Goodness-of-fit tests for imperfect maintenance models based on Martingale residuals, varentropy, and probability integral transform

Document Type : Research Paper

Authors

Department of Statistics, Faculty of Mathematical Sciences and Statistics, University of Birjand, Birjand, Iran

Abstract

In recent years, various goodness-of-fit tests have been developed to identify the underlying distribution of failure data. In this paper, we extend the application of such tests to evaluate the adequacy of imperfect maintenance models for engineering systems. Specifically, we investigate and compare three types of test statistics: those based on martingale residuals, the probability integral transform, and varentropy—a concept derived from information theory. The null hypothesis assumes that the failure times follow the $ARA_{\infty}$ model with a power law process (PLP) as the initial hazard rate. To evaluate the performance of the proposed tests, we conduct extensive simulation studies under different alternative maintenance models (e.g., $ARA_1$, $ARA_{\infty}$–Log Linear Process(LLP)) and varying parameter settings. Our findings show that the power of the tests varies depending on the nature of the alternatives, and varentropy-based statistics outperform others under certain conditions. Finally, we apply the proposed methods to a real dataset (Ambassador vehicle failure times) to assess their practical relevance. The results confirm the validity of the fitted model and demonstrate the usefulness of varentropy-based approaches for detecting subtle deviations in maintenance patterns.

