A Bayesian Approach to Parameter and Reliability Estimation in the Poisson Distribution

For life testing procedures, a Bayesian analysis is developed with respect to a random intensity parameter in the Poisson distribution. Bayes estimators are derived for the Poisson parameter and the reliability function based on uniform and gamma prior distributions of that parameter. A Monte Carlo procedure is implemented to make possible an empirical mean-squared error comparison between Bayes and existing minimum variance unbiased, as well as maximum likelihood, estimators. As expected, the Bayes estimators have mean-squared errors that are appreciably smaller than those of the other two.     

The zero-truncated negative binomial distribution as a failure model from the Bayesian approach

The zero-truncated negative binomial distribution is considered as a failure model from the Bayesian point of view. It is assumed that the location parameter is a stochastic variable with beta as its prior distribution. Under this assumption Bayes estimators are derived for the location parameter and reliability function. By means of Monte Carlo Simulation the minimum variance unbiased estimators (MVUEs) for the parameter and reliability function are compared to the corresponding Bayes estimators.     

Bayes Interval Estimation for the Shape Parameter of the Power Distribution

The power distribution is considered as failure model and uses a square-error loss function. Bayes credibility interval estimators for the shape parameter have been obtained assuming 1) the following priors for the shape parameter: Jeffrey's invariant prior, gamma, and inverted gamma; 2) the following priors for reliability: beta and log gamma function. It is straightforward to obtain estimators for reliability when the estimators for the shape parameter are known.     

Empirical Bayes Estimation in the Weibull Distribution

In part I empirical Bayes estimation procedures are introduced and employed to obtain an estimator for the unknown random scale parameter of a two-parameter Weibull distribution with known shape parameter. In part II, procedures are developed for estimating both the random scale and shape parameters. These estimators use a sequence of maximum likelihood estimates from related reliability experiments to form an empirical estimate of the appropriate unknown prior probability density function. Monte Carlo simulation is used to compare the performance of these estimators with the appropriate maximum likelihood estimator. Algorithms are presented for sequentially obtaining the reduced sample sizes required by the estimators while still providing mean squared error accuracy compatible with the use of the maximum likelihood estimators. In some cases whenever the prior pdf is a member of the Pearson family of distributions, as much as a 60% reduction in total test units is obtained. A numerical example is presented to illustrate the procedures.     

The logarithmic series distribution as a failure model from the Bayesian point of view

In this presentation the logorithmic series is studied as a failure model from the Bayesian point of view. It is assumed that the location parameter behaves as a random variable with beta as its prior distribution. Based on this assumption Bayes estimators for the location parameter and reliability function are derived. By using computer simulation we compare the Bayes estimator for the parameter with the corresponding minimum variance unbiased estimator (MVUE) and the Bayes estimator for the reliability with a corresponding unbiased estimator derived from the MVUE of the probability function.     

On Bayes Estimation of Reliability for the Birnbaum-Saunders Fatigue Life Model

Estimation of reliability for the Birnbaum-Saunders fatigue life distribution is considered. The scale parameter is also the median lifetime, and assuming that the scale parameter is known, Bayes estimators of the reliability function are obtained for a family of proper conjugate priors as well as for Jeffreys' vague prior for the shape parameter. When both parameters are unknown, a modified Bayes estimator of reliability is proposed using a moment estimator of the scale parameter. In addition to being computationally simpler than the MLE of reliability, Monte Carlo simulations for small samples show that the modified Bayes estimator is better than the MME for all values of the shape parameter and as good as the MLE for small values of the shape parameter in the sense of root mean squared errors.     

A Bayesian estimation of reliability model using the linex loss function

In this paper the problem of Bayes estimation of the reliability and the shape parameter p of a finite range failure time model is considered (assuming scale parameter θ is known). Following Zellner [A. Zellner, J. Am. Statist. Assoc. 81, 446–451 (1986)] the asymmetric loss function is used to obtain the Bayes estimators. Efficiencies of the proposed Bayes estimators are obtained with respect to the ordinary Bayes estimators and it was found that the proposed Bayes estimators are better than the ordinary Bayes estimators for quite a wide range of parameters.     

