Different Estimation Methods for Constant Stress Accelerated Life Test under the Family of the Exponentiated Distributions

The Accelerated Life Testing (ALT) has been used for a long time in several fields to obtain information on the reliability of product components and materials under operating conditions in a much shorter time. One of the main purposes of applying ALT is to estimate the failure time functions and reliability performance under normal conditions. This paper concentrates on the estimation procedures under ALT and how to select the best estimation method that gives accurate estimates for the reliability function. For this purpose, different estimation methods are used, such as maximum likelihood, least squares (LS), weighted LS, and probability weighted moment. Moreover, the reliability function under usual conditions is predicted. The estimation procedures are applied under the family of the exponentiated distributions in general, and for the exponentiated inverted Weibull (EIW) as a special case. Numerical analysis including simulated data and a real life data set is conducted to compare the performances between these four methods. It is found that the ML method gives the best results among other estimation methods. Finally, a comparison between the EIW and the Inverted Weibull (IW) distributions based on a real life data set is made using a likelihood ratio test. It is observed that the EIW distribution can provide better fitting than the IW in case of ALT. Copyright © 2015 John Wiley & Sons, Ltd.     

A novel MTTF estimator and associated parameter estimation method on heavily censoring data

Many reliability and maintenance decision problems need to estimate mean-time-to-failure (MTTF) of a particular product component and/or build its life distribution model as early as possible based on field failure data. The field failure data are often heavily censored. In this case, the exponential-assumption–based method can considerably overestimate the MTTF, and the classical parameter estimation methods such as the maximum likelihood method (MLM) cannot provide robust estimates. This paper aims to address this issue through proposing a novel method to estimate the MTTF and life distribution parameters. The proposed method first derives a nonparametric estimator of MTTF based on the decomposition of the integral expression of the theoretical MTTF and sample statistical characteristics. The estimated MTTF is then combined with a two-step single-parameter MLM to estimate the distribution parameters. A numerical experiment is carried out, and a real-world dataset is analyzed. The results show that the proposed method can provide accurate and robust estimates for the MTTF and distribution parameters. The method is applicable for any life distribution and offers practitioners an efficient tool for reliability analysis.     

Probabilistic anisomorphic constant fatigue life diagram approach for prediction of P–S–N curves for woven carbon/epoxy laminates at any stress ratio

A general engineering methodology to construct a family of anisomorphic constant fatigue life (CFL) diagrams with probability of failure as the parameter that allows efficiently predicting P–S–N curves at any stress ratios is developed and validated for a plain weave fabric carbon/epoxy laminate. Constant amplitude fatigue tests are first performed to obtain statistical samples of fatigue life at different stress levels and stress ratios, respectively. Static tensile and compressive strength data are also collected. The Kolmogorov–Smirnov and Anderson–Darling goodness-of-fit tests suggest that both two-parameter lognormal and Weibull distributions are acceptable as the distributions for the static strength and fatigue life data, respectively, at the significance level of 5%. Then, we attempt to develop a methodology for efficient construction of the anisomorphic CFL diagrams for different constant values of probability of failure. It requires the P–S–N curves for any percentile points of the distribution for the critical stress ratio. To come up with this requirement, a probabilistic scaling law is formulated. It takes account of the probability-of-failure dependence of the critical stress ratio and the stress-ratio dependence of the P–S–N curve for the critical stress ratio. Finally, the anisomorphic CFL diagrams for different constant values of probability of failure are predicted using the proposed methodology, and they are shown to be in good agreement with the experimental results. It is also demonstrated that the P–S–N curves can efficiently and accurately be predicted for the woven CFRP laminate at any stress ratios using the proposed probabilistic anisomorphic CFL diagram approach.     

