E-Bayesian Estimation Research Articles

Bayesian and E-Bayesian Estimation for Odd Generalized Exponential Inverted Weibull Distribution

Bayesian and E-Bayesian Estimation for Odd Generalized Exponential Inverted Weibull Distribution

Bayesian and E-Bayesian Estimation for a Modified Topp Leone–Chen Distribution Based on a Progressive Type-II Censoring Scheme

Bayesian and E-Bayesian Estimation for a Modified Topp Leone–Chen Distribution Based on a Progressive Type-II Censoring Scheme

Exponentiated gamma constant-stress partially accelerated life tests with unified hybrid censored data: Statistical inferences

This study considers one of the partially accelerated life testing models, mainly a constant-stress approach with unified hybrid censored data. The data represents the lifetime of units following the exponentiated gamma distribution (EGD). The estimation techniques for the parameter of EGD and the acceleration factor are discussed, along with the maximum likelihood estimation technique, the Bayesian and E-Bayesian estimation. Utilizing the squared error loss (SEL) and LINEX loss functions, the E-Bayesian and Bayesian estimates are generated. For deriving the estimates of Bayesian and E-Bayesian approaches, we use the MCMC method. Real-world data is also provided for illustration. Furthermore, unified hybrid censoring schemes (HCSs) are examined to determine their effectiveness in the light of the results. In the end, we compare among the proposed methods during their findings. The study's findings demonstrate how well the suggested techniques work when utilizing unified HCS. The study also makes clear how important it is to adopt the CSPALT model, as it yields more failed things in a shorter amount of time than items with typical use situations. In conclusion, it is suggested that this research be used to tasks like industrial product reliability testing.

E-Bayesian Estimation Using Spacing Function for Inverse Lindley Adaptive Type-I Progressively Censored Samples: Comparative Study with Applications.

For the first time, this paper offers the Bayesian and E-Bayesian estimation methods using the spacing function (SF) instead of the classical likelihood function. The inverse Lindley distribution, including its parameter and reliability measures, is discussed in this study through the mentioned methods, along with some other classical approaches. Six-point and six-interval estimations based on an adaptive Type-I progressively censored sample are considered. The likelihood and product of spacing methods are used in classical inferential setups. The approximate confidence intervals are discussed using both classical approaches. For various parameters, the Bayesian methodology is studied by taking both likelihood and SFs as observed data sources to derive the posterior distributions. Moreover, the E-Bayesian estimation method is considered by using the same data sources in the usual Bayesian approach. The Bayes and E-Bayes credible intervals using both likelihood and SFs are also taken into consideration. Several Monte Carlo experiments are carried out to assess the performance of the acquired estimators, depending on different accuracy criteria and experimental scenarios. Finally, two data sets from the engineering and physics sectors are analyzed to demonstrate the superiority and practicality of the suggested approaches.

The E-Bayesian and hierarchical Bayesian estimations for the reliability analysis of Kumaraswamy generalized distribution based on upper record values

This paper investigates the E-Bayesian and hierarchical Bayesian estimations of shape parameter and reliability function of Kumaraswamy generalized distribution based on upper record values. The classical estimation method is utilized to deduce the maximum likelihood estimation of unknown parameter and reliability function. Bayesian estimates are derived by using conjugate Gamma prior distributions under quadratic and general entropy loss functions. Furthermore, assuming that hyper-hyperparameters obey three prior distributions, the E-Bayesian estimates of unknown parameters and reliability functions are obtained. The hierarchical Bayesian estimates are obtained by using hierarchical prior distributions. We also explore some characteristics and size relationships of E-Bayesian and hierarchical Bayesian estimations. The performance of E-Bayesian, Hierarchical Bayesian, Bayesian, and maximum likelihood estimations is compared based on the minimum mean square error criterion. Finally, the proposed estimation methods are applied to evaluate the reliability of the specimen under ultrasonic fatigue testing, and the results align with their structures and profiles.

