E-Bayesian Estimation Research Articles

E-Bayesian estimation for Burr-X distribution based on Type-II hybrid censoring scheme

E-Bayesian estimation for Burr-X distribution based on Type-II hybrid censoring scheme

E-Bayesian Estimations of Reliability and Hazard Rate in the Generalized Inverted Exponential Distribution based on Type II Censoring

E-Bayesian Estimations of Reliability and Hazard Rate in the Generalized Inverted Exponential Distribution based on Type II Censoring

Comparison of Robust and Bayesian Methods for Estimating the Burr Type XII Distribution

This paper compares the robust and E-Bayesian estimations of the shape parameter for Burr XII distribution. BurrXII distribution was already reviewed by many researchers, as this distribution has gained special attention in recent times due to its complete applications, including the reliability field and failure time modeling. Burr distributions include 12 types of functions that produce a variety of probability density forms. We used two loss functions, quadratic, and LINEX with the E-Bayes method. The comparison conducted by simulation technique, and the absolute mean square error was measured to test the estimation methods' preference. In this study, many familiar distributions methods such as Weibull distribution, exponential logistic distribution, generalized logistic distribution, extreme value, and uniform distribution have been discussed accordingly by employing special cases and belonging to Burr distribution family. The current is dealing with the Bayesian method by depending on the parameter c that must be chosen to be close or not far from parameter b to ensure the robustness of the Bayesian estimator. Then the Bayesian expected of the parameter β under a quadratic loss function. It has been compared estimation for Burr-XII distribution by using the mentioned methods. We found many essential points, such as Robust estimates, in all cases, tend to be more efficient than the Bayes estimates. Also, by increasing the sample size, the robust estimates are still better than other estimation methods. When increasing the sample size, we notice a decrease in MAPE, which supports the statistical theory. We recommend using non-parametric methods to estimate Burr XII parameters. Thus, the main conclusion is that the robust process was the best.

E-Bayesian Estimation for the Weibull Distribution under Adaptive Type-I Progressive Hybrid Censored Competing Risks Data.

This article focuses on using E-Bayesian estimation for the Weibull distribution based on adaptive type-I progressive hybrid censored competing risks (AT-I PHCS). The case of Weibull distribution for the underlying lifetimes is considered assuming a cumulative exposure model. The E-Bayesian estimation is discussed by considering three different prior distributions for the hyper-parameters. The E-Bayesian estimators as well as the corresponding E-mean square errors are obtained by using squared and LINEX loss functions. Some properties of the E-Bayesian estimators are also derived. A simulation study to compare the various estimators and real data application is applied to show the applicability of the different estimators are proposed.

Study on the effect of the different prior distributions on E-Bayesian estimation of failure probability and its E-posterior risk

In reliability engineering, study the estimated risk of reliability parameters estimation is an important problem. This paper studies the E-Bayesian estimations and their E-posterior risk of the failure probability under different prior distributions. Moreover, the analytical expressions of E-posterior risk under different prior distributions of hyper parameters are obtained respectively. Monte Carlo simulation is performed to compare the performances of the proposed method of estimation, results are compared on basis of E-posterior risk and MSE (mean square error). Finally, used respectively E-Bayesian estimation method and MCMC method (also using OpenBUGS), combined with the aero engine problem are performed to calculation and analysis. When consider evaluating the E-Bayesian estimations under different prior distributions of hyper parameters, this paper proposed the E-posterior risk as an evaluation standard in reliability engineering.

E-Bayesian Estimation of Chen Distribution Based on Type-I Censoring Scheme.

In this paper, E-Bayesian estimation of the scale parameter, reliability and hazard rate functions of Chen distribution are considered when a sample is obtained from a type-I censoring scheme. The E-Bayesian estimators are obtained based on the balanced squared error loss function and using the gamma distribution as a conjugate prior for the unknown scale parameter. Also, the E-Bayesian estimators are derived using three different distributions for the hyper-parameters. Some properties of E-Bayesian estimators based on balanced squared error loss function are discussed. A simulation study is performed to compare the efficiencies of different estimators in terms of minimum mean squared errors. Finally, a real data set is analyzed to illustrate the applicability of the proposed estimators.

The E-Bayesian estimation and its E-MSE of Pareto distribution parameter under different loss functions

ABSTRACTThis paper studies the E-Bayesian estimations and their E-MSE of Pareto distribution parameter under different loss functions. In order to measure the estimated error, in the case of the two hyper parameters, the definition of E-MSE (expected mean square error) is introduced based on the definition of E-Bayesian estimation, and the formulas of E-Bayesian estimation and the formulas of E-MSE are given, respectively. Monte Carlo simulations are performed to analysis the performances of the proposed methods, results are compared on the basis of E-MSE. Finally, combined with the golfers income problem are performed to calculated, the maximum likelihood estimations of the shape parameter and scale parameter are given, respectively, and that compared and analysed with the E-Bayesian estimations; moreover, the Bayesian credible intervals of the shape parameter and scale parameter are also given, respectively, by MCMC method.