Two optimized models of life prediction based on luminance degradation for VFD under a conventional life test

Focusing on improving the accuracy of existing life prediction models for optoelectronic products, the three‐parameter Weibull right approximation method (TPWRAM) was employed to substitute exponential function based on the least square method in the analysis and two‐staged methods. Two optimized models were established (Model I and Model II), based on maximum likelihood estimation and the Monte Carlo method, respectively. One group of conventional life tests (CLTs) of vacuum fluorescent display (VFD) were conducted to collect luminance degradation data for each sample, and the two optimized models were applied to achieve VFD life prediction and obtain mean time to failure, median life, and confidence intervals. The results indicate that the CLT test design is correct and feasible, the amount of data on luminance degradation is large, and the test data selection method is reasonable. Model I and Model II optimized by TPWRAM both reflect the VFD luminance variation law well, and the predicted life approaches VFD service life from user feedback, proving that the two models are precise, and thus, can provide technical references for researchers and engineers regarding aspects of life prediction.     

Product Spacing of Stress–Strength under Progressive Hybrid Censored for Exponentiated-Gumbel Distribution

Maximum product spacing for stress–strength model based on progressive Type-II hybrid censored samples with different cases has been obtained. This paper deals with estimation of the stress strength reliability model R = P(Y < X) when the stress and strength are two independent exponentiated Gumbel distribution random variables with different shape parameters but having the same scale parameter. The stress–strength reliability model is estimated under progressive Type-II hybrid censoring samples. Two progressive Type-II hybrid censoring schemes were used, Case I: A sample size of stress is the equal sample size of strength, and same time of hybrid censoring, the product of spacing function under progressive Type-II hybrid censoring schemes. Case II: The sample size of stress is a different sample size of strength, in which the life-testing experiment with a progressive censoring scheme is terminated at a random time T ∈ (0,∞). The maximum likelihood estimation and maximum product spacing estimation methods under progressive Type-II hybrid censored samples for the stress strength model have been discussed. A comparison study with classical methods as the maximum likelihood estimation method is discussed. Furthermore, to compare the performance of various cases, Markov chain Monte Carlo simulation is conducted by using iterative procedures as Newton Raphson or conjugate-gradient procedures. Finally, two real datasets are analyzed for illustrative purposes, first data for the breaking strengths of jute fiber, and the second data for the waiting times before the service of the customers of two banks.     

Estimation of time‐to‐failure distribution derived from a degradation model using fuzzy clustering

Some life tests are terminated with few or no failures. In such cases, a recent approach is to obtain degradation measurements of product performance that may contain some useful information about product reliability. Generally degradation paths of products are modeled by a nonlinear regression model with random coefficients. If we can obtain the estimates of parameters under the model, then the time‐to‐failure distribution can be estimated. In some cases, the patterns of a few degradation paths are different from those of most degradation paths in a test. Therefore, this study develops a weighted method based on fuzzy clustering procedure to robust estimation of the underlying parameters and time‐to‐failure distribution. The method will be studied on a real data set. Copyright © 2000 John Wiley & Sons, Ltd.     

Generalized inverted exponential distribution under constant stress accelerated life test: Different estimation methods with application

Estimation of the parameters of generalized inverted exponential distribution is considered under constant stress accelerated life test. Besides the maximum likelihood method, nine different frequentist methods of estimation are used to estimate the unknown parameters. Moreover, the reliability function is estimated under use conditions based on different methods of estimation. We perform extensive simulation experiments to see the performance of the proposed estimators. As an illustration, a real data set is analyzed to demonstrate how the proposed methods may work in practice.     

