Reliability analysis of the dynamic system for the Chen model through sequential order statistics

Recently, the two-parameter Chen distribution has widely been used for reliability studies in various engineering fields. In this article, we have developed various statistical inferences on the composite dynamic system, assuming Chen distribution as a baseline model. In this dynamic system, failure of a component induces a higher load on the surviving components and thus increases component hazard rate through a power-trend process. The classical and Bayesian point estimates of the unknown parameters of the composite system are obtained by the method of maximum likelihood and Markov chain Monte Carlo techniques, respectively. In the Bayesian framework, we have used gamma priors to obtain Bayes estimates of unknown parameters under the squared error and generalized entropy loss functions. The interval estimates of the baseline reliability function are obtained by using the Fisher information matrix and Bayesian method. A parametric hypothesis test is presented to test whether the failed components change the hazard rate function. A compact simulation study is carried out to examine the behavior of the proposed estimation methods. Finally, one real data analysis is performed for illustrative purposes.     

Estimations of parameters in Pareto reliability model in the presence of masked data

Estimations of parameters included in the individual distributions of the life times of system components in a series system are considered in this paper based on masked system life test data. We consider a series system of two independent components each has a Pareto distributed lifetime. The maximum likelihood and Bayes estimators for the parameters and the values of the reliability of the system's components at a specific time are obtained. Symmetrical triangular prior distributions are assumed for the unknown parameters to be estimated in obtaining the Bayes estimators of these parameters. Large simulation studies are done in order: (i) explain how one can utilize the theoretical results obtained; (ii) compare the maximum likelihood and Bayes estimates obtained of the underlying parameters; and (iii) study the influence of the masking level and the sample size on the accuracy of the estimates obtained.     

含有屏蔽数据的截尾样本下部件的可靠性分析

基于屏蔽的系统寿命数据,讨论串联系统中Burr XII部件的可靠性估计问题.利用定数截尾样本,通过Bayes分析方法分别在平方损失、q-对称熵损失、Linex损失以及MLinex损失下给出了部件未知参数、可靠度函数和失效率函数的Bayes估计.最后通过Monte-Carlo方法进行随机模拟,研究截尾数和屏蔽水平对估计效果的影响并对各种估计进行了比较.     

Inference for dependence competing risks with partially observed failure causes from bivariate Gompertz distribution under generalized progressive hybrid censoring

Competing risks model is considered with dependence causes of failure in this paper. When the latent failure times are distributed by a bivariate Gompertz model, statistical inference for the unknown model parameters is studied from classical and Bayesian approaches, respectively. Under a generalized progressive hybrid censoring, maximum likelihood estimators of the unknown parameters together with the associated existence and uniqueness are established, and the approximate confidence intervals are also obtained based on asymptotic likelihood theory via the observed Fisher information matrix. Moreover, Bayes estimates and the highest posterior density credible intervals of the unknown parameters are also provided based on a flexible Gamma–Dirichlet prior, and Monte Carlo sampling method is also derived to compute associated estimates. Finally, simulation studies and a real-life example are given for illustration purposes.     

Evaluation of operational system structures based on reliability data

Systems of components have a structure that plays an important role in determining how the reliability of the individual components relates to the reliability of the system. The system reliability can be computed from component reliabilities using results from basic probability theory in the simplest case with all of the components assumed to act independently of one another. However, in the case of dependence, such calculations can be much more involved. When reliability data have been independently collected on both the system and each component in the system, it can be difficult to model any possible dependence between components. Established methods use the known structure of a system, along with these data, to assess whether the reliability of the individual components are mutually independent. In this paper, we expand this methodology to include an assessment of the type of dependence that may exist between the components. This is based on finding the system structure that would most likely produce the observed reliability data, under independence. In the frequentist setting, the likelihood approach is used to find these structures and an observed confidence measure is used to assess the strength of the statistical evidence in favor of each possible structure. In the Bayesian setting, posterior probabilities along with Bayes factors are used. An example demonstrates how these methods can be used in an applied setting.     

Estimating a Parametric Component Lifetime Distribution from a Collection of Superimposed Renewal Processes

Maintenance data can be used to make inferences about the lifetime distribution of system components. Typically, a fleet contains multiple systems. Within each system, there is a set of nominally identical replaceable components of particular interest (e.g., 2 automobile headlights, 8 dual in-line memory module (DIMM) modules in a computing server, 16 cylinders in a locomotive engine). For each component replacement event, there is system-level information that a component was replaced, but no information on which particular component was replaced. Thus, the observed data are a collection of superpositions of renewal processes (SRP), one for each system in the fleet. This article proposes a procedure for estimating the component lifetime distribution using the aggregated event data from a fleet of systems. We show how to compute the likelihood function for the collection of SRPs and provide suggestions for efficient computations. We compare performance of this incomplete-data maximum likelihood (ML) estimator with the complete-data ML estimator and study the performance of confidence interval methods for estimating quantiles of the lifetime distribution of the component. Supplementary materials for this article are available online.     

