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In this paper, we obtain the stress-strength reliability estimation under balanced joint Type-II progressive censoring scheme for independent samples from two different populations. We simultaneously place two independent samples where the experimental units follow Weibull distributions with common shape parameter β and different scale parameters α, λ, respectively. The maximum likelihood estimators of the unknown parameters are derived. Further, the Bayesian inference is considered using Lindley's approximation and Gibbs sampling method. Extensive simulations are performed to see the effectiveness of the proposed estimation methods. Further, we derive the optimal censoring scheme in the Bayesian framework by using the variable neighbourhood search method proposed by [Bhattacharya et al. On optimum life-testing plans under type-ii progressive censoring scheme using variable neighbourhood search algorithm. Test. 2016;25(2):309–330]. Further, some simulation schemes are provided to compare the performances of the estimations under the jointly censored samples versus two separate censored samples. [ABSTRACT FROM AUTHOR]

In this paper, the Bayesian and non-Bayesian estimation of $$\psi = P(X \gt Y)$$ ψ = P (X > Y) based on the progressively first-failure censored data is considered. The $$X$$ X and $$Y$$ Y are strength and stress random variables and follow the Topp-Leone distributions, respectively. The maximum likelihood and Bayes estimators of $$\psi $$ ψ are derived. The Bayes estimators under generalized entropy loss function are computed using Lindley's approximation and Gibbs sampling methods. Different interval estimates like asymptotic, bootstrap confidence, Bayesian credible, and highest posterior density credible intervals of $$\psi $$ ψ are constructed. Furthermore, a Monte Carlo numerical study is conducted to check the performance of various estimators developed. Finally, an application of algorithm real data is considered for illustrative purposes. [ABSTRACT FROM AUTHOR]