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References

[1] Agustin, M. Z. N., & Peña, E. A. (1999). Order statistic properties, random generation, and goodness-of-fit testing for a minimal repair model. Journal of the American Statistical Association, 94(445), 266–272. DOI:
https://www.tandfonline.com/doi/abs/10.1080/01621459.1999.10473842
[2] Ahn, C. W., Chae, K. C., & Clark, G. M. (1998). Estimating parameters of the powerlaw process with two measures of time. Journal of Quality Technology, 30(2), 127–132.https://doi.org/10.1080/00224065.1998.11979831
[3] Alizadeh, H., & Shafaei, M. (2024). Varentropy estimators applied to goodness of fit tests for the Gumbel distribution. São Paulo Journal of Mathematical Sciences, 2, 1944–1962. https://link.springer.com/article/10.1007/s40863-024-00456-1
[4] Alizadeh Noughabi, H., & Shafaei Noughabi, M. (2023). Varentropy estimators with applications in testing uniformity. Journal of Statistical Computation and Simulation, 93(15), 2582–2599. DOI: https://doi.org/10.1080/00949655.2023.2196627
[5] Andersen, P. K., Borgan, O., Gill, R. D., & Keiding, N. (1993). Statistical models based on counting processes. New York, NY, USA: Springer-Verlag.
[6] Bordes, L., & Mercier, S. (2013). Extended geometric processes: Semiparametric estimation and application to reliability. Journal of the Iranian Statistical Society, 12(1), 1–34.DOI: https://hal.science/hal-00867031/
[7] Brown, M., & Proschan, F. (1983). Imperfect repair. Journal of Applied Probability, 20(4), 851–859. DOI: https://doi.org/10.2307/3213596
[8] Chauvel, C., Dauxois, J. Y., Doyen, L., & Gaudoin, O. (2016). Parametric bootstrap goodness-of-fit tests for imperfect maintenance models. IEEE Transactions on Reliability, 65(3), 1343–1359. DOI: https://doi.org/10.1109/TR.2016.2578938
[9] Cook, R. J., & Lawless, J. F. (2007). The statistical analysis of recurrent events.DOI: https://link.springer.com/book/10.1007/978-0-387-69810-6
[10] Corset, F., Doyen, L., & Gaudoin, O. (2012). Bayesian analysis of ARA imperfect repair models. Communications in Statistics - Theory and Methods, 41(21), 3915–3941. DOI: https://doi.org/10.1080/03610926.2012.698688
[11] Crétois, E., Gaudoin, O., & El Aroui, M. A. (1999). U-plot method for testing the goodness-of-fit of the power-law process. Communications in Statistics - Theory and Methods, 28(7), 1731–1747. DOI: https://doi.org/10.1080/03610929908832382
[12] Crow, L. H. (1974). Reliability analysis for complex repairable systems. Reliability and Biometry, 13(6), 379–410. DOI: 10.1007/978-94-011-3482-8-4
[13] D’Agostino, R., & Stephens, M. (1986). Goodness-of-Fit Techniques. Boca Raton, FL, USA: CRC Press.
[14] Davison, A. C., & Hinkley, D. V. (1997). Bootstrap Methods and Their Application. Cambridge, UK: Cambridge University Press.
[15] Doyen, L., & Gaudoin, O. (2004). Classes of imperfect repair models based on reduction of failure intensity or virtual age. Reliability Engineering & System Safety, 84(1), 45–56. DOI: https://doi.org/10.1016/S0951-8320(03)00173-X
[16] Ebrahimi, N., Pflughoeft, K., & Soofi, E. S. (1994). Two measures of sample entropy. Statistics & Probability Letters, 20(3), 225–234. DOI: https://doi.org/10.1016/0167-7152(94)90046-9
[17] Efron, B. (1979). Bootstrap methods: Another look at the jackknife. The Annals of Statistics, 7(1), 1–26. DOI: https://link.springer.com/chapter/10.1007/978-1-4612-4380-9-41 
[18] Gaudoin, O. (1998). CPIT goodness-of-fit tests for the power-law process. Communications in Statistics - Theory and Methods, 27(1), 165–180. DOI: https://doi.org/10.1080/03610929808832658
[19] Guilda, M., & Pulcini, G. (2006). Bayesian analysis of repairable systems showing a bounded failure intensity. Reliability Engineering & System Safety, 91, 828–838. DOI: 10.1016/j.ress.2005.08.008
[20] Leonenko, N., Sun, Y., & Taufer, E. (2024). Varentropy Estimation via Nearest Neighbor Graphs. arXiv preprint, arXiv:2402.09374. Available at: https://arxiv.org/abs/2402.09374
[21] Lindqvist, B. H., Elvebakk, G., & Heggland, K. (2003). The trend-renewal process for statistical analysis of repairable systems. Technometrics, 45(1), 31–44. DOI: https://doi.org/10.1198/004017002188618671
[22] Lindqvist, B. H., & Rannestad, B. (2011). Monte Carlo exact goodness-of-fit tests for nonhomogeneous Poisson processes. Applied Stochastic Models in Business and Industry, 27(3), 329–341. DOI: https://doi.org/10.1002/asmb.841
[23] Liu, Y., Huang, H. Z., & Zhang, X. (2011). A data-driven approach to selectig imperfect maintenance models. IEEE Transactions on Reliability, 61(1), 101–112. DOI: https://doi.org/10.1109/TR.2011.2170252
[24] Park, W. J., & Kim, Y. G. (1992). Goodness-of-fit tests for the power-law process. IEEE Transactions on Reliability, 41(1), 107–111. DOI: https://doi.org/10.1109/24.121461
[25] Rosenblatt, M. (1952). Remarks on a multivariate transformation. The Annals of Mathematical Statistics, 23(3), 470-472. DOI: https://www.jstor.org/stable/2236692
[26] Saha, S., & Kayal, S. (2023). Weighted (residual) varentropy and its applications. Journal of Computational and Applied Mathematics, 442, 115710. DOI: https://doi.org/10.1016/j.cam.2023.115710
[27] Stute, W., Manteiga, W. G., & Quindimil, M. P. (1993). Bootstrap-based goodness-of-fit tests. Metrika, 40(1), 243–256. DOI: https://link.springer.com/article/10.1007/BF02613687
[28] Tibshirani, R. J., & Efron, B. (1993). An introduction to the bootstrap. Monographs on Statistics and Applied Probability, 57(1), 1–436. DOI: 10.1007/978-1-4899-4541-9
[29] Vere-Jones, D., & Daley, D. J. (2008). An Introduction to the Theory of Point Processes. New York, NY, USA: Springer. DOI: 10.1007/978-0-387-49835-5
[30] Wang, H., & Pham, H. (1996). A quasi-renewal process and its applications in imperfect maintenance. International Journal of Systems Science, 27(10), 1055–1062. DOI: https://doi.org/10.1080/00207729608929311
[31] Zhao, J., & Wang, J. (2005). A new goodness-of-fit test based on the Laplace statistic for a large class of NHPP models. Communications in Statistics - Simulation and Computation, 34(3), 725–736. DOI: https://doi.org/10.1081/SAC-200068389 

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  • Receive Date: 14 December 2024
  • Revise Date: 05 May 2025
  • Accept Date: 01 July 2025

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