A practical method for obtaining prior distributions in reliability

In this paper, we propose a comprehensive methodology to specify prior distributions for commonly used models in reliability. The methodology is based on characteristics easy to communicate by the user in terms of time to failure. This information could be in the form of intervals for the mean and standard deviation, or quantiles for the failure-time distribution. The derivation of the prior distribution is done for two families of proper initial distributions, namely s-normal-gamma, and uniform distribution. We show the implementation of the proposed method to the parameters of the s-normal, lognormal, extreme value, Weibull, and exponential models. Then we show the application of the procedure to two examples appearing in the reliability literature, and . By estimating the prior predictive density, we find that the proposed method renders consistent distributions for the different models that fulfill the required characteristics for the time to failure. This feature is particularly important in the application of the Bayesian approach to different inference problems in reliability, model selection being an important example. The method is general, and hence it may be extended to other models not mentioned in this paper.     

Bayesian Analysis of the Weibull Process With Unknown Scale and Shape Parameters

Previously, the Weibull process with an unknown scale parameter was examined as a model for Bayesian decision making. The analysis is extended by treating both the shape and scale parameters as unknown. It is not possible to find a family of continuous joint prior distributions on the two parameters that is closed under sampling, so a family of prior distributions is used that places continuous distributions on the scale parameter and discrete distributions on the shape parameter. Prior and posterior analyses are examined and seen to be no more difficult than for the case in which only the scale parameter is treated as unknown, but preposterior analysis and determination of optimal sampling plans are considerably more complicated in this case. To illustrate the use of the present model, an example is presented in which it is necessary to make probability statements about the mean life and reliability of a long-life component both before and after life testing.     

Reliability tests for Weibull distribution with varyingshape-parameter, based on complete data

The Weibull distribution indexed by scale and shape parameters is generally used as a distribution of lifetime. In determining whether or not a production lot is accepted, one wants the most effective sample size and the acceptance criterion for the specified producer and consumer risks. (μ₀ ≡ acceptable MTTF; μ₁ ≡ rejectable MTTF). Decide on the most effective reliability test satisfying both constraints: Pr{reject a lot | MTTF = μ₀} ⩽ α, Pr{accept a lot | MTTF = μ₁ } ⩽ β. α, β are the specified producer, consumer risks. Most reliability tests for assuring MTTF in the Weibull distribution assume that the shape parameter is a known constant. Thus such a reliability test for assuring MTTF in Weibull distribution is concerned only with the scale parameter. However, this paper assumes that there can be a difference between the shape parameter in the acceptable distribution and in the rejectable distribution, and that both the shape parameters are respectively specified as interval estimates. This paper proposes a procedure for designing the most effective reliability test, considering the specified producer and consumer risks for assuring MTTF when the shape parameters do not necessarily coincide with the acceptable distribution and the rejectable distribution, and are specified with the range. This paper assumes that α < 0.5 and β < 0.5. This paper confirms that the procedure for designing the reliability test proposed here applies is practical     

Life Distributions Derived from Stochastic Hazard Functions

Most of the familiar time-to-failure distributions used today are derived from hazard functions whose parameters are assumed constant. An unconditional time-to-failure distribution is derived here by assuming that a parameter of a classical failure distribution (viz., exponential and Weibull) is a random variable with a known distribution. With the use of the derived compound distributions and Bayesian techniques, it is possible to join the test data with prior information to arrive at a combined, and possibly superior, estimate of reliability. The prior distributions considered here are the two-point, the uniform, and the gamma. Conceptually, such a scheme may be a more realistic model for describing failure patterns under specific conditions.     

Weibull分布产品恒加应力缺失数据下的Bayes可靠性评估

在加速寿命试验过程中,由于试验设备、观测手段或其他方面的困难可能会造成某些试验数据丢失或未观测到.为解决Weibull分布产品在恒加应力试验中出现的小子样缺失数据情形下的可靠性评估问题,提出了可以综合利用多源信息的Bayes可靠性评估方法.首先通过概率元方法得到缺失数据的似然函数,同时根据似然函数中各未知参数的物理含义确定其验前分布类型,再利用第二类极大似然估计原理得到验前分布中超参数的估计.最后通过仿真实例说明了该评估方法在小子样缺失数据情形下的有效性.     

An engineering approach to Bayes estimation for the Weibull distribution

In this paper an engineering approach to Bayes reliability analysis of Weibull failure data collected under a randomly censored sampling is proposed. The posterior distribution of several decision variables, such as the meanlife, the reliability function, the reliable life, and the hazard rate, are derived, when either a prior information on the reliability or a prior information on the hazard rate is available. Point estimates of the selected decision variables are given, by assuming both symmetric and asymmetric loss functions. Finally, numerical examples are presented to illustrate the proposed estimation procedures.     

