Duane-LR模型下复杂系统的动态可靠性增长评定

基于复杂系统可靠性增长试验的特点，运用Duane可靠性增长模型结合数理统计中的线性回归方法对新批次产品的可靠性参数进行预测。结合产品的少量现场试验数据，利用Bayes方法对系统的可靠性增长试验结果进行评定。文中首先给出了可靠性增长分析的模型，然后运用历次阶段试验中的可靠性增长数据建立动态参数的递推估计模型，在此基础上，运用随机变量函数的分布，给出各阶段可靠性增长试验中可靠性参数的Bayes估计。文中对Weibull、指数和二项分布三种试验结果进行分析，给出计算公式。     

威布尔分布无失效数据的统计分析

本文对Weibull分布场合下的无失效数据(ti,ni),根据“平均剩余寿命”这一概念得到了参数的拟矩估计,进而将其转化至有一个或多个失效数据的情形,利用[1]中的结果给出了失效概率pi的多层Bayes估计,从而利用分布函数曲线拟合方法得到了未知参数的估计．并结合实际问题进行了计算．     

指数分布场合下无失效数据的统计分析

由于产品的可靠性愈来愈高，所以在可靠性寿命试验中，“无失效数据”的现象也愈来愈多．本文根据指数分布的无记忆性，给出可靠度的先验分布，进而在无失效数据的情况下，得到了平均寿命的Bayes估计。     

无失效数据情形可靠性参数的估计和调整

本文在无失效取样情形下,提出了产品可靠性参数的一种估计和调整的方法———加权多层Bayes估计法.在无失效数据情形下失效率的多层Bayes估计和引进失效信息后失效率的多层Bayes估计的基础上,对可靠性参数进行了估计和调整———给出了失效率和可靠度的加权多层Bayes估计.最后,结合发动机的实际问题进行了计算,结果表明本文提出的方法可行且便于应用.     

Weibull分布步进应力加速寿命试验的Bayes估计

本文研究了定时和定数截尾情形CE模型下Weibull分布场合步进应力加速寿命试验的Bayes估计.利用加速系数和加速方程将各种加速应力水平下的尺度参数换算为正常应力水平下的尺度参数,从而获得含正常应力下尺度参数的似然函数.在参数先验的选取时,尺度参数和加速系数分别取共轭先验和无信息先验,当形状参数m<1和m>1时分别取Beta分布和Gamma分布作为其先验.在平方损失下,利用Gibbs抽样和切片抽样给出了该模型参数的Bayes估计.最后,通过Monte Carlo模拟表明该Bayes估计是有效的.     

删失截断情形下Weibull分布多变点模型的参数估计

通过添加缺失的寿命变量数据,得到了删失截断情形下Weibull分布多变点模型的完全数据似然函数,研究了变点位置参数和形状参数以及尺度参数的满条件分布.利用Gibbs抽样与Metropolis-Hastings算法相结合的MCMC方法得到了参数的Gibbs样本,把Gibbs样本的均值作为各参数的Bayes估计.详细介绍了MCMC方法的实施步骤.随机模拟试验的结果表明各参数Bayes估计的精度都较高.     

第三讲:Weibull分布和对数正态分布情形的可靠性评定

对于产品寿命服从Weibull分布或对数正态分布的情形,针对几种不同类型的数据(例如随机截尾,定数截尾情形出现的数据),分别给出了可靠性参数(可靠度、可靠寿命)的点估计或置信下限。特别是在定时截尾出现零失效情形,给出了可靠性参数的置信下限的计算公式。     

Weibull过程将来第k次失效时间的Bayes预测区间

具Weibull强度函数的非齐次Poisson过程经常被用来分析可修系统的失效模式.基于极大似然估计,Engelhardt ＆ Bain(1978)导出了Weibull过程将来第k次失效时间的经典预测区间.在本文中,我们用无信息联合验前分布,根据Weibull过程的前n次失效时间,给出了建立将来第k次失效时间的Bayes预测区间的方法,并说明了如何应用这些方法。     

