Recently, a newly improved Type-II adaptive progressive censoring plan was devised, which can successfully ensure that the test length will not surpass a particular threshold period. In this study, we explore the statistical inference of the alpha power exponential distribution in the context of improved adaptive progressive Type-II censored data. The parameters, reliability, and hazard functions were estimated from both classical and Bayesian viewpoints using this censoring plan. To begin, we applied the maximum likelihood estimation approach to obtain parameter, reliability, and hazard function estimators. Following that, the approximate confidence intervals for the aforementioned metrics were derived, assuming the asymptotic normality traits of the maximum likelihood estimators. Additionally, by employing the Bayesian method via the Markov chain Monte Carlo technique, the point estimators and highest posterior density intervals of various parameters were created based on the symmetric squared error loss. A simulation study that incorporates numerous scenarios was used to assess the effectiveness of various estimation methodologies. The optimal progressive censorship plans are then discussed based on a set of criteria. Finally, three applications from the engineering and medical domains have been offered as examples.

In the statistical literature, one of the most important subjects that is commonly used is stress–strength reliability, which is defined as $\delta =P\left[W<V\right]$, where V and W are the strength and stress random variables, respectively, and $\delta$ is reliability parameter. Type-II progressive censoring with binomial removal is used in this study to examine the inference of $\delta =P\left[W<V\right]$ for a component with strength V and being subjected to stress W. We suppose that V and W are independent random variables taken from the Burr XII distribution and the Burr III distribution, respectively, with a common shape parameter. The maximum likelihood estimator of $\delta$ is derived. The Bayes estimator of $\delta$ under the assumption of independent gamma priors is derived. To determine the Bayes estimates for squared error and linear exponential loss functions in the lack of explicit forms, the Metropolis–Hastings method was provided. Utilizing comprehensive simulations and two metrics (average of estimates and root mean squared errors), we compare these estimators. Further, an analysis is performed on two actual data sets based on breakdown times for insulating fluid between electrodes recorded under varying voltages.

This paper addresses the estimation of the multi-component stress–strength reliability when both the strength variables and the stress variable follow the one-parameter Garhy distribution. Data are assumed to arise from a unified hybrid censoring scheme, which generalizes both Type-I and Type-II hybrid censoring. A closed-form expression for the reliability parameter $R_{m,k}=P(\mathrm{at}\mathrm{least}m\mathrm{of}(X_1,\dots ,X_k)>Y)$ is derived, enabling efficient computation. Three estimation procedures are developed: maximum likelihood estimation (MLE), Bayesian inference using Markov chain Monte Carlo (MCMC) with non-informative priors, and the Tierney–Kadane Laplace-type approximation for posterior moments. For each method, we provide complete mathematical derivations, including the likelihood function under unified hybrid censoring, the posterior conditionals, and the asymptotic distribution of the reliability via the Delta method. Furthermore, Bayesian estimation is extended to asymmetric loss functions, and posterior propriety is formally proven. To check the suitability of the proposed methods, a real data application on generator failure times in power systems is presented.

Central Composite Design (CCD) is often used in Response Surface Methodology (RSM) to fit second-order models. However, little is known about their behavior as the number of factors increases. The study develops a cross-dimensional evaluation methodology to investigate the behavior of three types of CCDs as the number of factors increases from $k=3$ to $k=10$. The CCD families considered in this study include Face-Centered CCD (FCCD), Rotatable CCD (RCCD), and Spherical CCD (SCCD). The designs are evaluated using the alphabetic optimality criteria of D-, A-, and G-optimality. A total of 1080 design configurations are generated by varying the replication levels of factorial points, axial points, and center points. The results show that classical CCDs encounter structural limitations in high-dimensional settings. When $k\ge 7$, the quadratic model matrix becomes rank deficient due to the fixed factorial core of 32 runs, which takes on a budget-constrained approach and uses regularization as a diagnostic tool to evaluate the stability of CCDs in the rank-deficient regime. In lower dimensions ($k\le 6$), RCCD consistently provides the highest efficiency, followed by SCCD and FCCD. In higher-dimensional settings, fixed regularization becomes less effective as the scale of the information matrix increases. To address this limitation, scaled regularization is introduced by adjusting the regularization parameter according to the average eigenvalue of the information matrix. The results indicate that rotatable designs provide greater efficiency and stability in high-dimensional settings compared with spherical and face-centered designs.