Bayesian sequential estimation of two parameters of a Weibull distribution

Weibull distribution is one of the most widely used model for failure data in reliability studies. In this paper a sequential estimation procedure for estimating the parameters of Weibull distribution is proposed, which is, in principle similar to Kalman filtering. The main advantage of this approach is that it shows the variation of parameters over a time as new failure data becomes available to the analyst for estimation. Also once an available data has been used, the method does not require that data for further processing as and when the new data becomes available for updating the estimates of parameters. Its use in Quality control asa control chart has been indicated and the procedure is illustrated with the help of examples.     

不完全先验信息条件下的可靠性评估

小子样产品可靠性的Bayes评估中通常需要用到主观经验信息,如可靠度均值或可信区间等,这些信息属于不完全先验信息,利用这些信息通常无法确定可靠性分布函数。基于Bayes理论,以贝塔分布作为先验分布类型,利用最大熵原理将不完全信息转化为完全型先验信息,得到产品可靠性的先验分布,再结合观测数据,利用Bayes公式得到产品可靠性后验分布。从仿真算例可以看出,给出的方法能够有效地处理不完全先验信息,提高产品可靠性评估的效率。     

An alternative degradation reliability modeling approach using maximum likelihood estimation

An alternative degradation reliability modeling approach is presented in this paper. This approach extends the graphical approach used by several authors by considering the natural ordering of performance degradation data using a truncated Weibull distribution. Maximum Likelihood Estimation is used to provide a one-step method to estimate the model's parameters. A closed form expression of the likelihood function is derived for a two-parameter truncated Weibull distribution with time-independent shape parameter. A semi-numerical method is presented for the truncated Weibull distribution with a time-dependent shape parameter. Numerical studies of generated data suggest that the proposed approach provides reasonable estimates even for small sample sizes. The analysis of fatigue data shows that the proposed approach yields a good match of the crack length mean value curve obtained using the path curve approach and better results than those obtained using the graphical approach.     

Effect of randomness of Cu-Sn intermetallic compound layerthickness on reliability of surface mount solder joints

A statistical reliability analysis on thermal fatigue lifetime of surface mount solder joints, considering randomness of Cu-Sn intermetallic compound (IMC) layer thickness, is presented. Based on published thermal fatigue life test data, the two-parameter Weibull distribution of the thermal fatigue lifetime for a fixed IMC layer thickness is found, and a K-S goodness-of-fit test is conducted to examine the goodness of fit of the assumed Weibull distribution. Then, the Weibull parameters as functions of IMC layer thickness are obtained. Considering the randomness of IMC layer thickness, the MTTF and reliability of surface mount solder joints on thermal cycles are analyzed. For surface mount solder joints formed under the same conditions and loaded during the same thermal cycling as stated in the publication, numerical results of the MTTF and reliability are presented. The results show that when the mean value of MC layer thickness is low (e.g., smaller than 1.5 μm), the effect of randomness of IMC layer thickness is significant; i.e., the MTTF has strong dependence on IMC layer thickness distribution; and the reliability is significantly different at high thermal cycles. When the mean value of IMC layer thickness is high (e.g., greater than 2.0 μm), the effect of randomness of IMC layer thickness is negligible. Therefore, the presented results are important to the reliability study of surface mount solder joints. Even though the validity of the presented results based on the test data remains to be verified from other sources of data, the proposed statistical method is generally applicable for thermal fatigue reliability analysis of surface mount solder joints. By combining the proposed method with the forming mechanism of IMC layer under varying manufacturing and loading conditions, a comprehensive reliability analysis on thermal fatigue lifetime of surface mount solder joints can be expected     

Weibull分布下的系统可靠性评估方法研究

在Weibull分布的定时截尾样本中,对可靠性、可靠寿命和失效率这3种参数的验前分布及形状参数的验前信息进行了分析。论述了结合熵损失函数来求得系统可靠度及寿命的Bayes点估计和置信下限,为大型系统的可靠性评估提供了一种重要的理论依据。     

The α-μ Distribution: A Physical Fading Model for the Stacy Distribution

This paper introduces a fading model, which explores the nonlinearity of the propagation medium. It derives the corresponding fading distribution-the alpha-mu distribution-which is in fact a rewritten form of the Stacy (generalized Gamma) distribution. This distribution includes several others such as Gamma (and its discrete versions Erlang and central Chi-squared), Nakagami-m (and its discrete version Chi), exponential, Weibull, one-sided Gaussian, and Rayleigh. Based on the fading model proposed here, higher order statistics are obtained in closed-form formulas. More specifically, level-crossing rate, average fade duration, and joint statistics (joint probability density function, general joint moments, and general correlation coefficient) of correlated alpha-mu variates are obtained, and they are directly related to the physical fading parameters     

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.     