无失效数据情形参数的综合估计

本对指数分布的无失效数据,在引进失效信息后,在先验分布为Gamma分布时,给出了失效率的多层Bayes估计和综合Bayes估计,并给出了无失效数据情形可靠度的综合估计,还结合实际问题进行了计算。     

随机时截尾寿命试验Weibull分布的Bayes统计分析

本文考虑形状参数为离散型随机变量,尺度参数为连续型随机变量,然后通过最大熵原则确定形状参数的先验分布,用共轭分布法确定形状参数给定的条件下尺度参数的条件先验分布,并利用随机时截尾寿命试验中所获的数据给出weibull分布参数,可靠度和失效率的Bayes点估计及其置信上、下限。     

威布尔分布的Bayes可靠性分析

本文针对两参数的威布尔分布的元件，采用Ｂａｙｅｓ方法对其可靠性进行分析，文中分别对两种假设情况进行了讨论，第一种情况是形状参数为离散取值，尺度参数为连续取值的情况，第二种情况假设形状参数的先验分布为均匀分布，尺度参数的先验分布为逆伽玛分布，推出了对应的可靠性估计和置信下限估计，并给出了计算算法，最后用实例对算法进行了验证。     

Reliability tests for Weibull distribution with variational shape parameter based on sudden death lifetime data

In the traditional design of reliability tests for assuring the mean time to failure (MTTF) in Weibull distribution with shape and scale parameters, it has been assumed that the shape parameter in the acceptable and rejectable populations is the same fixed number. For the purpose of expanding applicability of the reliability testing, Hisada and Arizono have developed a reliability sampling scheme for assuring MTTF in the Weibull distribution under the conditions that shape parameters in the both populations do not necessarily coincide, and are specified as interval values, respectively. Then, their reliability test is designed using the complete lifetime data. In general, the reliability testing based on the complete lifetime data requires the long testing time. As a consequence, the testing cost becomes sometimes expensive. In this paper, for the purpose of an economical plan of the reliability test, we consider the sudden death procedure for assuring MTTF in Weibull distribution with variational shape parameter.     

CE模型下Weibull分布序加试验的Bayes分析

序进应力加速寿命试验是一种最为有效而经济的寿命试验方法,随着其理论的日趋成熟,在实践中开始得到应用和推广．本文给出了逆幂律模型下Weilbull分布定时和定数场合序进应力加速寿命试验的一种Bayes统计分析,并利用Gibbs抽样方法解决了分布的形状参数取为连续先验时各参数的Bayes估计．这种先验意义更明确,实例表明这是一种非常有效的方法．     

Fitting the three-parameter Weibull distribution with Cross Entropy

The Weibull distribution is widely used in applications such as reliability and lifetime studies. Although this distribution has three parameters, for simplicity, literature pertaining to Weibull parameter estimation relaxes one of its parameters in order to estimate the other two. When the three-parameter Weibull distribution is of interest, the estimation procedure is complicated. For example, the likelihood function for a three-parameter Weibull distribution is hard to maximize. In this paper, a Cross Entropy (CE) method is developed in the context of maximum likelihood estimation (MLE) of a three-parameter Weibull distribution. Performing a simulation study, a comparative analysis between the newly developed method and two existing methods is conducted. The results show the proposed method has better performance in terms of accuracy, precision and run time for different parameter settings and sample sizes.     

Noninformative priors for the ratio of the shape parameters of two Weibull distributions

The Weibull distribution is one of the most widely used lifetime distributions in reliability engineering. Here, the noninformative priors for the ratio of the shape parameters of two Weibull models are introduced. The first criterion used is the asymptotic matching of the coverage probabilities of Bayesian credible intervals with the corresponding frequentist coverage probabilities. We develop the probability matching priors for the ratio of the shape parameters using the following matching criteria: quantile matching, matching of the distribution function, highest posterior density matching, and matching via inversion of the test statistics. We obtain one particular prior that meets all the matching criteria. Next, we derive the reference priors for different groups of ordering. Our findings show that some of the reference priors satisfy a first-order matching criterion and the one-at-a-time reference prior is a second-order matching prior. Lastly, we perform a simulation study and provide a real-world example.     