Reliability analysis of Weibull multicomponent system with stress-dependent parameters from accelerated life data

In this paper, reliability estimation of multicomponent system under a multilevel accelerated life testing. When the lifetime of components follows Weibull distribution, the problem of point and interval estimates are discussed from different perspectives. Under a general life-stress assumption that there are multiple nonconstant and stress-dependent scale and shape parameters, the maximum likelihood estimates of unknown parameters along with associated existence and uniqueness are established. Approximate confidence intervals are constructed correspondingly via expected Fisher information matrix. Furthermore, some pivotal quantities are constructed and alternative generalized point and interval estimates are also proposed for comparison. In addition, predictive intervals for the lifetime of the multicomponent system are discussed under classical and generalized pivotal approaches, respectively. The results show that the proposed generalized estimates are superior to the conventional likelihood approach in terms of the accuracy. A real data example is carried out to illustrate the implementations of the proposed methods.     

Estimating Component Failure Rates From Combined Component and Systems Data: Exponentially Distributed Component Lifetimes

The situation in which test results are available on systems consisting of some of k independent but nonidentical components in series is studied. The underlying distribution of the lifetime of the component is assumed to be an exponential. The method of maximum likelihood is used to obtain estimates, a chi squared approximation is used to approximate the mean and variance of the maximum likelihood estimate, and a method for reducing its bias is presented.     

Comparison of Two Methods of Obtaining Approximate Confidence Intervals for System Reliability

Some specific comparisons are made in this note between the use of the asymptotic Chi-square distribution of the likelihood ratio and the asymptotic normality of the maximum likelihood estimates to obtain confidence interval for reliabilities of arbitrary systems when only failure data on the components is known. In all the comparisons made, using moderate samples and systems of average complexity, the asymptotic Chi-square appears to give much more accurate confidence intervals. Although the asymptotic Chi-square method requires more computation for most systems than does the method based on asymptotic normality, these examples indicate the Chi-square method would yield superior results in most practical instances.     

Inference for two-parameter Rayleigh competing risks data under generalized progressive hybrid censoring

In this paper, a competing risks model based on the generalized progressive hybrid censored two-parameter Rayleigh distributions is studied under the assumption that the lifetime distributions of failure causes are identically distributed with same location and different scale parameters. We obtain maximum likelihood estimates of unknown parameters with associated existence uniqueness. The approximate confidence intervals are constructed using the asymptotic distribution of maximum likelihood estimates via the observed information matrix. Further, Bayes point estimates and the highest probability density credible intervals of unknown parameters are presented, and the Gibbs sampling technique is used to approximate corresponding estimates. A Monte Carlo simulation study is conducted to compare the accuracy of proposed estimates. Finally, a real-life example is presented for illustration purpose.     

Classical and Bayesian inference on progressive‐stress accelerated life testing for the extension of the exponential distribution under progressive type‐II censoring

In this paper, a progressive‐stress accelerated life test under progressive type‐II censoring is considered. The cumulative exposure model is assumed when the lifetime of test units follows an extension of the exponential distribution. The maximum likelihood and Bayes estimates of the model parameters are obtained. The approximate and credible confidence intervals of the estimators are derived. Furthermore, a real lifetime data set is analyzed to illustrate the proposed procedures. Finally, the simulation studies are used to compare between 2 different designs of the progressive‐stress test (simple and multiple ramp‐stress tests).     

Inference for a geometric‐poisson‐Rayleigh distribution under progressive‐stress accelerated life tests based on type‐I progressive hybrid censoring with binomial removals

Based on failures of a parallel‐series system, a new distribution called geometric‐Poisson‐Rayleigh distribution is proposed. Some properties of the distribution are discussed. A real data set is used to compare the new distribution with other 6 distributions. The progressive‐stress accelerated life tests are considered when the lifetime of an item under use condition is assumed to follow the geometric‐Poisson‐Rayleigh distribution. It is assumed that the scale parameter of the geometric‐Poisson‐Rayleigh distribution satisfies the inverse power law such that the stress is a nonlinear increasing function of time and the cumulative exposure model for the effect of changing stress holds. Based on type‐I progressive hybrid censoring with binomial removals, the maximum likelihood and Bayes (using linear‐exponential and general entropy loss functions) estimation methods are considered to estimate the involved parameters. Some point predictors such as the maximum likelihood, conditional median, best unbiased, and Bayes point predictors for future order statistics are obtained. The Bayes estimates are obtained using Markov chain Monte Carlo algorithm. Finally, a simulation study is performed, and numerical computations are performed to compare the performance of the implemented methods of estimation and prediction.     

Bayesian estimation of the 3-parameter inverse gaussian distribution

The three-parameter inverse Gaussian distribution is used as an alternative model for the three parameter lognormal, gamma and Weibull distributions for reliability problems. In this paper Bayes estimates of the parameters and reliability function of a three parameter inverse Gaussian distribution are obtained. Posterior variance estimates are compared with the variance of their maximum likelihood counterparts. Numerical examples are given.     