Bayesian and E-Bayesian estimation based on constant-stress partially accelerated life testing for inverted Topp–Leone distribution

Abstract Accelerated or partially accelerated life tests are particularly significant in life testing experiments since they save time and cost. Partially accelerated life tests are carried out when the data from accelerated life testing cannot be extrapolated to usual conditions. The constant-stress partially accelerated life test is proposed in this study based on a Type-II censoring scheme and supposing that the lifetimes of units at usual conditions follow the inverted Topp–Leone distribution. The Bayes and E-Bayes estimators of the distribution parameter and the acceleration factor are derived. The balanced squared error loss function, which is a symmetric loss function, and the balanced linear exponential loss function, which is an asymmetric loss function, are considered for obtaining the Bayes and E-Bayes estimators. Based on informative gamma priors and uniform hyper-prior distributions, the estimators are obtained. Finally, the performance of the proposed Bayes and E-Bayes estimates is evaluated through a simulation study and an application using real datasets.

Bayesian and E-Bayesian Estimation for the Generalized Rayleigh Distribution under Different Forms of Loss Functions with Real Data Application

This paper investigates the estimation of an unknown shape parameter of the generalized Rayleigh distribution using Bayesian and expected Bayesian estimation techniques based on type-II censoring data. Subsequently, these estimators are obtained using four different loss functions: the linear exponential loss function, the weighted linear exponential loss function, the compound linear exponential loss function, and the weighted compound linear exponential loss function. The weighted compound linear exponential loss function is a novel suggested loss function generated by combining weights with the compound linear exponential loss function. We use the gamma distribution as a prior distribution. In addition, the expected Bayesian estimator is obtained through three different prior distributions of the hyperparameters. Moreover, depending on the four distinct forms of loss functions, Bayesian and expected Bayesian estimation techniques are performed using Monte Carlo simulations to verify the effectiveness of the suggested loss function and to compare Bayesian and expected Bayesian estimation methods. Furthermore, the simulation results indicate that, depending on the minimum mean squared error, the Bayesian and expected Bayesian estimations corresponding to the weighted compound linear exponential loss function suggested in this paper have significantly better performance compared to other loss functions, and the expected Bayesian estimator also performs better than the Bayesian estimator. Finally, the proposed techniques are demonstrated using a set of real data from the medical field to clarify the applicability of the suggested estimators to real phenomena and to show that the discussed weighted compound linear exponential loss function is efficient and can be applied in a real-life scenario.

The Moore and Bilikam model and Burr XII sub-model under progressively type-II censoring scheme

The Moore and Bilikam family includes lifetime distributions, hence there is a need for a meticulous investigation of the proposed family. We evaluate different estimation procedures for both parameters and reliability function of the Moore and Bilikam family comprehensively, including the maximum likelihood, Bayesian and E-Bayesian estimation methods. The estimation methods of the Moore and Bilikam family are compared via the simulation data, whereas simulation results of the Burr XII sub-model are reported. Based on the simulation approach, we concluded the estimates of the Moore and Bilikam family are convergent to the corresponding parameters, and the root mean square error values derived by the E-Bayesian method are less than other estimators. The analysis of the time between failures of secondary reactor pumps data set has been represented for illustrative purposes, which confirmed simulation results.

Comparison of E-Bayesian Estimators in Burr XII Model Using E-PMSE Based on Record Values

In this paper, we consider the problem of E-Bayesian estimation and its expected posterior mean squared error (E-PMSE) in a Burr type XII model on the basis of record values. The Bayesian and E-Bayesian estimators are computed under different prior distributions for hyperparameters. The E-PMSE of E-Bayesian estimators are calculated in order to measure the estimated risk. Performances of the E-Bayesian estimators are compared using a Monte Carlo simulation. A real data set is analyzed for illustrating the estimation results.