A new approach for parameter estimation of finite Weibull mixture distributions for reliability modeling

The aim of this paper is to model lifetime data for systems that have failure modes by using the finite mixture of Weibull distributions. It involves estimating of the unknown parameters which is an important task in statistics, especially in life testing and reliability analysis. The proposed approach depends on different methods that will be used to develop the estimates such as MLE through the EM algorithm. In addition, Bayesian estimations will be investigated and some other extensions such as Graphic, Non-Linear Median Rank Regression and Monte Carlo simulation methods can be used to model the system under consideration. A numerical application will be used through the proposed approach. This paper also presents a comparison of the fitted probability density functions, reliability functions and hazard functions of the 3-parameter Weibull and Weibull mixture distributions using the proposed approach and other conventional methods which characterize the distribution of failure times for the system components. GOF is used to determine the best distribution for modeling lifetime data, the priority will be for the proposed approach which has more accurate parameter estimates.     

Bayesian Inference for Extremes: Accounting for the Three Extremal Types

The Extremal Types Theorem identifies three distinct types of extremal behaviour. Two different strategies for statistical inference for extreme values have been developed to exploit this asymptotic representation. One strategy uses a model for which the three types are combined into a unified parametric family with the shape parameter of the family determining the type: positive (Fréchet), zero (Gumbel), and negative (negative Weibull). This form of approach never selects the Gumbel type as that type is reduced to a single point in a continuous parameter space. The other strategy first selects the extremal type, based on hypothesis tests, and then estimates the best fitting model within the selected type. Such approaches ignore the uncertainty of the choice of extremal type on the subsequent inference. We overcome these deficiencies by applying the Bayesian inferential framework to an extended model which explicitly allocates a non-zero probability to the Gumbel type. Application of our procedure suggests that the effect of incorporating the knowledge of the Extremal Types Theorem into the inference for extreme values is to reduce uncertainty, with the degree of reduction depending on the shape parameter of the true extremal distribution and the prior weight given to the Gumbel type.     

BAYESIAN ANALYSIS OF DATA WITH ONLY ONE FAILURE

The hearings of a certain type have their lives following a Weibull distribution. In a life test with 20 sets of bearings, only one set failed within the specified time, and none of the remainder failed even after the time of to estimate the reliabilWith a set of testing data like that in Table 1, it is required to estimate the reliability at the mission time, In this paper, we first use hierarchical Bayesian method of determine the prior distribution and the Bayesian estimates of various probabilities of failures, pi‘s, then use the method of least squares to estimate the parameters of the Weibull distribution and the reliability. Actual computation shows that the estimates so obtained are rather robust. And the results have been adopted for practical use.     

基于混合Ⅰ型删失数据的威布尔模型精确推断（英文）

威布尔分布是可靠性和寿命测试试验中常用的模型.本文中,我们考虑了基于混合Ⅰ型删失数据的威布尔模型精确推断.我们得到了威布尔分布未知参数最大似然估计的精确分布以及基于精确分布的置信区间.由于精确分布函数较为复杂,我们也给出了未知参数的另外几种置信区间,比如,基于近似方法的置信区间,Bootstrap置信区间.为了评价本文的方法,我们给出了一些数值模拟的结果.     

On a proper way to select population failure distribution and a stochastic optimization method in parameter estimation

It is widely accepted that the Weibull distribution plays an important role in reliability applications. The reliability of a product or a system is the probability that the product or the system will still function for a specified time period when operating under some confined conditions. Parameter estimation for the three parameter Weibull distribution has been studied by many researchers in the past. Maximum likelihood has traditionally been the main method of estimation for Weibull parameters along with other recently proposed hybrids of optimization methods. In this paper, we use a stochastic optimization method called the Markov Chain Monte Carlo (MCMC) to carry out the estimation. The method is extremely flexible and inference for any quantity of interest is easily obtained.