Reliability estimation in generalized inverted exponential distribution under block progressive type-II censoring scheme

Abstract

In this paper, the estimation problem for the parameters and reliability characteristics of the generalized inverted exponential distribution is discussed when the observed failuretime data are progressively type-II censored, observed across multiple testing facilities, and facility effects are taken into account. Under classical estimation, maximum likelihood estimation and pivotal inference are used to obtain point estimates. The asymptotic confidence intervals based on the observed Fisher’s information matrix and generalized confidence intervals based on the generalized pivotal quantity are obtained. An algorithm for obtaining the pivotal samples, generalized point estimates, and generalized confidence intervals of the parameters and reliability characteristics is provided. In the Bayesian setup, a hierarchical structure is used to incorporate prior information about the parameters and obtain the corresponding posterior distribution. Further, an algorithm based on Markov chain Monte Carlo techniques is provided to obtain posterior samples, Bayes estimates, and highest-posterior-density credible intervals for the parameters and reliability characteristics. Finally, a simulation study is conducted to assess the performance of the estimates, followed by a real data illustration.

Keywords

• Bayes estimation
• block progressive censoring
• generalized inverted exponential distribution
• maximum likelihood estimation
• pivotal inference

References

1. [1] A. M. Abouammoh and A. M. Alshingiti, Reliability estimation of generalized inverted exponential distribution, J. Stat. Comput. Simul. 79 (11), 1301–1315, 2009.
2. [2] M. V. Ahmadi, M. Doostparast and J. Ahmadi, Block censoring scheme with twoparameter exponential distribution, J. Stat. Comput. Simul. 88 (7), 1229–1251, 2018.
3. [3] A. M. Almarashi, Inferences of generalized inverted exponential distribution based on partially constant-stress accelerated life testing under progressive Type-II censoring, Alex. Eng. J 63, 223–232, 2023.
4. [4] N. Balakrishnan and R. Aggarwala, Progressive Censoring: Theory, Methods, and Applications, Springer Science & Business Media, New York, 2000.
5. [5] H. A. Bayoud, F. B. Almathkour and M. Z. Raqab, Hybrid Block Censoring in Bathtub-Shaped Lifetime Models and Their Optimal Censoring Strategies, Qual. Technol. Quant. Manag. 1–15, 2025.
6. [6] G. Casella and R. Berger, Statistical Inference , CRC Press, 2002.
7. [7] H. Choudhary, H. Krishna, K. Nagar and K. Kumar, Estimation in generalized uniform distribution with progressively type-II censored sample, Life Cycle Reliab. Saf. Eng. 13 (3), 309–323, 2024.
8. [8] S. Dey and T. Dey, Generalized Inverted Exponential Distribution: Different Methods of Estimation, Am. J. Math. Manag. Sci. 33 (3), 194–215, 2014.