Prediction of Time Series Empowered with a Novel SREKRLS Algorithm

For the unforced dynamical non-linear statespace model, a new Q1 and efficient square root extended kernel recursive least square estimation algorithm is developed in this article. The proposed algorithm lends itself towards the parallel implementation as in the FPGA systems. With the help of an ortho-normal triangularization method, which relies on numerically stable givens rotation, matrix inversion causes a computational burden, is reduced. Matrix computation possesses many excellent numerical properties such as singularity, symmetry, skew symmetry, and triangularity is achieved by using this algorithm. The proposed method is validated for the prediction of stationary and non-stationary MackeyGlass Time Series, along with that a component in the x-direction of the Lorenz Times Series is also predicted to illustrate its usefulness. By the learning curves regarding mean square error (MSE) are witnessed for demonstration with prediction performance of the proposed algorithm from where it’s concluded that the proposed algorithm performs better than EKRLS. This new SREKRLS based design positively offers an innovative era towards non-linear systolic arrays, which is efficient in developing very-large-scale integration (VLSI) applications with non-linear input data. Multiple experiments are carried out to validate the reliability, effectiveness, and applicability of the proposed algorithm and with different noise levels compared to the Extended kernel recursive least-squares (EKRLS) algorithm.     

Robust Regression using Probability Plots for Estimating the Weibull Shape Parameter

The Weibull shape parameter is important in reliability estimation as it characterizes the ageing property of the system. Hence, this parameter has to be estimated accurately. This paper presents a study of the efficiency of using robust regression methods over the ordinary least‐squares regression method based on a Weibull probability plot. The emphasis is on the estimation of the shape parameter of the two‐parameter Weibull distribution. Both the case of small data sets with outliers and the case of data sets with multiple‐censoring are considered. Maximum‐likelihood estimation is also compared with linear regression methods. Simulation results show that robust regression is an effective method in reducing bias and it performs well in most cases. Copyright © 2006 John Wiley & Sons, Ltd.     

Classical and Bayesian inference of Cpy for generalized Lindley distributed quality characteristic

The process capability index (PCI) is a quality control–related statistic mostly used in the manufacturing industry, which is used to assess the capability of some monitored process. It is of great significance to quality control engineers as it quantifies the relation between the actual performance of the process and the preset specifications of the product. Most of the traditional PCIs performed well when process follows the normal behaviour. However, using these traditional indices to evaluate a non‐normally distributed process often leads to inaccurate results. In this article, we consider a new PCI, C_py, suggested by Maiti et al, which can be used for normal as well as non‐normal random variables. This article addresses the different methods of estimation of the PCI C_py from both frequentist and Bayesian view points of generalized Lindley distribution suggested by Nadarajah et al. We briefly describe different frequentist approaches, namely, maximum likelihood estimators, least square and weighted least square estimators, and maximum product of spacings estimators. Next, we consider Bayes estimation under squared error loss function using gamma priors for both shape and scale parameters for the considered model. We use Tierney and Kadane's method as well as Markov Chain Monte Carlo procedure to compute approximate Bayes estimates. Besides, two parametric bootstrap confidence intervals using frequentist approaches are provided to compare with highest posterior density credible intervals. Furthermore, Monte Carlo simulation study has been carried out to compare the performances of the classical and the Bayes estimates of C_py in terms of mean squared errors along with the average width and coverage probabilities. Finally, two real data sets have been analysed for illustrative purposes.     

最小二乘法原理及其处理方法的探讨

本文通过对最小二乘法测量方程与残余误差方程进行描述,说明了最小二乘法的原理,明确了其在数据处理和误差计算方面的适用范围,在工程实际和实验技术中具有重要作用。     

An analytical procedure for estimating field lifetime and failure rate of electronic packages

To understand the reliability characteristics of electronic packages under field conditions, accelerated life tests (ALT) with higher stress levels are needed in practice. Instead of the time-consuming and costly ALT, an analytical procedure based on finite element simulation and a Weibull statistical method to estimate the lifetime and failure rate of electronic packages subjected to thermal cycling loadings is proposed in the present study. To consider uncertainties, geometric parameters and material properties are assumed as random variables and incorporated into numerical simulation. The result shows that the mean time to failure (MTTF) of a studied electronic package under a specific thermal cycling loading condition can be predicted accurately. From either the proposed analysis or based on a particular model found in literature, the acceleration factor (AF) can be predicted accurately as well. Furthermore, according to the outcome from the Weibull statistical method, the failure rate under either the field or a particular test condition can be determined. Accordingly, the MTTF and failure rate of the package under field conditions can be estimated from the result of a simulated accelerated test as well as the AF model. The present study indicates that the proposed analytical procedure can help engineers evaluate the reliability of electronic packages rapidly and effectively.     

