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.Generalized Type-II HCS: Fix r ∈ {1, 2, · · · , n} and T 1 ,The experiment is terminated at T 1 , if r-th failure occurs before T 1 . When the r-th failure occurs between T 1 and T 2 , we terminate at Y r:n . In the third case, if r-th failure is observed after time T 2 , the experiment is terminated at T 2 . This scheme has been studied by many authors, such as [3], who presented details on censoring scheme developments in addition to generalized and unified HCS. Ref. [4] discussed Bayesian analysis and prediction based on generalized Type-II HCS for exponential and Pareto models. Ref.[5] studied maximum likelihood, Bayes and percentile bootstrap methods for unknown parameters, failure rate function, the survival function and the coefficient of variation of the exponential Rayleigh distribution with generalized Type-II HCS.Here, generalized Type-II HCS is considered. We observe one of these types of the censored data. Case 1:In this case, the r-th failure is obtained before T 1 , so the experiment is stopped at T 1 and d 1 number of failures is obtained at time T 1 .Case 2:In this case, the r-th failure occurs after T 1 , so the experiment is terminated at Y r:n , and r number of failures is obtained.Case 3:In this case, the r-th failure occurs after T 2 , so the experiment is ended at T 2 and d 2 number of failures is obtained at time T 2 . where T 1 and T 2 are time points determined by the experimenter according to how the experiment should continue based on the information about the product. Burr-X as a Lifetime ModelBurr-X model is part of Burr distribution family suggested by [6]. This distribution is important in many fields such as operations research and statistics. It is widely used in health, agriculture, and biology. For more details on the applications of this model, one can refer to [7], as they discussed the cumulative distribution function (CDF) of Burr-X distribution with the cumulative damage process and the shock model. Also, they assumed a mathematical model for the expected lifetime of AIDS patients, then fitted the observed data of infected persons for Burr-X distribution. The probability density function (PDF) of one-parameter...
Granite rocks are currently one of the foremost raw materials that can be used for various economic purposes such as ornamentation and building materials, because they do not possess radioactive concentrations and have good physical and mechanical properties. The granite rocks of north Um Taghir are connected to neoproterozoic rocks and integrated to the north Arabian-Nubian Shield (ANS), which lies in Northeast Africa. Inductively coupled plasma mass spectrometry (ICP-MS) and X-ray fluorescence analysis, concurrent to some statistical analysis, have been carried for major oxides and some trace elements to extract much fundamental information by following certain mathematical methods. The exposed granite rock units in north Um Taghir are classified into four rock units represented by tonalite, granodiorite, monzogranite, and alkali-feldspar granite which are cut by different types of dikes. The magma of tonalite and granodiorite is low-to-medium K calc-alkaline affinity, while the magma of monzogranite and alkali-feldspar granite is medium-to-high K calc-alkaline affinity, and of metaluminous to peraluminous nature. Granite rocks show a slightly depletion of fractionated patterns from light rare earth elements (LREEs) to heavy rare earth elements (HREEs) with slightly positive to negative Eu anomalies from tonalite to monzogranite and alkali-feldspar granites. The statistical criteria have been achieved to explore the significant differences of radiological hazard parameters among samples. It is obvious that there is no homogeneity among samples; furthermore, in Kruskal–Wallis test, Mann–Whitney test, and Pearson correlation coefficient, it can be noticed that there are significant differences between each pair of samples: tonalite, monzogranite; tonalite, alkali-feldspar granite; granodiorite, monzogranite; and granodiorite, alkali-feldspar granite. There is a strong direct relationship among granodiorite and both tonalite and alkali-feldspar granite, and among alkali-feldspar granite and tonalite and granodiorite. There is a strong inverse relationship among monzogranite and tonalite, granodiorite, and alkali-feldspar granite. As stated by all results, it can be mentioned that the granite rocks have a worthy result of mechanical and physical properties. So that they can be used for various economic purposes.
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