Reliability analysis of bathtub-shaped distribution using empirical Bayesian and E-Bayesian estimations under progressive Type-II censoring

This paper investigates the performance of empirical Bayesian and E-Bayesian estimation of the bathtub-shaped distribution based on the progressive Type-II censored samples. These two estimations overcome the selection of hyperparameters in the Bayesian method. The empirical Bayesian analysis employs the classical approach to determine the hyperparameters. The E-Bayesian estimation uses the prior distribution of hyperparameters to derive the expectation of Bayesian estimates. The estimates of parameters are derived under the squared error loss and LINEX loss functions. The extensive simulations show the results of the empirical Bayesian estimates and the E-Bayesain estimates. Further, the empirical Bayesian estimation is also presented to analyze the parameters of bathtub-shaped distribution based on the general progressive Type-II censored samples. The two datasets from the electromechanical field and medical survival analysis are analyzed using the empirical Bayesian and E-Bayesian methods.

Bayesian and E-Bayesian estimators in Burr type XII model based on censored data under reflected gamma loss function

Bayesian and E-Bayesian estimators in Burr type XII model based on censored data under reflected gamma loss function

E‐Bayesian estimation for the parameters and hazard function of Gompertz distribution based on progressively type‐II right censoring with application

AbstractE‐Bayesian inference is of incremental significance in the area related to the reliability of industrial products dealing with censored data sets. In this article, based on the progressive Type‐II right censored order statistics (PTIIRCOS), the E‐Bayesian estimates (E‐BE) for the parameters of Gompertz distribution (GD) and its hazard function are obtained and compared with the maximum likelihood (MLE) and Bayesian (BE) estimates. The squared error loss function (SE), linear‐exponential loss function (LINEX), and Al‐Bayyati loss function (ALB) are considered for the methods of BE and E‐BE. Some properties for the E‐BE estimates are discussed based on PTIIRCOS. Real data is established to clear the theoretical procedures. Akaike information criterion (AIC) is used to prove that GD is the best to model the data compared to other competitors and could potentially be very adequate in describing and modeling industrial products data. Finally, simulation study has been operated for comparing the E‐BEs with the MLEs and BEs.

Empirical E-Bayesian estimation of hierarchical poisson and gamma model using scaled squared error loss function

The hierarchical models have not only a major concern with developing computational schemes but also assist in inferring the multi-parameter problems. The E-Bayesian is the expected Bayesian estimation that can be found by taking the integrals of Bayesian estimator using a hyper-prior with respect to the hyper-parameters. This study introduces the empirical E-Bayesian estimation that is coalesced with hierarchical modeling which prior to this has not been investigated. The scaled squared error loss function (SELF) has been used to estimate the parameter of Hierarchical Poisson-Gamma (HPG) model using empirical E-Bayesian estimation. The empirical E-Posterior risk is considered to be the evaluation standard. In addition, the consistency along with the asymptotic normality of the posterior distribution have been discussed. Furthermore, the empirical Bayes method is used to estimate the values of hyper-parameters via Maximum Likelihood (ML) method. The Monte Carlo simulation is executed to assess the precision of proposed estimators and a real-data application has been analyzed for illustration and comparison purposes.

On E-Bayesian analysis of the hierarchical normal and inverse gamma model using different loss functions and its application

This paper introduces the idea of hierarchical modelling to derive the variance (i.e. squared scale) parameter of the hierarchical normal and inverse gamma model using E-Bayesian estimation. We propose the idea of a hierarchical probability density function instead of the traditional hierarchical prior density function. We aim to derive E-Bayesian estimators with respect to the conjugate prior distribution IG(a,b) on the basis of the Balanced squared error loss function (BSELF) and Stein's loss function (STLF) for the unknown squared scale parameter. We use E-Posterior Risk as an evaluation standard. This study also intends to reveal the relationship among the E-Bayesian estimates under three distinct bivariate independent prior distributions of hyperparameters. The asymptotic properties of the E-Bayesian estimators are also evaluated. Monte Carlo simulations are prosecuted to study the efficiency of E-Bayesian estimators empirically and also, a real data application is analysed for exemplifying purposes.