Mechanical fatigue of an electronic component under random vibration

Electronic equipment, which is widely used in military applications, must be able to survive harsh environments. The endurance of such equipment is defined by the durability of their internal sensitive components. In this study, vibration induced fatigue life analysis of an axial leaded aluminium capacitor is performed. Three point bending tests are performed for the composite FR‐4 printed circuit boards (PCBs) material in order to determine bending modulus. Experimental modal analysis is used to validate a simulation model of the PCB. Step stress tests (SSTs) of reinforced and unreinforced capacitors which are mounted on the test PCBs are performed. It is found that the failure locations on the test PCBs are compatible among themselves and all the failures are due to flexure stress developed at the lead wires and solder joints. Numerical fatigue analyses are performed to define failure in terms of damage index. In addition, the Weibull model is used to define mean time to failure (MTTF) values. The comparison between MTTF values shows that the fatigue lives are strongly increased by the eccobond reinforcement. The last stage in this work is to focus on the influence of some design parameters on the fatigue life. An exponential equation is proposed to find the relation between lead‐wire diameter and the fatigue damage. It is shown that fatigue damage becomes a maximum for a square shaped PCB and it appears that component body diameter is more effective than the body length in increasing fatigue life.     

基于质感要素的产品材质意象设计研究

目的为了探讨产品材质意象设计的有效方法,研究利用质感要素来构建材质意象评价体系。方法利用色彩理论、主成分分析法等方法,确定产品色彩、材料、表面处理工艺的质感要素、质感要素评价等级和质感要素评价系数。利用质感要素定量描述确定材质、质感要素、材质意象的数学表达式,并构建三者之间的关系模型。结果以水杯为例,利用最小二乘法确定水杯材质意象评价模型。经验证,水杯材质意象评价模型的准确率为80%。结论质感要素评价等级、质感要素评价系数为材质意象设计提供了参数化方法。产品材质意象关系模型较好地反映了材质、质感要素、材质意象之间的内在关系。利用提出的材质意象设计方法,可计算任意产品的材质意象值,为产品材质意象设计提供理论指导。     

Statistical Inference on Constant Stress Accelerated Life Tests under Generalized Gamma Lifetime Distributions

We will discuss the reliability analysis of the constant stress accelerated life tests when a parameter in the generalized gamma lifetime distribution is linear in the stress level. Statistical inference on the estimation of the underlying model parameters as well as the mean time to failure and the reliability function will be addressed on the basis of the maximum likelihood approach. Large sample theory will be derived for the goodness of fit of the data. Some simulation study and an illustrative real example will be presented to show the appropriateness of the proposed method. Copyright © 2012 John Wiley & Sons, Ltd.     

Random fatigue life prediction of carbon fibre-reinforced composite laminate based on hybrid time-frequency domain method

To evaluate fatigue life of composite laminate with hole under random loading, a random fatigue life prediction model is established by hybrid time-frequency domain method in this paper. Firstly, dynamic response of composite laminate is obtained from FE model in frequency domain. Secondly, root mean square of stress of six stress components of critical damage point in frequency domain are transferred to stresses in time domain. At last, 3D Tsai–Hill static failure criterion is adopted to convert the multiaxial stress into the uniaxial equivalent stress. Fatigue life is predicted by equivalent stress fatigue life code. The method is validated with the random vibration fatigue test of carbon fibre-reinforced composite laminate. Numerical results are compared with random fatigue experiments which show good agreement with numerical results.     