9. [9] S. Dey and M. Nassar, Generalized inverted exponential distribution under constant stress accelerated life test: Different estimation methods with application, Qual. Reliab. Eng. Int. 36 (4), 1296–1312, 2020.
10. [10] A. S. Hassan, A. Al-Omari and H. F. Nagy, Stress–Strength Reliability for the Generalized Inverted Exponential Distribution Using MRSS, Iran. J. Sci. Technol. Trans. Sci. 45 (2), 641–659, 2021.
11. [11] S. Dey and B. Pradhan, Generalized inverted exponential distribution under hybrid censoring, Stat. Methodol. 18, 101–114, 2014.
12. [12] M. A. Hussian, Estimation of stress-strength model for generalized inverted exponential distribution using ranked set sampling, Int. J. Adv. Eng. Technol. 6 (6), 2354– 2362, 2014.
13. [13] S. Kotz and S. Nadarajah, Extreme Value Distributions: Theory and Applications, World Scientific, 2000.
14. [14] H. Krishna and K. Kumar, Reliability estimation in generalized inverted exponential distribution with progressively type-II censored sample, J. Stat. Comput. Simul. 83 (6), 1007–1019, 2013.
15. [15] R. Kumari, Y. M. Tripathi, C. Lodhi and L. Wang, Pivotal based reliability inference for Burr-XII distribution under block progressive type-II censoring, Commun. Stat. Simul. Comput. 1–22, 2025.
16. [16] R. Kumari, Y. M. Tripathi, R. K. Sinha and L. Wang, Reliability estimation for bathtub-shaped distribution under block progressive censoring, Math. Comput. Simul. 213, 237–260, 2023.
17. [17] R. Kumari, Y. M. Tripathi, L. Wang and R. K. Sinha, Reliability estimation for Kumaraswamy distribution under block progressive type-II censoring, Statistics 58 (1), 142–175, 2024.
18. [18] A. O. Langlands, S. J. Pocock, G. R. Kerr and S. M. Gore, Long-term survival of patients with breast cancer: A study of the curability of the disease, BMJ 2 (6200), 1247–1251, 1979.
19. [19] J. F. Lawless, Statistical Models and Methods for Lifetime Data, John Wiley & Sons, New Jersey, 2003.
20. [20] C. Lodhi, A. K. Gangopadhyay and P. Bhattacharya, Likelihood and pivotal-based reliability inference for inverse exponentiated Rayleigh distribution under block progressive censoring, Qual. Technol. Quant. Manag. 1–26, 2024.
21. [21] S. Mondal and D. Kundu, Inference on Weibull parameters under a balanced twosample type-II progressive censoring scheme, Qual. Reliab. Eng. Int. 36 (1), 1–17, 2020.
22. [22] S. Nadarajah and S. Kotz, The exponentiated Fréchet distribution, Interstat 14, 01–07, 2003.
23. [23] S. Saini, Estimation of multi-stress strength reliability under progressive first failure censoring using generalized inverted exponential distribution, J. Stat. Comput. Simul. 94 (14), 3177–3209, 2024.
24. [24] S. Saini, R. Garg and C. Swaroop, Inference on multicomponent stress-strength model using progressively censored data from exponentiated Pareto distribution, Qual. Reliab. Eng. Int. 1–29, 2025.
25. [25] S. Saini, S. Tomer and R. Garg, Inference of multicomponent stress-strength reliability following Topp-Leone distribution using progressively censored data, J. Appl. Stat. 50 (7), 1538–1567, 2023.
26. [26] D. P. Singh, M. K. Jha, Y. Tripathi and L. Wang, Reliability estimation in a multicomponent stress-strength model for unit Burr-III distribution under progressive censoring, Qual. Technol. Quant. Manag. 19 (5), 605–632, 2022.
27. [27] K. Singh, Y. M. Tripathi, C. Lodhi and L. Wang, Inference for Unit Inverse Weibull Distribution Under Block Progressive Type-II Censoring, J. Stat. Theory Pract. 18 (3), 42, 2024.
28. [28] U. Singh, S. K. Singh and R. K. Singh, A comparative study of traditional estimation methods and maximum product spacings method in generalized inverted exponential distribution, J. Stat. Appl. Probab. 3 (2), 153, 2014.
29. [29] L.Wang, S.-J.Wu, H. Lin and Y. M. Tripathi, Inference for block progressive censored competing risks data from an inverted exponentiated exponential model, Qual. Reliab. Eng. Int. 39 (7), 2736–2764, 2023.
30. [30] L. Wang, C. Zhang, S.-J. Wu, S. Dey and Y. Lio, Estimation for partially observed left truncation and right censored competing risks data from a generalized inverted exponential distribution with illustrations, Qual. Technol. Quant. Manag. 21 (4), 525– 555, 2024.
31. [31] S. Weerahandi, Generalized inference in repeated measures: Exact methods in MANOVA and mixed models, John Wiley & Sons, New Jersey, 2004.
32. [32] W. Yan, P. Li and Y. Yu, Statistical inference for the reliability of Burr-XII distribution under improved adaptive type-II progressive censoring, Appl. Math. Model. 95, 38–52, 2021.
33. [33] T. Zhu, Reliability estimation for two-parameter Weibull distribution under block censoring, Reliab. Eng. Syst. Saf. 203, 107071, 2020.
34. [34] H. Zhang, J. Wu and W. Gui, Inferences on the Generalized Inverted Exponential Distribution in Constant Stress Partially Accelerated Life Tests Using Generally Progressively Type-II Censored Samples, Appl. Sci. 14 (14), 6050, 2024.
35. [35] A. A. Rather and G. Ozel, The Weighted Power Lindley distribution with applications on the life time data, Pak. J. Stat. Oper. Res. 16 (2), 225–237, 2020.
36. [36] Q. Chen and W. Gui, Statistical Inference of the Generalized Inverted Exponential Distribution under Joint Progressively Type-II Censoring, Entropy 24 (5), 576, 2022.
37. [37] A. A. Rather, M. Azeem, M. Alam, C. Subramanian, G. Ozel and I. Ali, Weighted Erlang-Truncated Exponential Distribution: System Reliability Optimization, Structural Properties, and Simulation, Lobachevskii J. Math. 45 (9), 4311–4337, 2024.
38. [38] C. Swaroop, S. Dutta, S. Saini and N. Tiwari, Estimation of stress-strength reliability for the generalized inverted exponential distribution based on improved adaptive Type- II progressive censoring, Comput. Stat. 40 (7), 3781–3817, 2025.
39. [39] A. Kumari, S. Kumar and K. Kumar, Inference for Reliability in a Multicomponent Stress-Strength Model from Generalized Inverted Exponential Lifetime Distribution under Progressive First Failure Censoring, J. Stat. Comput. Simul. 93 (6), 863–887, 2023.