Nonparametric Bayes-risk estimation

Two nonparametric methods to estimate the Bayes risk using classified sample sets are described and compared. The first method uses the nearest neighbor error rate as an estimate to bound the Bayes risk. The second method estimates the Bayes decision regions by applying Parzen probability-density function estimates and counts errors made using these regions. This estimate is shown to be asymptotically consistent in mean square. The results of experiments with these estimators on simulated and empirical data imply that the estimators both have acceptable small-sample properties; however, small-sample convergence of both estimators depends strongly on the choice of metric and local area or window size in the measurement space.     

Estimation of parameters of life from progressively censored data using Burr-XII model

Based on progressively Type-II censored samples, the maximum likelihood, and Bayes estimators for some lifetime parameters (reliability, and hazard functions), as well as the parameters of the Burr-XII model, are derived. The Bayes estimators are obtained using both the symmetric (Squared Error, SE) loss function, and asymmetric (LINEX, and General Entropy, GE) loss functions. This was done with respect to the conjugate prior for the one shape parameter, and discrete prior for the other parameter of this model. Also the existence, uniqueness, and finiteness of the ML parameter estimates for this type of censoring are discussed. A practical example consisting of data from an accelerated test on insulating fluid reported by Nelson (1982) was used for illustration, and comparison. Finally, some numerical results using simulation study concerning different sample sizes, and progressive censoring schemes were reported.     

Estimation of reliability in multi-component stress-strength model following a Burr distribution

This paper draws inferences about the reliability in a multi-component stress-strength system when both stress and strength are independently identically distributed (idd) Burr random variables. We consider both maximum likelihood and Bayes estimators of the system reliability. The two estimators are compared numerically by obtaining empirical efficiencies with respect to the maximum likelihood estimator (MLE) by generating 1000 random samples by a Monte Carlo simulation. It is found that the Bayes estimators are better than the corresponding MLEs for small samples (n_i ≤ 7; i = 1, 2). Moreover, the robustness of the Bayes estimators to the change of the prior parameters is also considered.     

Minimum Expected Loss Estimators of Reliability and Parameters of Certain Lifetime Distributions

The estimators of reliability and parameters of certain lifetime distributions which are widely used in reliability, repairability, and maintainability are obtained by using a different form of loss function and minimizing the s-expected loss with respect to the posterior distribution. These estimators are called MELO estimators. The applications, and the comparison between MELO, Bayes, and Maximum likelihood estimators are discussed.     

An Empirical Bayes Approach to Reliability

A Bayesian reliability estimation technique known as the ``empirical Bayes approach' is developed which uses previous experience nce to get a Bayesian point estimator. The techniques require no knowledge of the form of the unknown prior distribution and are robust to assumptions about its form. Empirical Bayes techniques are applicable to situations in which prior, independent observations of the random variable X from the random couple (?, X) are available where ? is the observed parameter of interest distributed in accordance with the unknown prior distribution. Performance comparisons of the empirical Bayes and other well established techniques are developed by examples for the binomial, exponential, Normal, and Poisson situations which often occur in reliability problems. In all cases the empirical Bayes estimator performed better than the classical estimator in minimizing the average squared error.     

Empirical Bayes Estimators of Reliability for Lognormal Failure Model

Empirical Bayes (EB) procedures are considered for estimating the reliability R(t;?,?) = gaufc[(ln t -?)/?] for the lognormal failure model. EB estimators are obtained for the 2 cases: i)? is unknown and ? is known, and both ? and ? are unknown. The empirical Cdf of the maximum likelihood estimators of the parameters is used to obtain the EB estimators. ii) A smooth EB estimator of R(t;?,?) is developed when ? is unknown and ? is known. A modification of this estimator is proposed for both ? and ? unknown. In both cases, EB estimators are obtained for complete samples at each testing stage. Monte Carlo simulations are presented to compare the EB estimators and the maximum likelihood (ML) estimators of R(t;?,?). The simulations indicate that the smooth EB estimators have smaller mean squared errors than the other EB estimators or the ML estimators.     

