The E-Bayesian and hierarchical Bayesian estimations for the system reliability parameter

Han introduced an E-Bayesian estimation method for estimating a system failure probability and revealed the relationship between the E-Bayesian estimates under three different prior distributions of hyperparameters in 2007. In this article, formulas of the hierarchical Bayesian estimation of a system failure probability are investigated and, furthermore, the relationship between hierarchical Bayesian estimation and E-Bayesian estimation is discussed. Finally, numerical example and application example are provided for illustrative purpose.     

E-Bayesian and hierarchical Bayesian estimations for the system reliability parameter based on asymmetric loss function

This paper explores properties of the E-Bayesian and hierarchical Bayesian estimations of the system reliability parameter. E-Bayesian estimation and hierarchical Bayesian estimation of Pascal distribution's parameter under two loss function, LINEX loss function and entropy loss function can be found. We obtained limits of that the E-Bayesian estimation and hierarchical Bayesian estimation are equal. A Monte Carlo simulation is used to compare performances of the two methods.     

The E-Bayesian and hierarchical Bayesian estimations of Pareto distribution parameter under different loss functions

In this paper, we study the E-Bayesian and hierarchical Bayesian estimations of the parameter derived from Pareto distribution under different loss functions. The definition of the E-Bayesian estimation of the parameter is provided. Moreover, for Pareto distribution, under the condition of the scale parameter is known, based on the different loss functions, formulas of the E-Bayesian estimation and hierarchical Bayesian estimations for the shape parameter are given, respectively, properties of the E-Bayesian estimation – (i) the relationship between of E-Bayesian estimations under different loss functions are provided, (ii) the relationship between of E-Bayesian and hierarchical Bayesian estimations under the same loss function are also provided, and using the Monte Carlo method simulation example is given. Finally, combined with the golfers income data practical problem are calculated, the results show that the proposed method is feasible and convenient for application.     

E-Bayesian estimation for the proportional hazard rate model based on record values

In this article, a new parameter estimation method, named E-Bayesian method, is considered to obtain the estimates of the unknown parameter and reliability function based on record values. The maximum likelihood, Bayesian, E-Bayesian, and hierarchical Bayesian estimates of the unknown parameter and reliability function are obtained when the underlying distribution belongs to the proportional hazard rate model. The Bayesian estimates are obtained based on squared error and linear-exponential loss functions. The previously obtained some relations for the E-Bayesian estimates are improved. The relationship between E-Bayesian and hierarchical Bayesian estimations are obtained under the same loss functions. The comparison of the derived estimates are carried out by using Monte Carlo simulations. Real data are analyzed for an illustration of the findings.     

The E-Bayesian and hierarchical Bayesian estimations for the proportional reversed hazard rate model based on record values

In this paper, E-Bayesian and hierarchical Bayesian estimations of the shape parameter, when the underlying distribution belongs to the proportional reversed hazard rate model, are considered. Maximum likelihood, Bayesian and E-Bayesian estimates of the unknown parameter and reliability function are obtained based on record values. The Bayesian estimates are derived based on squared error and linear–exponential loss functions. It is pointed out that some previously obtained order relations of E-Bayesian estimates are inadequate and these results are improved. The relationship between E-Bayesian and hierarchical Bayesian estimations is obtained under the same loss functions. The comparison of the derived estimates is carried out by using Monte Carlo simulations. A real data set is analysed for an illustration of the findings.     

基于分层贝叶斯分析的残疾率小域估计方法

小域估计（Small Area Estimation）是抽样调查领域里一个重要的研究方向,国计民生中的许多重要问题如失业率、传染病的发病率和民意测验等抽样调查都需要采用不同的小域估计方法。本文针对小域估计问题,以估计方法发展脉络为主线,以分层贝叶斯分析的小域估计为重点,对小域估计问题的理论、方法和最新进展进行简述,并利用澳大利亚残疾、老龄化和护理者（SDAC 2003）抽样调查实际数据,从分层贝叶斯分析角度对澳大利亚残疾率进行估计,最后对估计结果进行比较和讨论。     

Hierarchical Bayesian Models for the Estimation of Correlated Effects in Multilevel Data: A Simulation Study to Assess Model Performance

In this article, we aim at assessing hierarchical Bayesian modeling for the analysis of multiple exposures and highly correlated effects in a multilevel setting. We exploit an artificial data set to apply our method and show the gains in the final estimates of the crucial parameters. As a motivating example to simulate data, we consider a real prospective cohort study designed to investigate the association of dietary exposures with the occurrence of colon-rectum cancer in a multilevel framework, where, e.g., individuals have been enrolled from different countries or cities. We rely on the presence of some additional information suitable to mediate the final effects of the exposures and to be arranged in a level-2 regression to model similarities among the parameters of interest (e.g., data on the nutrient compositions for each dietary item).     

Hierarchical Bayesian Approach to a Multi-Site Hidden Markov Model

Multivariate data with a sequential or temporal structure occur in various fields of study. The hidden Markov model (HMM) provides an attractive framework for modeling long-term persistence in areas of pattern recognition through the extension of independent and identically distributed mixture models. Unlike in typical mixture models, the heterogeneity of data is represented by hidden Markov states. This article extends the HMM to a multi-site or multivariate case by taking a hierarchical Bayesian approach. This extension has many advantages over a single-site HMM. For example, it can provide more information for identifying the structure of the HMM than a single-site analysis. We evaluate the proposed approach by exploiting a spatial correlation that depends on the distance between sites.     