Reliability analysis for Weibull distribution with homogeneous heavily censored data based on Bayesian and least-squares methods

The reliability for Weibull distribution with homogeneous heavily censored data is analyzed in this study. The universal model of heavily censored data and existing methods, including maximum likelihood, least-squares, E-Bayesian estimation, and hierarchical Bayesian methods, are introduced. An improved method is proposed based on Bayesian inference and least-squares method. In this method, the Bayes estimations of failure probabilities are focused on for all the samples. The conjugate prior distribution of failure probability is set, and an optimization model is developed by maximizing the information entropy of prior distribution to determine the hyper-parameters. By integrating the likelihood function, the posterior distribution of failure probability is then derived to yield the Bayes estimation of failure probability. The estimations of reliability parameters are obtained by fitting distribution curve using least-squares method. The four existing methods are compared with the proposed method in terms of applicability, precision, efficiency, robustness, and simplicity. Specifically, the closed form expressions concerning E-Bayesian estimation and hierarchical Bayesian methods are derived and used. The comparisons demonstrate that the improved method is superior. Finally, three illustrative examples are presented to show the application of the proposed method.

Reliability computation for an uncertain PVC window production system using a modified bayesian estimation

Nowadays, Industries have been receiving much attention in Failure modelling and reliability assessment of repairable systems due to the fact that it plays a crucial role in risk and safety management of process. The primary purpose of this article is to present a methodology for discussing uncertainty in the reliability assessment if the production system. In fact, we discuss the fuzzy E-Bayesian estimation of reliability for PVC window production system. This approach is used to create the fuzzy E-Bayesian estimations of system reliability by introducing and applying a theorem called “Resolution Identity” for fuzzy sets. To be more specific, the model parameters are assumed to be fuzzy random variables. For this purpose, the original problem is transformed into a nonlinear programming problem which is divided into four sub-problems to simplify the computations. Finally, the results obtained for the sub-problems can be used to determine the membership functions of the fuzzy E-Bayesian estimation of system reliability. To clarify the proposed model, a practical example for PVC window production system is conducted.

E-Bayesian estimations for the cumulative hazard rate and mean residual life based on exponential distribution and record data

ABSTRACTEstimation of lifetime parameters such as reliability and hazard rate is necessary in many systems. Each method of estimation suffers from its own problems such as complexity of computations, high risk and etc. Toward this end, this study employed a new method, E-Bayesian, for estimating the parameters based on exponential distribution and record data. Furthermore, the asymptotic behaviours of E-Bayesian estimations and relations among them have been investigated. Finally, a comparison among the E-Bayesian estimations and older methods are made, using a real data and the Monte Carlo simulation. The computations show that the new method is more efficient than previous methods and is also easy to operate.

A Precautionary Loss-based Bayesian Algorithm for Zero Failure Analysis of Exponential Model

The aim of this paper is to study the Bayesian reliability analysis of the exponential distribution under a precautionary loss function based on zero failure data. In the Bayesian algorithm, the prior distribution of the parameter is assumed to be Gamma distribution. Bayesian and E-Bayesian estimators of the failure rate and reliability of exponential model are obtained under the precautionary loss function. Finally, a practical example is utilized to validate the effectiveness and robustness of the proposed estimators.

On E-Bayesian estimations for the cumulative hazard rate and mean residual life under generalized inverted exponential distribution and type-II censoring

Estimation of reliability and hazard rate is necessary in many applications. To this aim, different methods of estimation have been employed. Each method suffers from its own problems such as complexity of calculations, high risk and so on. Toward this end, this study employed a new method, E-Bayesian, for estimating the parametric functions of the Generalized Inverted Exponential distribution, which is one of the most noticeable distributions in lifetime studies. Relations are derived under a squared error loss function, type-II censoring and a conjugate prior. E-Bayesian estimations are obtained based on different priors of the hyperparameters to investigate the influence of different prior distributions on these estimations. The asymptotic behaviors of E-Bayesian estimations and relations among them have been investigated. Finally, a comparison among the maximum likelihood, Bayes and E-Bayesian estimations in different sample sizes are made, using a real data and the Monte Carlo simulation. Simulations show that the new presented method is more efficient than previous methods and is also easy to operate. Also, some comparisons among the results of Generalized Inverted Exponential distribution, Exponential distribution and Generalized Exponential distribution are provided.

E-Bayesian Estimation Based on Burr-X Generalized Type-II Hybrid Censored Data

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.

