Bayesian estimation for two parameter exponential distribution using linear transformation of reliability function

The estimation of two-parameter exponential distribution using Bayes approach need a prior distribution for the two parameters. It is difficult to know this joint prior distribution, so it requires sometimes the approximation or to some assumptions which depends on previous experience. An estimation method was proposed by using linear transformation of reliability function of two-parameter exponential distribution in order to get simple linear regression model. Its parameters can be estimated by using Bayes approach, and then get the estimated parameters of the distribution from the relationship between the distribution parameters and regression model parameters. Simulation experiments at different sample sizes were applied in order to make a comparison between Bayes estimators yield from approximation method and estimators from proposed method. The findings show that the proposed method estimators were more efficient that from approximation method estimators by using mean squares error (MSE) as a criterion for comparison. Also, the results of estimation methods were applied on actual data taken from Babil Tires Factory, where the data represents the working time (hours) between successive failures

Estimating rate of occurrence of rare events with empirical Bayes : a railway application

Classical approaches to estimating the rate of occurrence of events perform poorly when data are few. Maximum likelihood estimators result in overly optimistic point estimates of zero for situations where there have been no events. Alternative empirical-based approaches have been proposed based on median estimators or non-informative prior distributions. While these alternatives offer an improvement over point estimates of zero, they can be overly conservative. Empirical Bayes procedures offer an unbiased approach through pooling data across different hazards to support stronger statistical inference. This paper considers the application of Empirical Bayes to high consequence low-frequency events, where estimates are required for risk mitigation decision support such as as low as reasonably possible. A summary of empirical Bayes methods is given and the choices of estimation procedures to obtain interval estimates are discussed. The approaches illustrated within the case study are based on the estimation of the rate of occurrence of train derailments within the UK. The usefulness of empirical Bayes within this context is discusse

Empirical bayes estimates of development reliability for one shot devices

This article describes a method for estimating the reliability of a system under development that is an evolution of previous designs. We present an approach to making effective use of heritage data from similar operational systems to estimate reliability of a design that is yet to realise any data. The approach also has a mechanism to adjust initial estimates in the light of sparse data that becomes available in early stages of test. While the estimation approach, known as empirical Bayes is generic, we focus on one shot devices as this was the type of system which provided the practical motivation for this work and for which we illustrate an application

Item Response Theory for Peer Assessment

As an assessment method based on a constructivist approach, peer assessment has become popular in recent years. However, in peer assessment, a problem remains that reliability depends on the rater characteristics. For this reason, some item response models that incorporate rater parameters have been proposed. Those models are expected to improve the reliability if the model parameters can be estimated accurately. However, when applying them to actual peer assessment, the parameter estimation accuracy would be reduced for the following reasons. 1) The number of rater parameters increases with two or more times the number of raters because the models include higher-dimensional rater parameters. 2) The accuracy of parameter estimation from sparse peer assessment data depends strongly on hand-tuning parameters, called hyperparameters. To solve these problems, this article presents a proposal of a new item response model for peer assessment that incorporates rater parameters to maintain as few rater parameters as possible. Furthermore, this article presents a proposal of a parameter estimation method using a hierarchical Bayes model for the proposed model that can learn the hyperparameters from data. Finally, this article describes the effectiveness of the proposed method using results obtained from a simulation and actual data experiments

Empirical Bayes methodology for estimating equipment failure rates with application to power generation plants

Many reliability databases pool event data for equipment across different plants. Pooling may occur both within and between organizations with the intention of sharing data across common items within similar operating environments to provide better estimates of reliability and availability. Frequentist estimation methods can be poor when few, or no, events occur even when equipment operate for long periods. An alternative approach based upon empirical Bayes estimation is proposed. The new method is applied to failure data analysis in power generation plants and found to provide credible insights. A statistical comparison between the proposed and frequentist methods shows that empirical Bayes is capable of generating more accurate estimates

Merging expert and empirical data for rare event frequency estimation : pool homogenisation for empirical Bayes models

Empirical Bayes provides one approach to estimating the frequency of rare events as a weighted average of the frequencies of an event and a pool of events. The pool will draw upon, for example, events with similar precursors. The higher the degree of homogeneity of the pool, then the Empirical Bayes estimator will be more accurate. We propose and evaluate a new method using homogenisation factors under the assumption that events are generated from a Homogeneous Poisson Process. The homogenisation factors are scaling constants, which can be elicited through structured expert judgement and used to align the frequencies of different events, hence homogenising the pool. The estimation error relative to the homogeneity of the pool is examined theoretically indicating that reduced error is associated with larger pool homogeneity. The effects of misspecified expert assessments of the homogenisation factors are examined theoretically and through simulation experiments. Our results show that the proposed Empirical Bayes method using homogenisation factors is robust under different degrees of misspecification

Bayesian correction for covariate measurement error: a frequentist evaluation and comparison with regression calibration

Bayesian approaches for handling covariate measurement error are well established, and yet arguably are still relatively little used by researchers. For some this is likely due to unfamiliarity or disagreement with the Bayesian inferential paradigm. For others a contributory factor is the inability of standard statistical packages to perform such Bayesian analyses. In this paper we first give an overview of the Bayesian approach to handling covariate measurement error, and contrast it with regression calibration (RC), arguably the most commonly adopted approach. We then argue why the Bayesian approach has a number of statistical advantages compared to RC, and demonstrate that implementing the Bayesian approach is usually quite feasible for the analyst. Next we describe the closely related maximum likelihood and multiple imputation approaches, and explain why we believe the Bayesian approach to generally be preferable. We then empirically compare the frequentist properties of RC and the Bayesian approach through simulation studies. The flexibility of the Bayesian approach to handle both measurement error and missing data is then illustrated through an analysis of data from the Third National Health and Nutrition Examination Survey