using the observed information matrix. Bayesian estimation is carried out under squared error, LINEX, and generalized entropy loss functions using both the Tierney–Kadane approximation and Markov chain Monte Carlo methods. An extensive Monte Carlo simulation study is conducted to assess the finite-sample behavior of the proposed estimators under different sample sizes and progressive censoring schemes. The numerical results show that Bayesian procedures generally outperform the corresponding maximum likelihood estimators, particularly in small and moderately censored samples. A real-data application involving heart-disease patients demonstrates that the Gompertz-Lindley model provides a satisfactory fit and serves as a flexible alternative for competing-risks lifetime data.
- progressive Type-Ⅱ censoring,
- competing risks modeling,
- Gompertz-Lindley lifetime model,
- Bayesian estimation techniques,
- statistical survival analysis
Citation: Mahmoud H. Abu-Moussa, Ehab M. Almetwally, Abd El-Raheem M. Abd El-Raheem. Classical and Bayesian inference for progressively censored competing risks data under the Gompertz-Lindley model[J]. AIMS Mathematics, 2026, 11(4): 12064-12093. doi: 10.3934/math.2026495

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Abstract

This paper develops classical and Bayesian inferential procedures for progressively Type-Ⅱ censored competing-risks data when the latent failure times follow the Gompertz-Lindley distribution. Maximum likelihood estimators are derived for the model parameters, and asymptotic confidence intervals are constructed using the observed information matrix. Bayesian estimation is carried out under squared error, LINEX, and generalized entropy loss functions using both the Tierney–Kadane approximation and Markov chain Monte Carlo methods. An extensive Monte Carlo simulation study is conducted to assess the finite-sample behavior of the proposed estimators under different sample sizes and progressive censoring schemes. The numerical results show that Bayesian procedures generally outperform the corresponding maximum likelihood estimators, particularly in small and moderately censored samples. A real-data application involving heart-disease patients demonstrates that the Gompertz-Lindley model provides a satisfactory fit and serves as a flexible alternative for competing-risks lifetime data.

