Generalization of reliability in the odd Lindley-G family

Document Type : Original Scientific Paper

Authors

1 Department of Statistics‎, ‎Higher Education Complex of Bam‎, ‎Kerman‎, ‎Iran

2 Department of Statistics‎, ‎Yazd University, Yazd, Iran

10.22034/jsmta.2026.24914.1229

Abstract

This study presents a generalized numerical framework for evaluating the reliability criterion in the odd Lindley-G family of distributions. Although the odd Lindley-G family provides a flexible mechanism for extending classical baseline models, the existing analytical formulation for reliability estimation suffers from two critical limitations: it is restricted to cases where the strength and stress variables share an identical baseline distribution, and its series expansion diverges even under homogeneous baseline conditions. To address these limitations, we develop a robust numerical integration framework that accurately computes reliability for arbitrary and distinct baseline distributions within the odd Lindley-G family. The proposed method is validated through an extensive simulation study and a practical engineering application involving strength and stress data with different baseline distributions. Our findings confirm that the numerical framework not only generalizes the applicability of the odd Lindley-G family but also delivers a stable, accurate, and computationally efficient tool for advanced reliability analysis.

Keywords

Main Subjects

References

Afify, A.Z., Cordeiro, G.M., Ibrahim, N.A., Jamal, F., Elgarhy, M. and Nasir, M.A. (2021). The Marshall-Olkin odd Burr III-G family: Theory, estimation, and engineering applications. IEEE Access, 9:4376–4387.
Alizadeh, M., Afify, A.Z., Eliwa, M.S. and Ali, S. (2020). The odd log-logistic Lindley-G family of distributions: Properties, Bayesian and non-Bayesian estimation with applications. Computational Statistics, 35(1):281–308.
Alizadeh, M., Emadi, M., Doostparast, M., Cordeiro, G.M., Ortega, E.M.M. and Pescim, R.R. (2015). A new family of distributions: The Kumaraswamy odd log-logistic, properties and applications. Hacettepe Journal of Mathematics and Statistics, 44(6):1491–1512.
Almetwally, E.M., Sabry, M.A.H., Alharbi, R., Alnagar, D., Mubarak, S.M. and Hafez, E.H. (2021). Marshall-Olkin alpha power Weibull distribution: Different methods of estimation based on Type-I and Type-II censoring. Complexity, 2021(1):5533799.
Alzaatreh, A., Lee, C. and Famoye, F. (2013). A new method for generating families of continuous distributions. Metron, 71(1):63–79.
Bourguignon, M., Silva, R.B. and Cordeiro, G.M. (2014). The Weibull-G family of probability distributions. Journal of Data Science, 12(1):53–68.
Chaturvedi, A., Garg, R. and Saini, S. (2022). Estimation and testing procedures for the reliability characteristics of Kumaraswamy-G distributions based on the progressively first failure censored samples. OPSEARCH, 59(2):494–517.
Cordeiro, G.M., Alizadeh, M. and Ortega, E.M.M. (2014). The exponentiated half-logistic family of distributions: Properties and applications. Journal of Probability and Statistics, 2014(1):728396.
Cordeiro, G.M. and de Castro, M. (2011). A new family of generalized distributions. Journal of Statistical Computation and Simulation, 81(7):883–898.
Cüran, G. and Kızılaslan, F. (2025). Statistical inference on stress-strength reliability for the Marshall-Olkin family of distributions. Soft Computing, 30(1):89–208.
Eugene, N., Lee, C. and Famoye, F. (2002). Beta-normal distribution and its applications. Communications in Statistics-Theory and Methods, 31(4):497–512.
Ghitany, M.E., Atieh, B. and Nadarajah, S. (2008). Lindley distribution and its application. Mathematics and Computers in Simulation, 78(4):493–506.
Gómez-Déniz, E., Ghitany, M.E. and Al-Mutairi, D.K. (2024). New results for the Marshall-Olkin family of distributions. Mathematica Slovaca, 74(4):1023–1038.
Gomes-Silva, F., Percontini, A., de Brito, E., Ramos, M.W., Venâncio, R. and Cordeiro, G.M. (2017). The odd Lindley-G family of distributions. Austrian Journal of Statistics, 46(1):65–87.
Jafari, A.A., Tahmasebi, S. and Alizadeh, M. (2014). The Beta-Gompertz distribution. Revista Colombiana de Estadística, 37(1):141–158.
Kotz, S., Lumelskii, Y. and Pensky, M. (2003). The Stress-Strength Model and Its Generalizations: Theory and Applications. Singapore: World Scientific.
Lindley, D.V. (1958). Fiducial distributions and Bayes' theorem. Journal of the Royal Statistical Society: Series B (Methodological), 20(1):102–107.
Marshall, A.W. and Olkin, I. (1997). A new method for adding a parameter to a family of distributions with application to the exponential and Weibull families. Biometrika, 84(3):641–652.
R Core Team (2023). R: A Language and Environment for Statistical Computing. Austria, Vienna: R Foundation for Statistical Computing.
Pal, M. and Tiensuwan, M. (2015). Some properties of gamma generated distributions. Statistica, 75(4):391–403.
Piessens, R., de Doncker-Kapenga, E., Überhuber, C.W. and Kahaner, D.K. (1983). Quadpack: A Subroutine Package for Automatic Integration. Berlin: Springer Science & Business Media.
Saini, S. and Garg, R. (2022). Reliability inference for multicomponent stress-strength model from Kumaraswamy-G family of distributions based on progressively first failure censored samples. Computational Statistics, 37(4):1795–1837.
Samson, K., Abubakar, M.A., Asongo, A. and Alhaji, I.S. (2020). Odd Lindley-Kumaraswamy distribution: Model, properties and application. Journal of Advances in Mathematics and Computer Science, 35(6):42–58.
Sibuya, M. (1996). The shapes of a probability density function and its hazard function. In Lifetime Data: Models in Reliability and Survival Analysis, pp. 307–314. Boston, MA: Springer US.
Wang, L., Li, M. and Tripathi, Y.M. (2020). Inference for dependent competing risks from bivariate Kumaraswamy distribution under generalized progressive hybrid censoring. Communications in Statistics-Simulation and Computation, 51(6):3100–3123.
Yousof, H.M., Afify, A.Z., Hamedani, G.G. and Aryal, G. (2017). The Burr X generator of distributions for lifetime data. Journal of Statistical Theory and Applications, 16(3):325–340.
Yousof, H.M., Majumder, M., Jahanshahi, S.M.A., Ali, M.M. and Hamedani, G.G. (2018). A new Weibull class of distributions: Theory, properties and applications. Hacettepe Journal of Mathematics and Statistics, 47(6):1647–1671.
Zografos, K. and Balakrishnan, N. (2009). On families of beta- and generalized gamma-generated distributions and associated inference. Statistical Methodology, 6(4):344–362.
Journal of Statistical Modelling: Theory and Applications
Volume 7, Issue 1
January 2026
Pages 21-35

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History

  • Receive Date: 12 July 2026
  • Revise Date: 05 September 2026
  • Accept Date: 12 September 2026

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