E-Bayesian estimations of parameter and its evaluation standard: E-MSE (expected mean square error) under different loss functions

This paper is concerned with using the E-Bayesian method for computing estimates of Pareto index. In order to measure the estimated error, in the case of the one hyper parameter, the definition of E-MSE (expected mean square error) is proposed based on the definition of E-Bayesian estimation. Moreover, the formulas of E-Bayesian estimation and formulas of E-MSE are given respectively, these estimations are derived based on a conjugate prior distribution under different loss functions (including: squared error loss, weighted squared error loss and precautionary loss). Monte Carlo simulations are performed to compare the performances of the proposed methods of estimation, results are compared on basis of E-MSE. Finally, combined with the golfers income problem are performed to calculated (also using OpenBUGS), and for income inequality degree were performed to compared and analyzed. When considering evaluating the E-Bayesian estimations under different loss functions, this paper proposed the E-MSE as evaluation standard.

E-Bayesian Estimation for Burr-X Distribution Based on Generalized Type-I Hybrid Censoring Scheme

SYNOPTIC ABSTRACTThis 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.

E-Bayesian estimation and its E-posterior risk of the exponential distribution parameter based on complete and type I censored samples

This article studies E-Bayesian estimation and its E-posterior risk, for failure rate derived from exponential distribution, in the case of the two hyper parameters. In order to measure the estimated risk, the definition of E-posterior risk (expected posterior risk) is proposed based on the definition of E-Bayesian estimation. Moreover, under the different prior distributions of hyper parameters, the formulas of E-Bayesian estimation and formulas of E-posterior risk are given respectively, these estimations are derived based on a conjugate prior distribution for the unknown parameter under the squared error loss function. Monte Carlo simulations are performed to compare the performances of the proposed methods of estimation and a real data set have been analyzed for illustrative purposes, results are compared on the basis of E-posterior risk.

E-Bayesian estimations of the reliability and its E-posterior risk under different loss functions

This paper studies the E-Bayesian estimations of the reliability and their E-posterior risk under different loss functions (including: squared error loss, weighted squared error loss, precautionary loss and K-loss) were derived under condition of binomial distribution. In the case of the one hyper parameter in order to measure the estimated risk, the definition of E-posterior risk (expected posterior risk)is proposed based on the definition of E-Bayesian estimation, and the formulas of E-Bayesian estimation and the formulas of E-posterior risk are developed, respectively. And an simulation example was given by using the Monte Carlo method. Finally, two practical problems were also performed to compare under different E-posterior risks (also using OpenBUGS). The results show that the proposed method is feasible and convenient for application.

Parameter estimation for the Lomax distribution using the E-Bayesian method

In this paper, we will find the parameter estimator from Lomax distribution on one of censored data, that is right censored data type II. Lomax distribution is also called Pareto type II distribution. The parameter will be estimate is the shape parameter with the assumption of scale parameters (β) has been known. The parameter estimation method used in this study is E-Bayesian estimation method. E-Bayesian estimation is an expectation of Bayes estimation, in order to obtain Bayes estimation expectations is by calculating the mean of Bayes estimators. E-Bayesian estimation method is used to estimate failure rate. The estimation will use prior Gamma as conjugate prior distribution from Lomax distribution and Loss function will be used is balanced squared error loss function (BSELF). Thus, the main purpose of this study is to find the parameter estimator of the Lomax distribution on the right censored data type II using the E-Bayesian method. The final result of this study, we get the likelihood function from Lomax distribution on the right censored data type II and the parameter estimator from Lomax distribution on the right censored data type II using the E-Bayesian method.

Expectation Bayesian Estimation of System Reliability Based on Failures

This paper discusses a new approach for system reliability parameter. Actually, we provide expectation Bayesian (E-Bayesian) estimation of system reliability for Series and parallel systems based on Pascal distribution. The definition and properties of E-Bayesian estimation are given. Also we applied three different distributions for the parameters in prior distribution to investigate the influence of the different prior distributions on the E-Bayesian estimation. The confidence intervals of R, based on E-Bayesian and bootstrap methods, are developed. The performance of these confidence intervals is studied through extensive simulation. Two numerical practical examples, is presented to illustrate the implementation of the proposed procedure.

E-Bayesian estimation and its E-MSE under the scaled squared error loss function, for exponential distribution as example

ABSTRACTThis paper is concerned with using the E-Bayesian method for computing estimates of exponential distribution. In order to measure the estimated error, based on the E-Bayesian estimation, we proposed the definition of E-MSE(expected mean square error). Moreover, the formulas of E-Bayesian estimation and formulas of E-MSE are given respectively, these estimations are derived based on a conjugate prior distribution for the unknown parameter under the scaled squared error loss function. The properties of E-MSE under different scaled parameters are also provided. Monte Carlo simulations are performed to compare the performances of the proposed methods of estimation and a real data set have been analysed for illustrative purposes. Results are compared on the basis of E-MSE.