Non-parametric empirical Bayes procedure

In this paper we introduce an empirical Bayes procedure for estimating an unknown parameter, say θ. This procedure gives the empirical Bayes estimator for θ and its associated minimum posterior risk in closed forms without estimating the unknown prior density function of θ. In such procedure the posterior probability density function of θ is not required. A sufficient statistic for θ with conditional probability density function in the one parameter exponential family is required. Instead of estimating the unknown prior density function, the marginal density function of the sufficient statistic must be estimated. As special cases the empirical Bayes estimators and their respective minimum posterior risks of the failure rate for the exponential distribution, the unknown scale parameters of Weibull and gamma distributions are obtained in simple forms as special cases. Numerical results and a simulation study are introduced to (i) investigate how the number of available past experiments and the sample size of each influence the accuracy of the empirical Bayes estimator, (ii) make a comparison between the presented procedure and the Bayes procedure when the prior probability density function of the parameter θ is gamma.     

双参数Rayleigh分布环境因子的经验Bayes估计及应用

本文针对Rayleigh分布位置参数已知的情形,给出了Rayleigh分布环境因子的极大似然估计和经验Bayes估计,并将环境因子的估计结果应用于Rayleigh部件的可靠性评估,给出了该部件可靠度函数与失效率的估计。最后的随机模拟例子表明,经验Bayes估计优于极大似然估计,并且在考虑环境因子的情形下,Rayleigh部件可靠性指标的估计优于未考虑环境因子时的估计。     

Reliability estimation of a multicomponent stress-strength model for unit Gompertz distribution under progressive Type II censoring

We consider the problem of estimating multicomponent stress-strength (MSS) reliability under progressive Type II censoring when stress and strength variables follow unit Gompertz distributions with common scale parameter. We estimate MSS reliability under frequentist and Bayesian approaches. Bayes estimates are obtained by using Lindley approximation and Metropolis-Hastings algorithm methods. Further, we obtain uniformly minimum variance unbiased estimates of the reliability when common scale parameter is known. Asymptotic, bootstrap confidence interval and highest posterior density credible intervals have been constructed. We perform Monte Carlo simulations to compare the performance of proposed estimates and also present a discussion. Finally, three real data sets are analyzed for illustrative purposes.     

The Bayes procedure in exponential reliability family models using conjugate convex tent prior family

A Bayes procedure for estimating the unknown parameter indexed to some of one parameter exponential family distributions is presented. In such a procedure, we shall use a new conjugate prior family that we shall call a conjugate convex tent family. A member of this family could be constructed by assuming a little information about the unknown parameter. Some of the needed parameters for constructing prior density function are the values r, p and q. Bayes estimators for using a priori symmetrical convex tent density can be obtained as special cases of the present work. Numerical simulation study is introduced by using the Monte Carlo method. In this study we have investigated the influence of the prior parameters r, p and q on the accuracy of the Bayes estimators.     

Applied Statistics: A First Course in Inference

An important task of the U.S. Nuclear Regulatory Commission is to examine annual operating data from the nation's population of nuclear power plants for trends over time. We are interested here in trends in the scram rate at 66 commercial nuclear power plants based on annual observed scram data from 1984–1993. For an assumed Poisson distribution on the number of unplanned scrams, a gamma prior, and an appropriate hyperprior, a parametric empirical Bayes (PEB) approximation to a full hierarchical Bayes formulation is used to estimate the scram rate for each plant for each year. The PEB-estimated prior and posterior distributions are then subsequently smoothed over time using an exponentially weighted moving average. The results indicate that such bidirectional shrinkage is quite useful for identifying reliability trends over time.     

Bayesian system reliability assessment under fuzzy environments

The Bayesian system reliability assessment under fuzzy environments is proposed in this paper. In order to apply the Bayesian approach, the fuzzy parameters are assumed as fuzzy random variables with fuzzy prior distributions. The (conventional) Bayes estimation method will be used to create the fuzzy Bayes point estimator of system reliability by invoking the well-known theorem called ‘Resolution Identity’ in fuzzy sets theory. On the other hand, we also provide the computational procedures to evaluate the membership degree of any given Bayes point estimate of system reliability. In order to achieve this purpose, we transform the original problem into a nonlinear programming problem. This nonlinear programming problem is then divided into four subproblems for the purpose of simplifying computation. Finally, the subproblems can be solved by using any commercial optimizers, e.g. GAMS or LINGO.     