An iterative approach for estimating component reliability from masked system life data

Life data from systems of components are often analysed to estimate the reliability of the individual components. These estimates are useful since they reflect the reliability of the components under actual operating conditions. However, owing to the cost or time involved with failure analysis, the exact component causing system failure may be unknown or ‘masked’. That is, the cause may only be isolated to some subset of the system's components. We present an iterative approach for obtaining component reliability estimates from such data for series systems. The approach is analogous to traditional probability plotting. That is, it involves the fitting of a parametric reliability function to a set of nonparametric reliability estimates (plotting points). We present a numerical example assuming Weibull component life distributions and a two-component series system. In this example we find estimates with only 4 per cent of the computation time required to find comparable MLEs.     

Optimum combined test plans for systems and components

Since mathematical models based on component reliabilities are frequently used for prediction of system reliability, it stands to reason that cost-effective inferences on the reliability of a system could be made on the basis of tests of its constituent components. Prior research in the area of system-based component testing has for the most part addressed the development of plans that test only the components. From a practitioner's point of view, this is an issue of concern since system failures are often caused by imperfect interfaces and other causes that are not directly attributable to component failures. The exclusion of system tests may thus be an erroneous approach. This paper addresses the development of test plans that explicitly consider the possibility of interface failures. The paper analyzes a series system to determine when testing should be performed on the system alone, on the components only, and on both, depending on test costs and interface reliabilities. Optimum test plans are: derived by solving a two-stage mathematical program.     

Overdispersion in modelling accidents on road sections and in empirical bayes estimation

In multivariate statistical models of road safety one usually finds that the accident counts are ‘overdispersed’. The extent of the overdispersion is itself subject to estimation. It is shown that the assumption one makes about the nature of overdispersion will affect the maximum likelihood estimates of model parameters. If one assumes that the same overdispersion parameter applies to all road sections in the data base, then, the maximum likelihood estimate of parameters will be unduly influenced by very short road sections and insufficiently influenced by long road sections. The same assumption about the overdispersion parameter also leads to an inconsistency when one estimates the safety of a road section by the Empirical Bayes method. A way to avoid both problems is to estimate an overdispersion parameter (φ) that applies to a unit length of road, and to set the overdispersion parameter for a road section of length L to φL. How this would change the estimates of regression parameters for road section models now in use requires examination. Safety estimation by the Empirical Bayes method is altered substantially.     

Statistical inference on progressive-stress accelerated life testing for the logistic exponential distribution under progressive type-II censoring

The accelerated life testing (ALT) is an efficient approach and has been used in several fields to obtain failure time data of test units in a much shorter time than testing at normal operating conditions. In this article, a progressive-stress ALT under progressive type-II censoring is considered when the lifetime of test units follows logistic exponential distribution. We assume that the scale parameter of the distribution satisfying the inverse power law. First, the maximum likelihood estimates of the model parameters and their approximate confidence intervals are obtained. Next, we obtain Bayes estimators under squared error loss function with the help of Metropolis-Hasting (MH) algorithm. We also derive highest posterior density (HPD) credible intervals of the model parameters. Monte Carlo simulations are performed to compare the performances of the proposed methods of estimation. Finally, one data set has been analyzed for illustrative purposes.     

Inference with progressively censored <Emphasis Type="Italic">k</Emphasis>-out-of-<Emphasis Type="Italic">n</Emphasis> system lifetime data

A system with n independent components which works if and only if a least k of its n components work is called a k-out-of-n system. For exponentially distributed component lifetimes, we obtain point and interval estimators for the scale parameter of the component lifetime distribution of a k-out-of-n system when the system failure time is observed only. In particular, we prove that the maximum likelihood estimator (MLE) of the scale parameter based on progressively Type-II censored system lifetimes is unique. Further, we propose a fixed-point iteration procedure to compute the MLE for k-out-of-n systems data. In addition, we illustrate that the Newton–Raphson method does not converge for any initial value. Finally, exact confidence intervals for the scale parameter are constructed based on progressively Type-II censored system lifetimes.     

An alternative perspective on the mixture estimation problem

The paper presents an alternative perspective on the mixture estimation problem. First, observations are counted into a histogram. Secondly, rough and enhanced parameter estimation followed by the separation of observations is done. Finally, the residue is distributed between the components by the Bayes decision rule. The number of components, the mixture component parameters and the component weights are modelled jointly, no initial parameter estimates are required, the approach is numerically stable, the number of components has no influence upon the convergence and the speed of convergence is very high. The alternative perspective is compared to the EM algorithm and verified through several data sets. The presented algorithm showed significant advantages compared to the competitive methods and has already been successfully applied in reliability and fatigue analyses.     

An additive Chen-Weibull distribution and its applications in reliability modeling

In this article, we introduce a new lifetime distribution with increasing and bathtub-shaped failure rates. Some statistical properties of the proposed distribution are studied. We use the method of maximum likelihood for estimating the model parameters and reliability characteristics and discuss the interval estimates using asymptotic confidence intervals and bootstrap confidence intervals on one hand, and we provide Bayes estimators and highest posterior density intervals for the parameters via Hamiltonian Monte Carlo simulation method on the other hand. We demonstrate the superiority of the proposed distribution by fitting two reliability data sets well-known from references.