Series-system reliability-estimation using very small binomialsamples

This investigation explored the effect of incorporating prior information into series-system reliability estimates, where the inferences are made using very small sets (less than 10 observations) of binomial test-data. To capture this effect, the performance of a set of Bayes interval estimators was compared to that of a set of classical estimators over a wide range of subsystem beta prior-distribution parameters. During a Monte Carlo simulation, the Bayes estimators tended to provide shorter interval estimators when the mean of the prior system-reliability differed from the true reliability by 20 percent of less, but the classical estimators dominated when the difference was greater. Based on these results, the authors conclude that there is no clear advantage to using Bayes interval estimation for sample sizes less than 10 unless the poor mean system reliability is believed to be within 20 percent of the true system reliability. Otherwise, the Lindstrom-Madden estimator, a useful classical alternative for very small samples, should be used     

Modified `practical Bayes-estimators' [reliability theory]

This paper presents a new formulation of `practical Bayes-estimators' (PBE) for the 2-parameter Weibull model when both parameters are unknown. Overcoming some limitations of the first formulation gave rise to this work, but the results are beyond this intent. These estimators are a tool to improve technical knowledge by using a few experimental data. In this case, the controversy about whether to use Bayes or classical methods is surmounted since estimators, like maximum likelihood, give estimates that often appear unlikely on the basis of technical knowledge of the engineers. A Monte Carlo study supports the following conclusions: if the shape parameter is greater than one, modified PBE maintain the good properties of practical Bayes estimators; otherwise the modified PBE are much better and do not suffer from the past limitation regarding the formulation of the prior interval on the shape parameter itself; and when there are very few data the modified PBE work as a filter that always improves (on average) the prior information if it is poor, or substantially confirms it if it is good. From this viewpoint, Bayes theorem allows statistics to help engineering and not vice versa     

逐次截尾样本下电子元件混联系统可靠性指标的EB估计

在逐次截尾样本下,研究电子元件混联系统可靠性指标的估计问题。将Bayes方法和极大似然法相结合,在平方损失下,获得部件失效率、系统可靠度和平均寿命的经验Bayes估计。最后给出随机模拟例子,说明该方法的正确性。结果表明可靠性指标的经验Bayes估计值精度较高。     

Bayes and classical estimation of environmental factors for thebinomial distribution

An environmental factor converts reliability test results at one environmental condition into equivalent “failure” information at other environments. This paper studies environmental-factor estimation for the binomial distribution. Under general conditions, Bayes point estimates and credibility limits for environmental factors are derived. Classical point and confidence interval estimates are introduced and compared with the Bayes estimators. The characteristics of Bayes and classical estimators for the binomial distribution are summarized through numerical computation and theoretical analysis. A numerical example of reliability assessment by means of environmental factors is presented     

Estimation of constrained parameters in a linear model withmultiplicative and additive noise

Estimation of a Hilbert-space valued parameter in a linear model with compact linear transformation is considered with both multiplicative and additive noise present. The unknown parameter is assumed a priori to lie in a compact rectangular parallelepiped oriented in a certain way in the Hilbert space. Linear estimators are devised that minimize reasonable upper bounds on mean-squared error depending on conditions on the noise. Under prescribed conditions the estimators are minimax in the class of linear estimators. With the prior constraint on the unknown parameter removed, the estimation problem is ill-posed. Restricting the unknown provides a regularization of the basically ill-posed estimation. It turns out the estimators developed here belong to a well-known class of regularized estimators. With the interpretation that the constraint is soft, the procedure is applicable to many signal-processing problems     

Rethinking biased estimation

In this lecture note we discuss methods to improve the accuracy of unbiased estimators used in many signal processing problems. Our approach is based on introducing a bias as a means of reducing the mean-squared error (MSE). The important aspect of our framework is that the reduction in MSE is guaranteed for all values of the unknown parameter.