Analysis of reliability index $\mathfrak{R}=P(Y<X)$ for newly extended xgamma progressively first-failure censored samples with applications

The stress-strength index measures the likelihood that a system's strength exceeds its stress. This study focuses on deducting the stress-strength index, denoted as $\mathfrak{R}=P(Y<X)$, where the strength $(X)$ and stress $(Y)$ are independent random variables following new extended xgamma distributions. Inferences are made based on progressively first-failure censored samples. Both maximum likelihood and Bayesian estimation approaches, including point and interval estimations, are considered. The estimations take into account the model parameters as well as the reliability index. The Bayes estimates are obtained using the Markov chain Monte Carlo sampling procedure with the squared error loss function. Additionally, the approximate confidence intervals and Bayes credible intervals are developed. A simulation experiment is conducted to assess the different estimates presented in this paper. Precision metrics such as root mean square error, mean relative absolute bias, and interval length are used to evaluate the efficiency of various point and interval estimates. Two insulating fluid data sets are analyzed to demonstrate the relevance and applicability of the proposed estimation methods.