Bayesian reliability analysis when multiple early failures may be present

Summary This paper discusses the Bayesian reliability analysis for an exponential failure model on the basis of some ordered observations when the firstp observations may represent “early failures” or “outliers”. The Bayes estimators of the mean life and reliability are obtained for the underlying parametric model referred to as theSB(p) model under the assumption of the squared error loss function, the inverted gamma prior for the scale parameter and a generalized uniform prior for the nuisance parameter.     

一类连续型单参数指数族参数的经验Bayes检验函数的收敛速度

本文研究了一类连续型单参数指数族参数的经验Bayes检验问题．利用独立同分布样本下概率密度函数的递归核估计和Bayes检验函数的单调性,重新构造一类连续型单参数指数族参数的EB检验函数．通过修改EB检验函数的构造方法,构造了单调的Bayes检验函数．在一定的条件下,获得了EB检验函数的收敛速度,改进了现有文献中的收敛速度阶的结果．最后给出满足定理条件的数值算例．     

刻度参数的贝叶斯经验贝叶斯估计

我们采用贝叶斯经验贝叶斯方法估计了一个特殊的指数分布族中的刻度参数,证明了刻度参数的贝叶斯经验贝叶斯估计几乎处处收敛到它的贝叶斯估计且获得了它的渐近最优性。最后,提出了一个模拟试验去验证贝叶斯经验贝叶斯估计的渐近最优性。     

Bayes estimation of residual life by fusing multisource information

Residual life estimation is essential for reliability engineering. Traditional methods may experience difficulties in estimating the residual life of products with high reliability, long life, and small sample. The Bayes model provides a feasible solution and can be a useful tool for fusing multisource information. In this study, a Bayes model is proposed to estimate the residual life of products by fusing expert knowledge, degradation data, and lifetime data. The linear Wiener process is used to model degradation data, whereas lifetime data are described via the inverse Gaussian distribution. Therefore, the joint maximum likelihood (ML) function can be obtained by combining lifetime and degradation data. Expert knowledge is used according to the maximum entropy method to determine the prior distributions of parameters, thereby making this work different from existing studies that use non-informative prior. The discussion and analysis of different types of expert knowledge also distinguish our research from others. Expert knowledge can be classified into three categories according to practical engineering. Methods for determining prior distribution by using the aforementioned three types of data are presented. The Markov chain Monte Carlo is applied to obtain samples of the parameters and to estimate the residual life of products due to the complexity of the joint ML function and the posterior distribution of parameters. Finally, a numerical example is presented. The effectiveness and practicability of the proposed method are validated by comparing it with residual life estimation that uses non-informative prior. Then, its accuracy and correctness are proven via simulation experiments.     

Bayesian Inference and Life Testing Plan for the Weibull Distribution in Presence of Progressive Censoring

This article deals with the Bayesian inference of unknown parameters of the progressively censored Weibull distribution. It is well known that for a Weibull distribution, while computing the Bayes estimates, the continuous conjugate joint prior distribution of the shape and scale parameters does not exist. In this article it is assumed that the shape parameter has a log-concave prior density function, and for the given shape parameter, the scale parameter has a conjugate prior distribution. As expected, when the shape parameter is unknown, the closed-form expressions of the Bayes estimators cannot be obtained. We use Lindley's approximation to compute the Bayes estimates and the Gibbs sampling procedure to calculate the credible intervals. For given priors, we also provide a methodology to compare two different censoring schemes and thus find the optimal Bayesian censoring scheme. Monte Carlo simulations are performed to observe the behavior of the proposed methods, and a data analysis is onducted for illustrative purposes.     