E-Bayesian estimation of the exponentiated distribution family parameter under LINEX loss function

This paper is concerned with using the E-Bayesian method for computing estimates of the exponentiated distribution family parameter. Based on the LINEX loss function, formulas of E-Bayesian estimation for unknown parameter are given, these estimates are derived based on a conjugate prior. Moreover, property of E-Bayesian estimation—the relationship between of E-Bayesian estimations under different prior distributions of the hyper parameters are also provided. A comparison between the new method and the corresponding maximum likelihood techniques is conducted using the Monte Carlo simulation. Finally, combined with the golfers income data practical problem are calculated, the results show that the proposed method is feasible and convenient for application.     

Hierarchical modelling of orthopaedic hip replacement damage accumulation and reliability

Summary.  We describe the modelling of crack initiation and growth in a group of orthopaedic hip replacements. A hierarchical model is proposed and fitted with Bayesian methods, implemented by the Gibbs sampler. Our interest is in predicting the failure of the replacements, which is predominately caused by crack activity in the cement that fixates the metal prosthesis to the bone. Simulation from the model based on posterior distribution samples allows us to approx-imate the reliability function and to assess the effect on reliability of the properties of the replacement. We discuss the ideas behind this approach, which can be generalized to any group of physical systems and any function of their state that interests us.     

E-Bayesian estimation for system reliability and availability analysis based on exponential distribution

The main purpose of this article is to introduce the E-Bayesian approach to gain flexibility in the reliability-availability system estimation. This approach will be used in series systems, parallel systems, and k-out-of-m systems, based on exponential distribution under squared error loss function, when time is continuous. We use three prior distributions to investigate its impact on the E-Bayesian approach, those prior distributions cover a big spectrum of possibilities. We show in real examples and also by simulations, how the procedure behaves. In the simulation study also we explore the impact on this estimation approach, when the number of components of the system increases.     

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

This 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.     

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

Abstract

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.     

Rates of Convergence for a Bayesian Level Set Estimation

Abstract.  We are interested in estimating level sets using a Bayesian non-parametric approach, from an independent and identically distributed sample drawn from an unknown distribution. Under fairly general conditions on the prior, we provide an upper bound on the rate of convergence of the Bayesian level set estimate, via the rate at which the posterior distribution concentrates around the true level set. We then consider, as an application, the log-spline prior in the two-dimensional unit cube. Assuming that the true distribution belongs to a class of Hölder, we provide an upper bound on the rate of convergence of the Bayesian level set estimates. We compare our results with existing rates of convergence in the frequentist non-parametric literature: the Bayesian level set estimator proves to be competitive and is also easy to compute, which is of no small importance. A simulation study is given as an illustration.     

A Bayesian semiparametric model for non negative semicontinuous data

When the target variable exhibits a semicontinuous behavior (a point mass in a single value and a continuous distribution elsewhere), parametric “two-part models” have been extensively used and investigated. The applications have mainly been related to non negative variables with a point mass in zero (zero-inflated data). In this article, a semiparametric Bayesian two-part model for dealing with such variables is proposed. The model allows a semiparametric expression for the two parts of the model by using Dirichlet processes. A motivating example, based on grape wine production in Tuscany (an Italian region), is used to show the capabilities of the model. Finally, two simulation experiments evaluate the model. Results show a satisfactory performance of the suggested approach for modeling and predicting semicontinuous data when parametric assumptions are not reasonable.     

Robust Bayesian estimation of a two-parameter exponential distribution under generalized Type-I progressive hybrid censoring

A generalized Type-I progressive hybrid censoring scheme was proposed recently to overcome the limitations of the progressive hybrid censoring scheme. In this article, we provide a robust Bayesian method to estimate the unknown parameters of the two-parameter exponential distribution of a generalized Type-I progressive hybrid censored sample. For each parameter, we derive the marginal posterior density functions and the corresponding Bayesian estimators under the squared error loss function. To assess the proposed method, Monte Carlo simulations are performed using a real dataset.     

A Bayesian Hierarchical Model for Multiple Comparisons in Mixed Models

We propose a Bayesian hierarchical model for multiple comparisons in mixed models where the repeated measures on subjects are described with the subject random effects. The model facilitates inferences in parameterizing the successive differences of the population means, and for them, we choose independent prior distributions that are mixtures of a normal distribution and a discrete distribution with its entire mass at zero. For the other parameters, we choose conjugate or vague priors. The performance of the proposed hierarchical model is investigated in the simulated and two real data sets, and the results illustrate that the proposed hierarchical model can effectively conduct a global test and pairwise comparisons using the posterior probability that any two means are equal. A simulation study is performed to analyze the type I error rate, the familywise error rate, and the test power. The Gibbs sampler procedure is used to estimate the parameters and to calculate the posterior probabilities.     

A Practical Implementation and Bayesian Estimation in Franklin's Randomized Response Procedure

A Bayesian estimator based on Franklin's randomized response procedure is proposed for proportion estimation in surveys dealing with a sensitive character. The method is simple to implement and avoids the usual drawbacks of Franklin's estimator, i.e., the occurrence of negative estimates when the population proportion is small. A simulation study is considered in order to assess the performance of the proposed estimator as well as the corresponding credible interval.