This article deals with Bayesian and E-Bayesian (expectation of the Bayesian estimate) estimation methods of the parameter and the reliability function of Burr-X distribution based on a generalized Type-I hybrid censoring scheme. Bayesian and E-Bayesian estimates are obtained under LINEX and squared error loss functions. By applying Markov chain Monte Carlo techniques, Bayesian and E-Bayesian estimates based on a generalized Type-I hybrid censoring scheme are derived. Also, credible intervals for Bayesian and E-Bayesian estimates are computed. Examples of generalized Type-I hybrid censored samples and real data sets are presented for the purpose of illustration. Finally, a comparison between Bayesian and E-Bayesian estimation methods is conducted.

In this article, we are concerned with the E-Bayesian (the expectation of Bayesian estimate) method, the maximum likelihood and the Bayesian estimation methods of the shape parameter, and the reliability function of one-parameter Burr-X distribution. A hybrid generalized Type-II censored sample from one-parameter Burr-X distribution is considered. The Bayesian and E-Bayesian approaches are studied under squared error and LINEX loss functions by using the Markov chain Monte Carlo method. Confidence intervals for maximum likelihood estimates, as well as credible intervals for the E-Bayesian and Bayesian estimates, are constructed. Furthermore, an example of real-life data is presented for the sake of the illustration. Finally, the performance of the E-Bayesian estimation method is studied then compared with the performance of the Bayesian and maximum likelihood methods.

The widespread usage of granite in the building sector motivated us to conduct this research and examine the material’s sustainability in terms of the investigated characteristics. The purpose of this paper is to discuss the statistical analysis results for the mineralogical impact on radiological hazards indices, such as the equivalent of radium, absorbed gamma dose rate, annual effective dose, internal and external hazard indices, as well as the gamma-ray index, that were calculated to estimate the environmental risks associated with these granites used as building materials, to protect the public from excessive radioactivity exposure. We focused primarily on statistical significance at a 95% confidence level. We employed a non-parametric test (Kruskal–Wallis Test) rather than a one-way ANOVA, to determine the statistical significance of the samples due to the lack of homogeneity or normality among them. To assess the difference between the samples, we used the Mann–Whitney Test on each pair of samples. Additionally, Pearson correlation coefficients for all the mineralogical results are computed. The presence of K-rich minerals (Kefeldspars, biotite) and accessories such as uranophane, uranothorite, allanite, xenotime, fergusonite, aeschynite, zircon, cassiterite, apatite, and sphene, which are mostly found in granitic rocks, determines the level of natural radioactivity of the investigated granites. Most of the rock samples analyzed have indicators of radioactive dangers that are within the acceptable level range, indicating that they are suitable for use as building materials. On the other hand, some samples have environmental criteria that are higher than international standards, indicating that they are unsuitable for use as construction materials.

This paper is concerned with a constant-stress partially accelerated life test (CSPALT) model whose items are allocated at either normal use condition or under accelerated use condition. The lifetimes of items are assumed to follow Burr-X distribution based on Type-I hybrid censoring scheme. Two methods of estimation namely, E-Bayesian and Bayesian estimation methods are used for estimating the distribution parameter and the acceleration factor. These estimates are derived by using LINEX and squared error loss (SEL) functions. To obtain estimates of Bayesian and E-Bayesian methods, Markov chain Monte Carlo (MCMC) method is used. For illustration, a real-life data set is presented and applied to the CSPALT model. In the end, a comparison between proposed methods is carried out to investigate the performance of methods.

The Bayesian prediction of future failures from Lomax distribution is the subject of this research. The observed data is censored using a Type‐I hybrid censoring scheme under a step‐stress partially accelerated life test. There are two types of sampling schemes considered: one‐sample and two‐sample. We create predictive intervals for failure observations in the future. Bayesian prediction intervals are constructed using MCMC algorithms. After all, two numerical examples, simulation study and a real‐life example are provided for both one‐sample and two‐sample methods for the purpose of illustration.

Numerous studies have already been attempted to explore the reliability of systems considering mask data, though the mass of them has largely focused on basic series or parallel systems, where component failures are assumed to follow an exponential or Weibull distribution. However, most electrotonic products and systems are made up of numerous components integrated in parallel-series, series-parallel, and other bridge hybrid structures, and the number of studies in the area of accelerated life testing (ALT) employing masked data for hybrid systems is limited. In this paper, the constant-stress ALT (CSALT) is explored based on type-II progressive censoring scheme (TIIPCS) for a four-component hybrid system using geometric process (GmP). The failure times of the components of the system are assumed to follow the generalized Pareto (GP) distribution. The maximum likelihood estimate (MLE) technique is used to establish statistical inference for the model's unknown parameters under the premise that the failure reasons are unknown for the hybrid system. In addition, the asymptotic confidence intervals (ACIs) are also obtained by inverting the fisher information matrix. Finally, a simulation study is given to explain the proposed techniques and to evaluate the performance of the estimates. The performance of MLEs is assessed in terms of root mean square errors (RMSEs) and relative absolute biases (RABs), whereas the performance of ACIs is assessed in terms of their interval length (IL) and coverage probabilities (CPs). The findings show that the technique can deliver good estimation performance with small and intermediate sample sizes, and the estimates are more accurate when more failures are observed, showing the estimation method's efficiency.