Time-variant reliability global sensitivity analysis with single-loop Kriging model combined with importance sampling

This paper develops a novel failure probability-based global sensitivity index by introducing the Bayes formula into the moment-independent global sensitivity index to approximate the effect of input random variables or stochastic processes on the time-variant reliability. The proposed global sensitivity index can estimate the effect of uncertain inputs on the time-variant reliability by comparing the difference between the unconditional probability density function of input variables and the conditional probability density function in failure state of input variables. Furthermore, a single-loop active learning Kriging method combined with metamodel-based importance sampling is employed to improve the computational efficiency. The accuracy of the results obtained by Kriging model is verified by the reference results provided by the Monte Carlo simulation. Four examples are investigated to demonstrate the significance of the proposed failure probability-based global sensitivity index and the effectiveness of the computational method.     

Variability and uncertainty of biokinetic model parameters: the discrete empirical Bayes approximation

In the Bayesian approach to internal dosimetry, uncertainty and variability of biokinetic model parameters need to be taken into account. The discrete empirical Bayes approximation replaces integration over biokinetic model parameters by discrete summation in the evaluation of Bayesian posterior averages using Bayes theorem. The discrete choices of parameters are taken as best-fit point determinations of model parameters for a study subpopulation with extensive data. A simple heuristic model is constructed to numerically and theoretically study this approximation. The heuristic example is the measurement of heights of a group of people, say from a photograph where measurement uncertainty is significant. A comparison is made of posterior mean and standard deviation of height after a measurement, (i) using the exact prior describing the distribution of true height in the population and (ii) using the approximate discrete empirical Bayes prior obtained from measurements of some study subpopulation.     

Bayesian estimation of a normal mean parameter using the Linex loss function and robustness considerations

Summary This paper presents a derivation of an explicit analytical form for the Bayes estimator of the normal location parameter using the Linex loss function with a general class of prior distributions. Exact and approximate results based on Pericchi and Smith’s paper (1992) are given, where the priors are double-exponential and Studentt, respectively. The results of this paper provide a link between the robust Bayesian analysis for the normal location parameter when adopting either the Linex loss function or the squared error loss function.     

E-Bayesian estimation of reliability characteristics of two-parameter bathtub-shaped lifetime distribution with application

This article presents the expected Bayesian (E-Bayesian) estimation of the scale parameter, reliability and failure rate functions of two-parameter bathtub-shaped lifetime distribution under type-II censoring data with. Squared error loss function and gamma distribution as a conjugate prior distribution for the unknown parameter are used to obtain the E-Bayesian estimators. Also, three different prior distributions for the hyperparameters for the E-Bayesian estimators are considered. Some properties of the E-Bayesian estimators are studied. Using minimum mean square error criteria, a simulation study is conducted to compare the performance of the E-Bayesian estimators and the corresponding Bayes and maximum likelihood estimators. A real data set is analysed to show the applicability of the different proposed estimators. The numerical results show that the E-Bayesian estimators perform better than the classical and Bayesian estimators.     

Validation of reliability computational models using Bayes networks

This paper proposes a methodology based on Bayesian statistics to assess the validity of reliability computational models when full-scale testing is not possible. Sub-module validation results are used to derive a validation measure for the overall reliability estimate. Bayes networks are used for the propagation and updating of validation information from the sub-modules to the overall model prediction. The methodology includes uncertainty in the experimental measurement, and the posterior and prior distributions of the model output are used to compute a validation metric based on Bayesian hypothesis testing. Validation of a reliability prediction model for an engine blade under high-cycle fatigue is illustrated using the proposed methodology.     

Mixing Bayes and empirical Bayes inference to anticipate the realization of engineering concerns about variant system designs

Mixing Bayes and Empirical Bayes inference provides reliability estimates for variant system designs by using relevant failure data - observed and anticipated - about engineering changes arising due to modification and innovation. A coherent inference framework is proposed to predict the realization of engineering concerns during product development so that informed decisions can be made about the system design and the analysis conducted to prove reliability. The proposed method involves combining subjective prior distributions for the number of engineering concerns with empirical priors for the non-parametric distribution of time to realize these concerns in such a way that we can cross-tabulate classes of concerns to failure events within time partitions at an appropriate level of granularity. To support efficient implementation, a computationally convenient hypergeometric approximation is developed for the counting distributions appropriate to our underlying stochastic model. The accuracy of our approximation over first-order alternatives is examined, and demonstrated, through an evaluation experiment. An industrial application illustrates model implementation and shows how estimates can be updated using information arising during development test and analysis.     

