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.     

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 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.     

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.     

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 Hierarchical Bayesian Estimation of Failure Probability

This article introduces a new parameter estimation method, named E-Bayesian estimation, to estimate failure probability. The method is suitable for the censored or truncated data with small sample sizes and high reliability. The definition, properties and related simulation study of the E-Bayesian estimation are given. A real data set is also discussed. Through the examples, the efficiency and easiness of operation of this method are commended.     

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.     

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.     

Estimating E-bayesian and hierarchical bayesian of scalar parameter of gompertz distribution under type II censoring schemes based on fuzzy data

In this study, the E-Bayesian and hierarchical Bayesian of the scalar parameter of a Gompertz distribution under Type II censoring schemes were estimated based on fuzzy data under the squared error (SE) loss function and the efficiency of the proposed methods was compared with each other and with the Bayesian estimator using Monte Carlo simulation.     

Estimation of the location parameter and the average worth of the selected subset of two parameter exponential populations under LINEX loss function

Suppose a subset of populations is selected from k exponential populations with unknown location parameters θ₁, θ₂, …, θ_k and common known scale parameter σ. We consider the estimation of the location parameter of the selected population and the average worth of the selected subset under an asymmetric LINEX loss function. We show that the natural estimator of these parameters is biased and find the uniformly minimum risk-unbiased (UMRU) estimator of these parameters. In the case of k = 2, we find the minimax estimator of the location parameter of the smallest selected population. Furthermore, we compare numerically the risk of UMRU, minimax, and the natural estimators.     

On robust Bayesian estimation under some asymmetric and bounded loss function

The problem of Bayesian and robust Bayesian estimation with some bounded and asymmetric loss function ABL is considered for various models. The prior distribution is not exactly specified and covers the conjugate family of prior distributions. The posterior regret, most robust and conditional Γ-minimax estimators are constructed and a preliminary comparison with square-error loss and LINEX loss is presented.     

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.     

Sufficient conditions for the admissibility under the LINEX loss function in non-regular case

Consider an estimation problem under the LINEX loss function in one-parameter non-regular distributions where the endpoint of the support depends on an unknown parameter. The purpose of this paper is to give sufficient conditions for a generalized Bayes estimator of a parametric function to be admissible. Also, it is shown that the main result in this paper is an extension of the quadratic loss case. Some examples are given.     

Bayesian multivariate normal analysis under the extended reflected normal loss function

This paper considers the Bayesian analysis of the multivariate normal distribution under a new and bounded loss function, based on a reflection of the multivariate normal density function. The Bayes estimators of the mean vector can be derived for an arbitrary prior distribution of [d]. When the covariance matrix has an inverted Wishart prior density, a Bayes estimator of[d] is obtained under a bounded loss function, based on the entropy loss. Finally the admissibility of all linear estimators c[d]+ d for the mean vector is considered     

Estimation of covariance matrices based on hierarchical inverse-Wishart priors

This paper focuses on Bayesian shrinkage methods for covariance matrix estimation. We examine posterior properties and frequentist risks of Bayesian estimators based on new hierarchical inverse-Wishart priors. More precisely, we give the conditions for the existence of the posterior distributions. Advantages in terms of numerical simulations of posteriors are shown. A simulation study illustrates the performance of the estimation procedures under three loss functions for relevant sample sizes and various covariance structures.     

Estimation of parameters of exponential distribution in the truncated space using asymmetric loss function

This paper is concerned with estimation of location and scale parameters of an exponential distribution when the location parameter is bounded above by a known constant. We propose estimators which are better than the standard estimators in the unrestricted case with respect to the suitable choice of LINEX loss. The admissibility of the modified Pitman estimators with respect to the LINEX loss is proved. Finally the theory developed is applied to the problem of estimating the location and scale parameters of two exponential distributions when the location parameters are ordered.     

Inferential analysis for the reliability parameter based on the three-parameter Lindley distribution

ABSTRACT

In this article, we consider the estimation of R = P(Y < X), when Y and X are two independent three-parameter Lindley (LI) random variables. On the basis of two independent samples, the modified maximum likelihood estimator along its asymptotic behavior and conditional likelihood-based estimator are used to estimate R. We also propose sample-based estimate of R and the associated credible interval based on importance sampling procedure. A real life data set involving the times to breakdown of an insulating fluid is presented and analyzed for illustrative purposes.     

Some Bayesian lower bounds on reliability in the lognormal distribution

The two-parameter lognormal distribution with density function f(y: γ, σ²) = [(2πσ²)^1/2y] ¹exp[?(ln y ? γ)²/2σ²], y > 0, is important as a failure-time model in life testing. In this paper, Bayesian lower bounds for the reliability function R(t: γ, σ²) = ?[(γ ? ln t)/σ] are obtained for two cases. First, it is assumed that γ is known and σ² has either an inverted gamma or “general uniform” prior distribution. Then, for the case that both γ and σ² are unknown, the normal-gamma prior and Jeffreys' vague prior are considered. Some Monte Carlo simulations are given to indicate some of the properties of the Bayesian lower bounds.     

Testimator of the scale parameter of the exponential distribution using linex loss function

The present paper investigates the properties of a testimator of scale of an exponential distribution under Linex loss function. The risk function of testimator is derived and compared with that of an admissible estimator relative to Linex loss function. The shrinkage testimator is proposed which is the extension of testimator and its properties have been discussed. The level of significance of testimator is decided on the basis of Akaike information criterion following Hirano (1977, 1978). It is found that the testimator and shrinkage testimator dominates the admissible estimator in terms of risk in certain parametric space.     

Bayesian and maximum likelihood estimations of the inverse Weibull parameters under progressive type-II censoring

In this paper, the statistical inference of the unknown parameters of a two-parameter inverse Weibull (IW) distribution based on the progressive type-II censored sample has been considered. The maximum likelihood estimators (MLEs) cannot be obtained in explicit forms, hence the approximate MLEs are proposed, which are in explicit forms. The Bayes and generalized Bayes estimators for the IW parameters and the reliability function based on the squared error and Linex loss functions are provided. The Bayes and generalized Bayes estimators cannot be obtained explicitly, hence Lindley's approximation is used to obtain the Bayes and generalized Bayes estimators. Furthermore, the highest posterior density credible intervals of the unknown parameters based on Gibbs sampling technique are computed, and using an optimality criterion the optimal censoring scheme has been suggested. Simulation experiments are performed to see the effectiveness of the different estimators. Finally, two data sets have been analysed for illustrative purposes.