Inference of stress-strength reliability based on adaptive progressive type-Ⅱ censing from Chen distribution with application to carbon fiber data

In this paper, we used the maximum likelihood estimation (MLE) and the Bayes methods to perform estimation procedures for the reliability of stress-strength $R=P(Y<X)$ based on independent adaptive progressive censored samples that were taken from the Chen distribution. An approximate confidence interval of $R$ was 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. Additionally, the asymptotic distribution theory and delta approach were used to generate the approximate confidence interval. Further, the Bayesian estimation of $R$ was obtained based on the balanced loss function, which came 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 distribution were assumed to be independently distributed and to have informative gamma priors. Additionally, a mixture of Gibbs sampling algorithm and Metropolis-Hastings algorithm was used to compute the Bayes estimate of $R$ and the associated highest posterior density credible interval. In the end, simulation research was used to assess the general overall performance of the proposed estimators and a real dataset was provided to exemplify the theoretical results.