Estimation of Mission Reliability from Multiple Independent Grouped Censored Samples

Aircraft or missiles are flown for missions of varying durations. Data are collected at the end of each mission which indicate the mission duration and whether the equipment failed. The data are considered as multiple s-independent grouped censored samples with failure times unknown. The underlying failure model considered is the 2-parameter Weibull distribution. Maximum likelihood estimates are derived. The exponential distribution is used for comparison. Monte Carlo simulations are used to compare s-efficiency of estimates for grouped data with estimates if failure times were known. The asymptotic variance-covariance matrix was computed for the sampling conditions studied and was used to obtain lower s-confidence bounds on the system reliability.     

Estimation of Weibull Parameters With Competing-Mode Censoring

Existing results are reviewed for the maximum likelihood (ML) estimation of the parameters of a 2-parameter Weibull life distribution for the case where the data are censored by failures due to an arbitrary number of independent 2-parameter Weibull failure modes. For the case where all distributions have a common but unknown shape parameter the joint ML estimators are derived for i) a general percentile of the j-th distribution, ii) the common shape parameter, and iii) the proportion of failures due to failure mode j. Exact interval estimates of the common shape parameter are constructable in terms of the ML estimates obtained by using i) the data without regard to failure mode, and ii) existing tables of the percentage points of a certain pivotal function. Exact interval estimates for a general percentile of failure-mode-j distribution are calculable when the failure proportion due to failure-mode-j is known; otherwise a joint s-confidence region for the percentile and failure proportion is calculable. It is shown that sudden death endurance test results can be analyzed as a special case of competing-mode censoring. Tabular values for the construction of interval estimates for the 10-th percentile of the failure-mode-j distribution are given for 17 combinations of sample size (from 5 to 30) and number of failures.     

威布尔分布无故障数据的可靠性评估

对于威布尔分布无故障数据可靠性评估方法中形状参数已知和未知的两种方法,通过一个例子进行对比分析,指出当形状参数毫无所知时,所得到的基本可靠度置信下限估计最为保守。通过相似产品的信息和工程经验对形状参数作出一个较为精确的估计是可行的。     

Uniqueness of maximum likelihood estimators of the 2-parameterWeibull distribution

We present a simple statistic, calculated from either complete failure data or from right-censored data of type-I or -II. It is useful for understanding the behavior of the parameter maximum likelihood estimates (MLE) of a 2-parameter Weibull distribution. The statistic is based on the logarithms of the failure data and can be interpreted as a measure of variation in the data. This statistic provides: (a) simple lower bounds on the parameter MLE, and (b) a quick approximation for parameter estimates that can serve as starting points for iterative MLE routines; it can be used to show that the MLE for the 2-parameter Weibull distribution are unique     

Estimation of Reliability from Multiple Independent Grouped Censored Samples with Failure Times Known

Failure times of one type aircraft-engine component were recorded. In addition, life times are periodically recorded for unfailed engine components. The data are considered as multiple s-independent grouped censored samples with failure times known. The assumed failure model is the 2-parameter Weibull distribution. Maximum likelihood estimates are derived. The exponential model is used for comparison. Monte Carlo simulation is used to derive s-bias and mean square error of the estimates. The asymptotic covariance matrix was computed for the sampling conditions studied. The maximum likelihood estimates of the reliability were obtained as a function of component operating time since overhaul.     

An Evaluation of Exponential and Weibull Test Plans

MIL-STD-781B gives sampling plans (sequential and fixed length) for reliability tests under the assumption of a constant failure rate. Using Monte Carlo techniques, the authors compare s-expected time to a decision and producer and consumer risks for some of these plans. It is shown that plans which assume an exponential distribution are not robust to departures from that assumption. A simple modification of these plans for use when life has a Weibull distribution with known shape parameter not equal to one, and an adaptive test procedure for use when life has a Weibull distribution with unknown shape parameter are proposed. The modified plans for a Weibull distribution with known shape parameter have the same designated producer and consumer risks, but different s-expected time to a decision than the corresponding exponential plans. Using Monte Carlo techniques, the authors determine s-expected time to a decision and producer and consumer risks for various forms of the adaptive procedure.     