Bayesian and E-Bayesian Estimation for Rayleigh Distribution Using Unified Progressive Hybrid Censored Samples

Bayesian and E-Bayesian Estimation for Rayleigh Distribution Using Unified Progressive Hybrid Censored Samples

E-Bayesian estimation and E-posterior risk of failure probability under different loss functions and its applications in reliability analysis

This paper, based on the E-Bayesian estimation (expected Bayesian estimation), introduced the definition of E-posterior risk (expected posterior risk); we will derive the expressions of E-Bayesian estimation and their E-posterior risk for failure probability under the compound symmetric LINEX loss function, and will compare the results with the corresponding results under the square loss function. Monte Carlo simulations compared the performances of the proposed the estimation method, and compared the results on the basis of E-posterior risk. Finally, combined with the life testing data from a type of aero engine, the calculation and reliability analysis are performed, with E-Bayesian estimation and hierarchical Bayesian estimation, respectively (also using the MCMC method). When considering evaluating the E-Bayesian estimations under different loss functions, in this paper the E-posterior risk as an the evaluation standard is proposed.

Reliability Estimation for Dependent Left-Truncated and Right-Censored Competing Risks Data with Illustrations

In this paper, a competing risks model with dependent causes of failure is considered under left-truncated and right-censoring scenario. When the dependent failure causes follow a Marshall–Olkin bivariate exponential distribution, estimation of model parameters and reliability indices are proposed from classic and Bayesian approaches, respectively. Maximum likelihood estimators and approximate confidence intervals are constructed, and conventional Bayesian point and interval estimations are discussed as well. In addition, E-Bayesian estimators are proposed and their asymptotic behaviors have been investigated. Further, another objective-Bayesian analysis is also proposed when a noninformative probability matching prior is used. Finally, extensive simulation studies are carried out to investigate the performance of different methods. Two real data examples are presented to illustrate the applicability.

The E-Bayesian Methods for the Inverse Weibull Distribution Rate Parameter Based on Two Types of Error Loss Functions

Given a sample, E-Bayesian estimates, which are the expected Bayesian estimators over the joint distributions of two hyperparameters in the prior distribution, are developed for the inverse Weibull distribution rate parameter under the scaled squared error and linear exponential error loss functions, respectively. The corresponding expected mean square errors, EMSEs, of E-Bayesian estimators based on the sample are derived. Moreover, the theoretical properties of EMSEs are established. A Monte Carlo simulation study is conducted for the performance comparison. Finally, three data sets are given for illustration.

Reliability Assessment of Heavily Censored Data Based on E-Bayesian Estimation

The classic E-Bayesian estimation methods can only derive point estimation of the reliability parameters. In this paper, an improved E-Bayesian estimation method is proposed to evaluate product reliability under heavily censored data, which can achieve both point and confidence interval estimation for the reliability parameters. Firstly, by analyzing the concavity & convexity and function characteristics of the Weibull distribution, the value of product failure probability is limited to a certain range. Secondly, an improved weighted least squares method is utilized to construct the confidence interval estimation model of reliability parameters. Simulation results show that the proposed approach can significantly improve the calculation speed and estimation accuracy with just very few robustness reductions. Finally, a real-world case study of the sun gear transmission mechanism is used to validate the effectiveness of the presented method.

E–Bayesian Estimation for the Inverted Topp – Leone Distribution Based on Type-II Censored Data

The objective of this paper is to estimate the unknown parameter for inverted Topp – Leone (ITL) distribution using the E-Bayesian estimation method under type-II censored sampling. The squared error loss function and the LINEX loss function are used to estimate the unknown parameter. The effectiveness of the E-Bayesian estimators in comparison to the Bayesian and maximum likelihood estimators based on mean square error is investigated using a Monte Carlo simulation.