Estimation of the Parameters of a Mixture of Exponential and Weibull Distributions with Progressive Censoring

A method is proposed for estimating the parameters of a mixture of exponential and Weibull distributions using censored samples. Preliminary estimates obtained by graphical analysis are refined by the method of maximum likelihood. The efficiency of the method is confirmed by the results of a statistical modeling.     

A Universal Distribution for Component Failure Times

It is proposed to use a mixture of Weibull and exponential distributions to describe component reliability on the basis of a realistic model for resource consumption.     

Maximum likelihood Estimation of Parameters in the Inverse Gaussian Distribution,With Unknown Origin

Maximum likelihood estimation is applied to the three-parameter Inverse Gaussian distribution, which includes an unknown shifted origin parameter. It is well known that for similar distributions in which the origin is unknown, such as the lognormal, gamma, and Weibull distributions, maximum likelihood estimation can break down. In these latter cases, the likelihood function is unbounded and this leads to inconsistent estimators or estimators not asymptotically normal. It is shown that in the case of the Inverse Gaussian distribution this difticulty does not arise. The likelihood remains bounded and maximum likelihood estimation yields a consistent estimator with the usual asymptotic normality properties. A simple iterative method is suggested for the estimation procedure. Numerical examples are given in which the estimates in the Inverse Gaussian model are compared with those of the lognormal and Weibull distributions.     

Parameter and Quantile Estimation for the Generalized Pareto Distribution

The generalized Pareto distribution is a two-parameter distribution that contains uniform, exponential, and Pareto distributions as special cases. It has applications in a number of fields, including reliability studies and the analysis of environmental extreme events. Maximum likelihood estimation of the generalized Pareto distribution has previously been considered in the literature, but we show, using computer simulation, that, unless the sample size is 500 or more, estimators derived by the method of moments or the method of probability-weighted moments are more reliable. We also use computer simulation to assess the accuracy of confidence intervals for the parameters and quantiles of the generalized Pareto distribution.     

Maximum Likelihood Parameter Estimation for the Quartic Exponential Distribution

The quartic exponential (QE) distribution defined by the probability density function of the type

is examined in detail.

The problem of obtaining maximum likelihood point estimates of the population parameters reduces to that of identifying the α as functions of the population moments μ _r ′, r = 1, 2.3.4.

The invalidity is explained of methods proposed by previous authors to deal with the nonlinear relationships involved, and a new algorithm is developed which overcomes these objections. The new algorithm is applied to practical data, and the resulting distributions fitted to observed frequencies are shown to compare favourably with those obtained by previous Methods.     

Zubair Lomax Distribution: Properties and Estimation Based on Ranked Set Sampling

In this article, we offer a new adapted model with three parameters, called Zubair Lomax distribution. The new model can be very useful in analyzing and modeling real data and provides better fits than some others new models. Primary properties of the Zubair Lomax model are determined by moments, probability weighted moments, Renyi entropy, quantile function and stochastic ordering, among others. Maximum likelihood method is used to estimate the population parameters, owing to simple random sample and ranked set sampling schemes. The behavior of the maximum likelihood estimates for the model parameters is studied using Monte Carlo simulation. Criteria measures including biases, mean square errors and relative efficiencies are used to compare estimates. Regarding the simulation study, we observe that the estimates based on ranked set sampling are more efficient compared to the estimates based on simple random sample. The importance and flexibility of Zubair Lomax are validated empirically in modeling two types of lifetime data.     

Stress‐Strength Reliability Analysis with Extreme Values based on q‐Exponential Distribution

When dealing with practical problems of stress–strength reliability, one can work with fatigue life data and make use of the well‐known relation between stress and cycles until failure. For some materials, this kind of data can involve extremely large values. In this context, this paper discusses the problem of estimating the reliability index R = P(Y < X) for stress–strength reliability, where stress Y and strength X are independent q‐exponential random variables. This choice is based on the q‐exponential distribution's capability to model data with extremely large values. We develop the maximum likelihood estimator for the index R and analyze its behavior by means of simulated experiments. Moreover, confidence intervals are developed based on parametric and nonparametric bootstrap. The proposed approach is applied to two case studies involving experimental data: The first one is related to the analysis of high‐cycle fatigue of ductile cast iron, whereas the second one evaluates the specimen size effects on gigacycle fatigue properties of high‐strength steel. The adequacy of the q‐exponential distribution for both case studies and the point and interval estimates based on maximum likelihood estimator of the index R are provided. A comparison between the q‐exponential and both Weibull and exponential distributions shows that the q‐exponential distribution presents better results for fitting both stress and strength experimental data as well as for the estimated R index. Copyright © 2016 John Wiley & Sons, Ltd.     