Linear Estimation of the Parameters of the Extreme-Value Distribution Based on Suitably Chosen Order Statistics

Best linear unbiased estimates (BLUEs) based on a few order statistics are found for the location and scale parameters of the extreme-value distribution (Type-I asymptotic distribution of smallest values), when one or both parameters are unknown, such that the estimates have maximum efficiencies among the BLUEs based on the same number of order statistics. These estimates are then compared with the BLUEs and asymptotically best linear estimates (ABLEs) based on a few order statistics whose ranks were determined from the spacings that maximize the asymptotic efficiencies of the ABLEs. An application to the Weibull distribution is given.     

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.     

A new model for step-stress testing

The mathematical intractability of the Weibull cumulative exposure model (CE-M) has impeded the development of statistical procedures for step-stress accelerated life tests. Our new model (KH-M) is based on a time transformation of the exponential CE-M. The time-transformation enables the reliability engineer to use known results for multiple-step, multiple-stress models that have been developed for the exponential step-stress model. KH-M has a realistically appealing proportional-hazard property. It is as flexible as the Weibull CE-M for fitting data, but its mathematical form makes it easier to obtain parameter estimates and standard deviations. Maximum likelihood estimates are given for test plans with unknown shape parameter. The mathematical similarity to the constant-stress Weibull model is shown. Chi-square goodness of fit tests are performed on simulated data to compare the fit of the models     

Estimating Weibull Parameters for a General Class of Devices from Limited Failure Data

Relatively simple approaches to estimating Weibull parameters for a general class of devices are developed through regression models. It is assumed that data are collected on a number of device types belonging to a general class. For each device type, the only information available is the number of devices being observed, the total time observed and the total number of failures. By assuming a constant shape parameter and a scale parameter that may vary with the characteristics of the device-type, the least squares method is used to provide estimates of the parameters of a two-parameter Weibull distribution for both replacement and nonreplacement data. An approach is also suggested for dealing with troublesome cases of zero failure occurrences. A numerical example is provided to illustrate the approach.     

On estimating parameters in a discrete Weibull distribution

Two discrete Weibull distributions are discussed, and a simple method is presented to estimate the parameters for one of them. Simulation results are given to compare this method with the method of moments. The estimates obtained by the two methods appear to have almost similar properties. The discrete Weibull data arise in reliability problems when the observed variable is discrete. The modeling of such a random phenomenon has already been accomplished. Estimation of parameters in these models is considered. Since the usual methods of estimation are not easy to apply, a simple method is suggested to estimate the unknown parameters. The estimates obtained by this method are comparable to those obtained by the method of moments. The method can be applied in most inferential problems. Though the authors have restricted themselves to type I distribution, their method of proportions for the estimation of parameters can be easily applied to the type II distribution as well     

An improved modeling for life prediction of high-power white LED based on Weibull right approximation method

Aiming at precisely predicting the life of the high-power white light LED (HPWLED), a three-parameter Weibull function and the right approximation method were employed to establish the luminance degradation model. The lumen maintenance data collected according to the IES LM-80-08 lumen maintenance test standard were fitted with and without error corrections, and the pseudo failure time of each HPWLED sample was extrapolated. The statistical analysis on the failure time was achieved by using Weibull distribution, normal distribution, lognormal distribution and Akaike Information Criterion (AIC). Then the life information was acquired. The results indicate that Weibull right approximation luminance degradation model (WRALDM) accurately reflects the variation of the lumen law with time. The failure time is accurately obtained. The best life distributions before and after the error correction to the lumen maintenance data are identified, based on AIC, as Weibull distribution and lognormal distribution, respectively. It is further confirmed by comparing the widths of life confidence interval and the life provided by the IES TM-21-11 method that the HPWLED life using WRALDM has a better accuracy. The optimized model provides researchers and manufacturers with significant guidelines for the further development of life prediction methodology.     