Bayes credibility intervals for the left-truncated exponential distribution

In this paper, a Bayes approach for statistical inference on life characteristics is proposed, when the underlying lifetime distribution has the left-truncated exponential density function. The proposed Bayes procedure provides credibility intervals on several life characteristics of great interest to the applied reliability engineer, when the experimental data are collected under a randomly censored sampling. The prior technical knowledge is expressed in the form of a prior density on the reliability level at a prefixed time in conjunction with an upper bound on the location parameter. The statistical properties of the proposed Bayes procedure are compared, via Monte Carlo simulation, with those of the Bayes procedure under the noninformative prior, when both correct and uncorrect prior information on the reliability is available. A numerical example is used for illustration and comparison.     

基于LSSVM的威布尔分布形状参数估计

固态介质击穿寿命特性通常用威布尔分布来描述,形状参数卢反应了固态介质的失效特征,因而需要精确估计β值.提出了在小样本情况下基于最小二乘支持向量机(LSSVM)的参数评估方法,并给出了LSSVM在MOS电容与时间有关的击穿寿命分布评估中的应用实例,并与常规的最小二乘评估方法相比,得到的结果表明LSSVM的评估精度更高(均方误差更小)、鲁棒性更好,在小样本情况下能更精确地确定威布尔分布的形状参数.     

Bayesian Estimation of Parameters of Life Distributions and Reliability from Type II Censored Samples

The Bayesian approach to reliability estimation from Type II censored samples is discussed here with emphasis on obtaining natural conjugate prior distributions. The underlying sampling distribution from which the censored samples are drawn follows a generalized life model (GLM) which includes a model proposed by Epstein and Sobel, Weibull, exponential, and Rayleigh distributions as special cases. Results are given for the Type II asymptotic distribution of largest values, Pareto, and Limited distribution. The natural conjugate prior, Bayes estimate for the generalized scale parameter, posterior risk, Bayes risk and Bayes estimate of the reliability function were derived for the distributions studied. In every case the natural conjugate prior is a 2-parameter family which provides a wide range of possible prior knowledge. Conjugate diffuse priors were derived. A diffuse prior, also called a quasi-pdf, is not a pdf because its integral is not unity. It represents roughly an informationless prior state of knowledge. The proper choice of the parameter for the diffuse prior leads to maximum likelihood, classical uniform minimum-variance unbiased estimator, and an admissible biased estimator with minimum mean square error as the generalized Bayes estimate. A feature of the GLM is the increasing function g(·) with possible applications in accelerated testing. KG(·) is a s-complete s-sufficient statistic for ?, and KG(·)/m is a maximum likelihood estimate for ?. Similar results were obtained for the Pareto, Type II asymptotic distribution of extremes, Pareto (associated with Pearl-Reed growth distribution) and others.     

A Bayes method for assessing product-reliability during developmenttesting

A fully Bayes approach is presented for analyzing product reliability during the development phase. Based on a Bayes version of the Barlow-Scheuer reliability-growth model, it is assumed that the product goes through a series of test/modification stages, where each product test yields attribute (pass-fail) data, and failure types are classified as fixable or nonfixable. Relevant information on both the failure probabilities and the reliability-growth process is used to motivate the prior joint distribution for the probability of each failure type over the specified range of testing. Results at a particular test-stage can be used to update the knowledge about the probability of each failure type (and thus product reliability) at the current test-stage as well as at subsequent test-stages, and at the end of the development phase. A relative ease of incorporation of prior information and a tractability of the posterior analysis are accomplished by using a Dirichlet distribution as the prior distribution for a transformation of the failure probabilities     

The use of imprecise component reliability distributions inreliability calculations

System reliability prediction can be needed when detailed data concerning each component are unavailable. This prevents data from various sources from being integrated by considering each source as a sample from the same population. For the Weibull distribution, fuzzy arithmetic is used to synthesize a single fuzzy probability distribution for a component from several possible distributions by considering particular features that map to the Weibull parameters. The use of fuzzified probabilities in fault trees has been considered by others; once the fuzzy component failure probabilities have been calculated, a fuzzy system reliability distribution can be obtained which portrays the inexact nature of the data on which the result is based     

Inverse gaussian distribution and its application to reliability

This paper discusses the problem of estimating parameters for the inverse Gaussian distribution when early failures dominate the problem. Since the Weibull distribution is a well-known competitor of the inverse Gaussian distribution, the estimate for the reliability function can be obtained by using the Weibull distribution. Simulation studies are performed to compare reliability by using both inverse Gaussian and Weibull distributions. Various implications of these results are discussed.     

Properties of a multivariate survival distribution generated by aWeibull and inverse-Gaussian mixture

The authors consider the influence of the work environment on a system of nonrenewable components. The failure times for the components are Weibull distributed and the work environment has an inverse Gaussian distribution. A multivariate Weibull and inverse Gaussian mixture distribution is derived. Several pertinent properties for this multivariate distribution are discussed that shed some light on the nature of the distribution. The authors account for the operating environment and its changing nature by averaging over a parameter corresponding to the environment. The distribution is applied to find the mean number of components working at some mission time and the reliability for k-out-of-n components