聚氯乙烯薄膜使用寿命预测

文中对聚氯乙烯(PVC)薄膜在实际应用环境中的使用寿命进行了快速可靠的预测。通过加速寿命试验,以断裂伸长率下降至其初始值的50%为失效指标,对PVC薄膜在户外环境因子(紫外光和热)协同作用下的老化动力学进行了研究,利用威布尔统计方程分析老化失效数据,并使用Temperature-non-thermal模型建立外推关系式,确立了PVC薄膜的寿命-环境因子统计学关系式,模拟了PVC薄膜寿命分布在环境因子(热和强紫外光)协同作用下的变化规律。最后,应用上述寿命-环境因子关系式,对PVC薄膜在上海地区的户外使用寿命进行了预测,得出150μm厚的PVC薄膜曝露于上海地区时,户外使用寿命为13个月。     

基于GM-噪声SVR方法的矿山设备可靠性参数估计

为了有效估计小子样条件下矿山设备的三参数威布尔分布可靠性模型参数,提出基于GM-噪声SVR的参数估计方法。该方法以灰色估计法(GM)为基础估计模型的位置参数,采用基于训练样本数量和噪声参数寻优的ε - 带支持向量回归机(ε-SVR)估计尺度参数和形状参数,并通过拟合的三参数威布尔分布函数分析预测和解决设备的可靠性问题。算例结果表明,GM-噪声SVR方法可以很好地用于矿山设备可靠性模型参数估计,估计某带式输送机三参数威布尔分布可靠性模型的位置参数、尺度参数和形状参数依次为3.1525、188.3763、1.0476,平均无故障时间为188 h,标准均方根误差NRMSE为0.0519。这表明该方法的可行性和有效性。     

Iterative ultrasonic signal and image deconvolution for estimation of the complex medium response

The ill‐conditioned inverse problem of estimating ultrasonic medium responses by deconvolution of RF signals is investigated. The primary difference between the proposed method and others is that the medium response function is assumed to be complex‐valued rather than restricted to being real‐valued. Derived from the complex medium model, complex Wiener filtering is presented, and a Hilbert transform related limitation to inverse filtering type methods is discussed. We introduce a nonparametric iterative algorithm, the least squares method with point count regularization (LSPC). The algorithm is successfully applied to simulated and experimental data and demonstrates the capability of recovering both the real and imaginary parts of the medium response. The simulation results indicate that the LSPC method can outperform Wiener filters and improve the resolution of the ultrasound system by factors as high as 3.7. Experimental results using a single element transducer and a conventional medical ultrasound system with a linear array transducer show that despite the errors in pulse estimation and the noise in the RF signals, excellent results can be obtained, demonstrating the stability and robustness of the algorithm. © 2006 Wiley Periodicals, Inc. Int J Imaging Syst Technol, 15, 266–277, 2005     

Optimization research on S‐N curve of ring welding structure based on structural stress method

In engineering, △F‐N curves are usually used to predict the fatigue life of ring welding, which is time‐consuming, laborious, and not universal. To improve the above inadequacies, an S‐N curve for evaluating the fatigue life of the ring welded specimen is proposed. The fatigue life of ring welded specimens with different materials, plate thicknesses, and hole diameters is obtained by tensile and shear fatigue tests. Shell elements, CBar beam elements (a kind of beam element in Nastran that can simulate bending), and rigid elements are used to establish the finite element model of ring welding. The stress of the ring welding structure under tensile shear load is calculated according to the structural stress method. The stress range △ σs of the welding core is taken as the longitudinal coordinate and the experimental life N as the horizontal coordinate, using two‐parameter log‐log model and the least square method of the fatigue data for linear fitting to obtain the S‐N curve equation of fatigue life evaluation. Most of the data are located within five times of the life span, which proves that the predicted life is close to the actual life of the test, and it can provide a certain reference for design and life prediction of the ring welding structures.