Weibull accelerated life testing with unreported early failures

Situations arise in life testing where early failures go unreported, e.g. a technician believes an early failure is “his fault” or “premature” and must not be recorded. Consequently, the reported data come from a truncated distribution and the number of unreported early failures is unknown. Inferences are developed for a Weibull accelerated life-testing model in which transformed scale and shape parameters are expressed as linear combinations of functions of the environment (stress). Coefficients of these combinations are estimated by maximum likelihood methods which allow point, interval, and confidence bound estimates to be computed for such quantities of interest for a given stress level as the shape parameter, the scale parameter, a selected quantile, the reliability at a particular time, and the number of unreported early failures. The methodology allows lifetimes to be reported as exact, right censored, or interval-valued, and to be subject optionally to testing protocols which establish thresholds below which lifetimes go unreported. A broad spectrum of applicability is anticipated by virtue of the substantial generality accommodated in both stress modeling and data type     

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     

Moment estimators for the 3-parameter Weibull distribution

Weibull moments are defined generally and then calculated for the 3-parameter Weibull distribution with non-negative location parameter. Sample estimates for these moments are given and used to estimate the parameters. The results of a simulation investigation of the properties of the parameter estimates are discussed briefly. A simple method of deciding whether the location parameter can be considered zero is described     

Relation of a Physical Process to the Reliability of Electronic Components

An ensemble of electronic components having a random variation of some parameter, such as surface contamination, is considered. A physical process is postulated which leads to a change in one of the operating characteristics of the device. When this operating characteristic attains a value outside an acceptable range, the device is considered to have failed. The failure rate is calculated directly from the time behavior of the physical process and compared, for illustration, to the Weibull failure law. The parameters of the Weibull law are then related to the parameters of the physical process and the distribution of starting parameters.     

The exponentiated Weibull family: a graphical approach

The exponentiated Weibull family extends the two-parameter Weibull distribution. The shape of the Weibull plotting-paper plots are discussed, and a parametric characterization of the pdf and the failure rate for the exponentiated Weibull family are carried out. Such a study is very relevant to deciding if a given data set can be adequately modeled by such a distribution     

Weibull component reliability-prediction in the presence of maskeddata

Analysts are often interested in obtaining component reliabilities by analyzing system-life data. This is generally done by making a series-system assumption and applying a competing-risks model. These estimates are useful because they reflect component reliabilities after assembly into an operational system under true operating conditions. The fact that most new systems under development contain a large proportion of old technology also supports the approach. In practice, however, this type of analysis is often confounded by the problem of masking (the exact cause of system failure is unknown). This paper derives a likelihood function for the masked-data case and presents an iterative procedure (IMLEP) for finding maximum likelihood estimates and confidence intervals of Weibull component life-distribution parameters. The approach is illustrated with a simple numerical example     

A simplified method for evaluating time dependent fault-trees,using Weibull distribution

A new formula for calculating the failure rate of the mode failure for a time dependent fault tree is derived. The time-to-failure distribution for each component (primary failure) is assumed to be Weibull model with different values of parameters for each component. By using this formula, the TOP failure rate for the fault tree can be easily calculated. By appropriate choice of the two parameters of the Weibull model, a wide range of hazard curves can be approximated. The equations are derived by applying the “Kinetic Tree Theory” mentioned by Vesely [1]. The assumptions of the methodology are the independence of the primary failures and that the mode failures are previously known.     

Accelerated Life Testing?Step-Stress Models and Data Analyses

This paper presents statistical models and methods for analyzing accelerated life-test data from step-stress tests. Maximum likelihood methods provide estimates of the parameters of such models, the life distribution under constant stress, and other information. While the methods are applied to the Weibull distribution and inverse power law, they apply to many other accelerated life test models. These methods are illustrated with step-stress data on time to breakdown of an electrical insulation.