A Neural Network Approach to Find The Cumulative Failure Distribution: Modeling and Experimental Evidence

The failure prediction of components plays an increasingly important role in manufacturing. In this context, new models are proposed to better face this problem, and, among them, artificial neural networks are emerging as effective. A first approach to these networks can be complex, but in this paper, we will show that even simple networks can approximate the cumulative failure distribution well. The neural network approach results are often better than those based on the most useful probability distribution in reliability, the Weibull. In this paper, the performances of multilayer feedforward basic networks with different network configurations are tested, changing different parameters (e.g., the number of nodes, the learning rate, and the momentum). We used a set of different failure data of components taken from the real world, and we analyzed the accuracy of the approximation of the different neural networks compared with the least squares method based on the Weibull distribution. The results show that the networks can satisfactorily approximate the cumulative failure distribution, very often better than the least squares method, particularly in cases with a small number of available failure times. Copyright © 2015 John Wiley & Sons, Ltd.     

竞争失效场合Weibull产品的可靠性分析

在竞争失效场合下,建立了Weibull产品的具有二项移走的逐步Ⅱ型截尾寿命试验模型.针对一般近似算法在小样本情形下精度较低的问题,以Gamma分布综合产品的先验信息,采用Gibbs抽样建立了参数的Bayes估计.利用Bootstrap方法、Bayes后验分布法分别构造了参数的置信区间.最后,利用随机模拟方法对估计结果进行了比较,结果表明在小样本情形下,Bayes比MLEs估计效果更好.     

The Weibull log-logistic mixture distributions: Model,theory and application to lifetime data

Lifetime data collected from reliability tests are among data that often exhibit significant heterogeneity caused by variations in manufacturing, which makes standard lifetime models inadequate. Finite mixture models provide more flexibility for modeling such data. In this paper, the Weibull-log-logistic mixture distributions model is introduced as a new class of flexible models for heterogeneous lifetime data. Some statistical properties of the model are presented including the failure rate function, moments generating function, and characteristic function. The identifiability property of the class of all finite mixtures of Weibull-log-logistic distributions is proved. The maximum likelihood estimation (MLE) of model parameters under the Type I and Type II censoring schemes is derived. Some numerical illustrations are performed to study the behavior of the obtained estimators. The model is applied to the hard drive failure data made by the Backblaze data center, where it is found that the proposed model provides more flexibility than the univariate life distributions (Weibull, Exponential, logistic, log-logistic, Frechet). The failure rate of hard disk drives (HDDs) is obtained based on MLE estimates. The analysis of the failure rate function on the basis of SMART attributes shows that the failure of HDDs can have different causes and mechanisms.     

Estimation of the Parameters of the Gamma Distribution by Sample Quantiles

This paper deals with large sample estimation of the location parameter (α₁ and the scale parameter α₂ in the gamma distribution with known shape parameter. Best linear unbiased estimates based on k sample quantiles are used. For a given k, the optimum spacings of the sample quantiles can be replaced by simpler “nearly optimum” spacings at virtually no loss of asymptotic efficiency. The theory behind the nearly optimum spacings is briefly reviewed. The major part of the paper concerns estimation of α₂ when α₂ is known. Nearly optimum spacings together with the coefficients to be used in computing the estimates are presented in a number of tables for k = 1(1) 10, and various values of the shape parameter. The paper also contains brief discussions of estimation of α₁, when α₂ is known, and simultaneous estimation of α₁ and α₂.     

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.     

Nonparametric estimation of the offspring distribution and the mean for a controlled branching process

In this paper, under a nonparametric context, some estimators for the mean and the offspring distribution of a controlled branching process are provided. Their conditional and unconditional moments and their asymptotic properties (consistency, normality) are investigated. Finally, as an illustration, a simulated example is presented. Research supported by the Plan Nacional de Investigación Científica, Desarrollo e Innovación Tecnológica, grant BFM2000-0356 and the Consejería de Educación, Ciencia y Tecnología de la Junta de Extremaduray y el Fondo Social Europeo, grant IPR00A056.     