Bayesian estimation of the 3-parameter inverse gaussian distribution

The three-parameter inverse Gaussian distribution is used as an alternative model for the three parameter lognormal, gamma and Weibull distributions for reliability problems. In this paper Bayes estimates of the parameters and reliability function of a three parameter inverse Gaussian distribution are obtained. Posterior variance estimates are compared with the variance of their maximum likelihood counterparts. Numerical examples are given.     

Bayes approach in RDT using accelerated and long-term life data

A common problem of reliability demonstration testing (RDT) is the magnitude of total time on test required to demonstrate reliability to the consumer’s satisfaction, particularly in the case of high reliability components. One solution is the use of accelerated life testing (ALT) techniques. Another is to incorporate prior beliefs, engineering experience, or previous data into the testing framework. This may have the effect of reducing the amount of testing required in the RDT in order to reach a decision regarding conformance to the reliability specification. It is in this spirit that the use of a Bayesian approach can, in many cases, significantly reduce the amount of testing required.We demonstrate the use of this approach to estimate the acceleration factor in the Arrhenius reliability model based on long-term data given by a manufacturer of electronic components (EC). Using the Bayes approach we consider failure rate and acceleration factor to vary randomly according to some prior distributions. Bayes approach enables for a given type of technology the optimal choice of test plan for RDT under accelerated conditions when exacting reliability requirements must be met. These requirements are given by a hypothetical consumer by two different ways. The calculation of posterior consumer’s risk is demonstrated in both cases.The test plans are optimum in that they take into account Var{λ|data}, posterior risk, E{λ|data}, Median λ or other percentiles of λ at data observed at the accelerated conditions. The test setup assumes testing of units with time censoring.     

Investigating the effects of the fixed and varying dispersion parameters of Poisson-gamma models on empirical Bayes estimates

Traditionally, transportation safety analysts have used the empirical Bayes (EB) method to improve the estimate of the long-term mean of individual sites; to correct for the regression-to-the-mean (RTM) bias in before-after studies; and to identify hotspots or high risk locations. The EB method combines two different sources of information: (1) the expected number of crashes estimated via crash prediction models, and (2) the observed number of crashes at individual sites. Crash prediction models have traditionally been estimated using a negative binomial (NB) (or Poisson-gamma) modeling framework due to the over-dispersion commonly found in crash data. A weight factor is used to assign the relative influence of each source of information on the EB estimate. This factor is estimated using the mean and variance functions of the NB model. With recent trends that illustrated the dispersion parameter to be dependent upon the covariates of NB models, especially for traffic flow-only models, as well as varying as a function of different time-periods, there is a need to determine how these models may affect EB estimates. The objectives of this study are to examine how commonly used functional forms as well as fixed and time-varying dispersion parameters affect the EB estimates. To accomplish the study objectives, several traffic flow-only crash prediction models were estimated using a sample of rural three-legged intersections located in California. Two types of aggregated and time-specific models were produced: (1) the traditional NB model with a fixed dispersion parameter and (2) the generalized NB model (GNB) with a time-varying dispersion parameter, which is also dependent upon the covariates of the model. Several statistical methods were used to compare the fitting performance of the various functional forms. The results of the study show that the selection of the functional form of NB models has an important effect on EB estimates both in terms of estimated values, weight factors, and dispersion parameters. Time-specific models with a varying dispersion parameter provide better statistical performance in terms of goodness-of-fit (GOF) than aggregated multi-year models. Furthermore, the identification of hazardous sites, using the EB method, can be significantly affected when a GNB model with a time-varying dispersion parameter is used. Thus, erroneously selecting a functional form may lead to select the wrong sites for treatment. The study concludes that transportation safety analysts should not automatically use an existing functional form for modeling motor vehicle crashes without conducting rigorous analyses to estimate the most appropriate functional form linking crashes with traffic flow.