International Journal of Nonlinear Analysis and Applications

On the dependency in stress-strength models

Articles in Press

Document Type : Research Paper

Author

Department of Statistics, Semnan University, Semnan, Iran

Abstract

In reliability studies, the probability $R = P(Y < X)$ is called “reliability”,  and  $X, Y$  are typically considered independent. However, in many applications, such an assumption may be unrealistic. In this paper, we consider the stress-strength reliability $R = P(Y < X)$ where the stress $Y$ and strength $X$ are dependent. We use a copula to describe dependence among random variables $X$ and $Y$. We obtain $R$ for several copula functions with exponential marginal and plot graphs of $R$ versus the dependence parameter to study the effect of (positive and negative) dependency on $R$.

Keywords

[1] T. Emura and C.H. Pan, Parametric likelihood inference and goodness-of-fit for dependently left-truncated data, a copula-based approach, Statist. Papers 61 (2020), 479–501.
[2] R.C. Gupta, M.E. Ghitany, and D.K. Al-Mutairi. Estimation of reliability from a bivariate log-normal data, J. Statist. Comput. Simul. 83 (2012), no. 6, 1068–1081.
[3] H. Joe, Multivariate Models and Multivariate Dependence Concepts, Chapman and Hall/CRC, 1997.
[4] M. Jovanovic, Estimation of P(X < Y ) for geometric-exponential model based on complete and censored samples communications in statistics, Simul. Comput. 46 (2017), no. 4, 3050–3066.
[5] M. Jovanovi´c and V. Rajic. 2014. Estimation of P(X < Y ) for Gamma exponential model, Yugoslav J. Oper. Res. 24 (2014), no. 2, 283–291.
[6] S. Kotz, Y. Lumelski, and M. Pensky, The Stress-Strength Model and its Generalizations Theory and Applications, World Scientific, 2003.
[7] S. Nadarajah and S. Kotz, Reliability for some bivariate exponential distributions, Prob. Engin. Mech. 21 (2005), no. 4, 338–351.
[8] R.B. Nelsen, An Introduction to Copulas, 2nd Edition, Springer, 2006.
[9] D. Patil, U.V. Naik-Nimbalkar, and M.M. Kale, Effect of dependency on the estimation of P[Y > X] in exponential stress-strength models, Aust. J. Statist. 51 (2022), 10–34.
[10] S. Rezaei, R. Tahmasbi, and M. Mahmoodi. Estimation of P(Y < X) for generalized Pareto distribution, J. Statist. Plann. Infer. 140 (2010), no. 2, 480–494.
[11] S. Sengupta, Unbiased estimation of P(X > Y ) for two-parameter exponential populations using order statistics, Statistics 45 (2011), no. 2, 179–188.
International Journal of Nonlinear Analysis and Applications

Articles in Press, Corrected Proof
Available Online from 08 July 2025
  • Receive Date: 25 May 2024
  • Revise Date: 25 June 2024
  • Accept Date: 13 July 2024