Inference of Stress-Strength Reliability Based on Adaptive Progressive Type-II Censing from Bath Tube Distribution with Application to Carbon Fiber Data

In this paper, we will use the maximum likelihood estimation (MLE) and the Bayes methods to perform estimation procedures for the reliability of stress-strength ℜ=P[X>Y] based on independent adaptive progressive censored samples that were taken from the Chen distributions. An approximate confidence interval of ℜ is constructed using a variety of classical techniques, such as the normal approximation of the MLE, the normal approximation of the log-transformed MLE, and the percentile bootstrap (Boot-p) procedure. The Bayesian estimation of ℜ is obtained based on the balanced loss function, which comes in two versions here, the symmetric balanced squared error (BSE) loss function and the asymmetric balanced linear exponential (BLINEX) loss function. When estimating ℜ using the Bayesian approach, all the unknown parameters of the Chen distributions are assumed to be independently distributed and to have informative gamma priors. Further, a mixture of Gibbs sampling algorithm and Metropolis-Hastings algorithm is used to compute the Bayes estimate of ℜ and the associated highest posterior density credible interval. In the end, simulation research are used to assess the general overall performance of the proposed estimators and a real data set is provided to exemplify the theoretical results.