In the analysis of failure time data, researchers often encounter situations in which events arise from multiple mutually exclusive causes. This framework is commonly referred to as competing risks. Traditional methods are inadequate in such settings, as they typically assume the presence of a single type of failure and do not account for the influence of other competing events. This study investigates the competing risks model under an improved adaptive progressive Type-Ⅱ censoring scheme, which is particularly beneficial in contexts where the duration of testing is critical. The lifetimes associated with competing risks are assumed to follow independent Burr-X distributions, a flexible model capable of accommodating a variety of data types. Both classical and Bayesian estimation methods are utilized to estimate the model parameters and the reliability function, a key metric in reliability assessment. Maximum likelihood estimates are computed numerically, and approximate confidence intervals are derived. For Bayesian inference, squared error and linear-exponential loss functions are employed. Given the complexity of the posterior distribution, Markov chain Monte Carlo techniques are utilized to obtain Bayesian estimates and construct Bayesian credible intervals. A comprehensive numerical analysis, which includes a simulation study and the examination of real competing risk datasets, is conducted to evaluate and compare the performance of the proposed estimation methods.

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.