References

[1]	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
[2]	S. A. Salem, O. E. Abo-Kasem, A. A. Khairy, Inference for generalized progressive hybrid type-ii censored weibull lifetimes under competing risk data, Comput. J. Math. Statist. Sci., 3 (2024), 177–202. https://doi.org/10.21608/CJMSS.2024.256760.1035 doi: 10.21608/CJMSS.2024.256760.1035
[3]	R. Al-Jarallah, M. Ghitany, D. Kundu, Bayesian inference on gompertz-lindley distribution based on different loss functions, Int. J. Appl. Math., 37 (2024), 187–204. http://dx.doi.org/10.12732/ijam.v37i2.5 doi: 10.12732/ijam.v37i2.5
[4]	F. Almathkour, A. Alothman, M. Ghitany, R. C. Gupta, J. Mazucheli, A comparative study of various methods of estimation for gompertz-lindley distribution, Int. J. Appl. Math., 35 (2022), 347. https://doi.org/10.12732/ijam.v35i2.12 doi: 10.12732/ijam.v35i2.12
[5]	E. M. Almetwally, Uncertainty in lifetime data quantified with bayesian adaptive hamiltonian methods, Measurement, 276 (2026), 121455. https://doi.org/10.1016/j.measurement.2026.121455 doi: 10.1016/j.measurement.2026.121455
[6]	E. M. Almetwally, Statistical inference for a one-parameter lifetime model under unified progressive hybrid censoring with binomial random removals, AIMS Math., 11 (2026), 211–242. https://doi.org/10.3934/math.2026009 doi: 10.3934/math.2026009
[7]	E. M. Almetwally, A. H. Tolba, D. A. Ramadan, N. Sayed-Ahmed, Comparative analysis of three distributions under studying the regression competing risks model with hiv to aids infection application, Appl. Math. Inf. Sci., 19 (2025), 921–944. https://doi.org/10.18576/amis/190417 doi: 10.18576/amis/190417
[8]	N. Balakrishnan, R. Aggarwala, Progressive censoring: theory, methods, and applications, Springer Science & Business Media, 2000.
[9]	N. Balakrishnan, E. Cramer, Art of progressive censoring, New York: Springer Science & Business Media, 2014. https://doi.org/10.1007/978-0-8176-4807-7
[10]	D. R. Cox, The analysis of exponentially distributed life-times with two types of failure, J. Roy. Stat. Soc. 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
[11]	M. J. Crowder, Classical competing risks, New York: Chapman and Hall/CRC, 2001. https://doi.org/10.1201/9781420035902
[12]	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
[13]	R. B. Geskus, Data analysis with competing risks and intermediate states, New York: Chapman and Hall/CRC, 2015. https://doi.org/10.1201/b18695
[14]	M. E. Ghitany, S. M. Aboukhamseen, A. A. Baqer, R. C. Gupta, Gompertz-lindley distribution and associated inference, Commun. Stat. Simul. C., 51 (2022), 2599–2618. https://doi.org/10.1080/03610918.2019.1699113 doi: 10.1080/03610918.2019.1699113
[15]	H. Haj Ahmad, E. M. Almetwally, D. A. Ramadan, Competing risks in accelerated life testing: A study on step-stress models with tampered random variables, Axioms, 14 (2025), 32. https://doi.org/10.3390/axioms14010032 doi: 10.3390/axioms14010032
[16]	A. Hassan, S. Maiti, R. Mousa, N. Alsadat, M. Abu-Moussa, Analysis of competing risks model using the generalized progressive hybrid censored data from the generalized lomax distribution, AIMS Math., 9 (2024), 33756–33799. https://doi.org/10.3934/math.20241611 doi: 10.3934/math.20241611
[17]	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), 2479–2492. https://doi.org/10.1134/S1995080222120149 doi: 10.1134/S1995080222120149
[18]	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
[19]	J. D. Kalbfleisch, R. L. Prentice, The statistical analysis of failure time data, John Wiley & Sons, 2002. https://doi.org/10.1002/9781118032985
[20]	A. Koley, D. Kundu, On generalized progressive hybrid censoring in presence of competing risks, Metrika, 80 (2017), 401–426. https://doi.org/10.1007/s00184-017-0610-9 doi: 10.1007/s00184-017-0610-9
[21]	A. Koley, D. Kundu, Analysis of progressive type-Ⅱ censoring in presence of competing risk data under step stress modeling, Stat. Neerl., 75 (2021), 115–136. https://doi.org/10.1111/stan.12226 doi: 10.1111/stan.12226
[22]	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
[23]	A. K. Mahto, C. Lodhi, Y. M. Tripathi, L. Wang, Inference for partially observed competing risks model for kumaraswamy distribution under generalized progressive hybrid censoring, J. Appl. Stat., 49 (2022), 2064–2092. https://doi.org/10.1080/02664763.2021.1889999 doi: 10.1080/02664763.2021.1889999
[24]	S. G. Nassr, E. M. Almetwally, W. S. A. El Azm, Statistical inference for the extended weibull distribution based on adaptive type-ii progressive hybrid censored competing risks data, Thail. Statist., 19 (2021), 547–564.
[25]	M. Pintilie, Competing risks: a practical perspective, John Wiley & Sons, 2006. https://doi.org/10.1002/9780470870709
[26]	D. A. Ramadan, E. M. Almetwally, A. H. Tolba, Statistical inference to the parameter of the akshaya distribution under competing risks data with application hiv infection to aids, Ann. Data Sci., 10 (2023), 1499–1525. http://doi.org/10.1007/s40745-022-00382-z doi: 10.1007/s40745-022-00382-z
[27]	L. Tierney, J. B. Kadane, Accurate approximations for posterior moments and marginal densities, J. Am. Stat. Assoc., 81 (1986), 82–86. https://doi.org/10.1080/01621459.1986.10478240 doi: 10.1080/01621459.1986.10478240
[28]	S. Yang, D. Meng, H. Wang, C. Yang, A novel learning function for adaptive surrogate-model-based reliability evaluation, Philos. Trans. A Math. Phys. Eng. Sci., 382 (2024), 20220395. https://doi.org/10.1098/rsta.2022.0395 doi: 10.1098/rsta.2022.0395
[29]	S. Yang, D. Meng, M. Alfouneh, B. Keshtegar, S. P. Zhu, A robust-weighted hybrid nonlinear regression for reliability based topology optimization with multi-source uncertainties, Comput. Method. Appl. M., 447 (2025), 118360. https://doi.org/10.1016/j.cma.2025.118360 doi: 10.1016/j.cma.2025.118360
[30]	S. Yang, D. Meng, H. Yang, B. Keshtegar, A. M. De Jesus, S. P. Zhu, Adaptive kriging-assisted enhanced sparrow search with augmented-lagrangian first-order reliability method for highly efficient structural reliability analysis, Reliab. Eng. Syst. Safe., 267 (2025), 111916. https://doi.org/10.1016/j.ress.2025.111916 doi: 10.1016/j.ress.2025.111916

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