Parametric and nonparametric estimation stress strength model based on copula function under first-failure progressively unified hybrid censoring schemes

In the traditional interference of stress strength model, it is generally supposed that the strength variable and the stress variable are independent. However, in many engineering applications strength variable and the stress variable are dependent. To solve the situation where the strength variable and the stress variable are dependent, in this paper the dependent between strength and stress is considered, the correlation between strength and stress is described by Farlie–Gumbel–Morgenstern copula function. Three estimate methods (maximum likelihood estimation, Bayes estimator, kernel density estimation) are used to estimate the reliability for \(\delta =P(X<Y)\) $\delta =P(X<Y)$ under first failure progressively unified hybrid censoring samples. Also, bootstrap confidence interval is constructed using percentile bootstrap method. The effectiveness of the proposed estimation method is demonstrated by numerical simulation. Finally, the practicability of the proposed estimation method is verified by data of breakdown time of an insulating fluid.