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In this study, we introduce a novel estimation technique for assessing the reliability parameter R = P (Y < X) of the uniform truncated negative binomial distribution (UTNBD) in the context of stress–strength analysis. We base our inferences on the assumption that both the strength (X) and stress (Y) random variables follow a UTNBD with identical first shape and scale parameters. In the presence of a progressive type-II censoring scheme, we employ maximum likelihood, two parametric bootstrap methods, and Bayesian estimation approaches to derive the estimators. Due to the complexity introduced by censoring, the estimators are not available in explicit forms and are instead obtained through numerical approximation techniques. Furthermore, we compute the highest posterior density credible intervals and determine the asymptotic variance-covariance matrix. To assess the performance of our proposed estimators, we conduct a Monte Carlo simulation study and provide a comparative analysis. Finally, we illustrate the practical applicability of our study with an engineering application. [ABSTRACT FROM AUTHOR]

Clustered data frequently occur in biomedical research fields and clinical trials. The log-rank tests are widely used for two-independent samples of clustered data tests. The randomized block design and truncated binomial design are used for forcing balance in clinical trials and reducing selection bias. In this paper, survival clustered data are randomized by generalized randomized block, and subsequently clustered data in each block are randomized by truncated binomial design. Consequently, the p-values of the null permutation distribution of log-rank tests for clustered data are approximated via the double saddlepoint approximation method. Comprehensive numerical studies are carried out to assess the accuracy of the saddlepoint approximation. This approximation has a great accuracy over the asymptotic normal approximation. [ABSTRACT FROM AUTHOR]