飞机抖振响应数据处理及分析方法

为了编制抖振疲劳谱、估算抖振疲劳寿命等,必须预先对时域的抖振响应数据进行处理与分析。考虑到抖振响应的随机特性,特别是其显著的分散性,建立了统计模型来对其分析处理。针对同一飞行状态下数据仓中的抖振响应数据,将其划分为若干子数据块,以子数据块中数据统计特征来描述对应子数据块均方根下的响应分布情况,而以对应飞行状态下各子数据块的均方根分布情况来描述该状态下抖振响应的总体分布趋势以及选择其关键响应状态水平。首先,采用威布尔分布假设,运用极大似然估计法对子数据块的数据进行分布参数估计,并给出了分布假设的检验方法;然后,采用“三步进”经验函数来描述抖振响应均方根的分布规律。在本文研究的基础上,根据给出的抖振数据处理与分析结果使用流程,即只需根据确定的几个关键均方根水平,定位到相应的子数据块,再结合子数据块数据的统计模型得到对应飞行状态下的响应分布,就可用于飞机设计与强度校核。由实际飞行试验抖振数据的处理与分析表明,此方法具备一定的合理性     

Estimation of the variance for a controlled branching process

In this paper we obtain estimators for the variance of the offspring distribution of a controlled branching process and we derive, for these estimators, some moments and asymptotic properties as consistency and limiting distribution. Research supported by the Ministerio de Ciencia y Tecnología and the FEDER through the Plan Nacional de Investigación Científica, Desarrollo e Innovación Tecnológica, grant BFM2003-06074.     

Three-parameter vs. two-parameter Weibull distribution for pultruded composite material properties

The three-parameter and two-parameter Weibull distributions are compared using 26 mechanical property data sets of fiber-reinforced polymeric (FRP) composite materials manufactured by the pultrusion process. Both strength and stiffness properties were examined. The probability distributions were compared on the basis of goodness of fit, nominal design values, and allowable load to achieve uniform reliability. It is recommended that the two-parameter Weibull distribution be used to characterize FRP composite material properties. The primary basis for this recommendation is small differences in nominal design values and small differences in allowable loads between the two-parameter and three-parameter Weibull distributions. Other supporting reasons for the recommendation are similar observed significance levels in distribution fitting, computational efficiency, and the fact that the location parameter of the three-parameter Weibull distribution is near the first order statistic.     

Comparison of Methods to Estimate the Time‐to‐failure Distribution in Degradation Tests

Degradation tests are alternative approaches to lifetime tests and accelerated lifetime tests in reliability studies. Based on a degradation process of a product quality characteristic over time, degradation tests provide enough information to estimate the time‐to‐failure distribution. Some estimation methods, such as analytical, the numerical or the approximated, can be used to obtain the time‐to‐failure distribution. They are chosen according to the complexity of the degradation model used in the data analysis. An example of the application and analysis of degradation tests is presented in this paper to characterize the durability of a product and compare the various estimation methods of the time‐to‐failure distribution. The example refers to a degradation process related to an automobile's tyre, and was carried out to estimate its average distance covered and some percentiles of interest. Copyright © 2004 John Wiley & Sons, Ltd.     

A Statistical Physics Method of Electronic Device Reliability Testing from Working Data

A combined method is used to estimate the parameters for a mixture of exponential and Weibull distributions. Examples are given of the method applied in reliability analysis for high-power electronic devices.     

Bayesian approach to parameter estimation of the generalized pareto distribution

Several methods have been used for estimating the parameters of the generalized Pareto distribution (GPD), namely maximum likelihood (ML), the method of moments (MOM) and the probability-weighted moments (PWM). It is known that for these estimators to exist, certain constraints have to be imposed on the range of the shape parameter,k, of the GPD. For instance, PWM and ML estimators only exist fork>−0.5 andk≤1, respectively. Moreover, and particularly for small sample sizes, the most efficient method to apply in any practical situation highly depends on a previous knowledge of the most likely values ofk. This clearly suggests the use of Bayesian techniques as a way of using prior information onk. In the present work, we address the issue of estimating the parameters of the GPD from a Bayesian point of view. The proposed approach is compared via a simulation study with ML, PWM and also with the elemental percentile method (EPM) which was developed by Castillo and Hadi (1997). The estimation procedure is then applied to two real data sets.     

多元Marshall～Olkin型指数分布的特征及其参数估计

导出了多元Marshall～Olkin型指数分布的一个特征,利用该特征,获得了多元Marshall～Olkin型指数分布参数的最大似然估计及矩估计.