Inference on Weibull Percentiles and Shape Parameter from Maximum Likelihood Estimates

Four functions of the maximum likelihood estimates of the Weibull shape parameter and any Weibull percentile are found. The sampling distributions are independent of the population parameters and depend only upon sample size and the degree of (Type II) censoring. These distributions, once determined by Monte Carlo methods, permit the testing of the following hypotheses: 1) that the Weibull shape parameter is equal to a specified value; 2) that a Weibull percentile is equal to a specified value; 3) that the shape parameters of two Weibull populations are equal; and 4) that a specified percentile of two Weibull populations are equal given that the shape parameters are. The OC curves of the various tests are shown to be readily computed. A by-product of the determination of the distribution of the four functions are the factors required for median unbiased estimation of 1) the Weibull shape parameter, 2) a Weibull percentile, 3) the ratio of shape parameters of two Weibull distributions, and 4) the ratio of a specified percentile of two Weibull distributions.     

Parameter compatibility relations for accelerated testing

It is pointed out that the use of acceleration studies implies that stress does not change the mechanisms of failure but only contracts the time to reach failure level. This implication is translated into a requirement that the parameter set for the failure distribution does not change, but values of the parameters may. The generality of the relationship between parameter values at various stress levels and the time transformation that connects the respective distributions is emphasized. These are termed compatibility relations. An example of compatibility relations for acceleration studies described by a Weibull distribution is given in detail. The compatibility relations between the coefficients in the time transformation in acceleration studies based on exponential, gamma, log-normal, and Weibull distributions and respective parameter values are discussed generally. Compatibility relations can be used to improve estimation procedures, check the consistency of parameter estimation during analysis of acceleration test data, and check the validity of the use of acceleration studies in the first place     

Robust minimum-distance estimation using the 3-parameter Weibulldistribution

Maximum-likelihood and minimum-distance estimates were compared for the three-parameter Weibull distribution. Six estimation techniques were developed by using combinations of maximum-likelihood and minimum-distance estimation. The minimum-distance estimates were made using both the Anderson-Darling and Cramer-Von Mises goodness-of-fit statistics. The estimators were tested by Monte Carlo simulation. For each set of parameters and sample size, 1000 data sets were generated and evaluated. Five evaluation criteria were calculated; they measured both the precision of estimating the population parameters and the discrepancy between the estimated and population Cdfs. The robustness of the estimation techniques was tested by fitting Weibull Cdfs to data from other distributions. Whether the data were Weibull or generated from other distributions, minimum-distance estimation using the Anderson-Darling goodness-of-fit statistic on the location parameter and maximum likelihood on the shape and scale parameters was the best or close to the best estimation technique     

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     

Interval estimation from censored and masked system-failure data

The author addresses confidence interval (CI) estimation in a competing risk (or multiple failure mode) framework where sample data are singly time-censored on the right and partially masked. A three-component series system with exponentially-distributed component-failure times is considered in order to represent cases involving full as well as partial masking. The approximate CIs considered are based on: asymptotic-normal theory for maximum likelihood estimators; cube-root transformation of the exponential distribution rate parameter; and inverted likelihood ratio tests. The small-sample coverage properties of these approximate CIs are estimated via computer simulation. These results also apply to models where component-failure times are Weibull distributed with known shape parameters     

Asymptotic sampling distribution of inversecoefficient-of-variation and its applications

This paper develops the asymptotic sampling distribution of the inverse of the coefficient of variation (InvCV). This distribution is used for making statistical inference about the population CV (coefficient of variation) or InvCV without making an assumption about the population distribution. It applies to making inferences (point and interval estimation, and hypothesis-testing) about the shape parameter of some popular lifetime distributions like the Gamma, Weibull, and log-normal, when the scale parameter is unknown. The test procedure is used to test exponentiality against a Gamma or a Weibull alternative. The results are compared with those in the literature     

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.     

恒加试验下LED照明灯的三参数Weibull分布的参数估计

三参数Weibull分布拟合LED照明灯寿命的优势较为明显,但要得到三参数Weibull分布参数较为精确的点估计较为困难。目前常用的参数估计方法有极大似然法、矩估计法、Bayes估计法等,由于其计算的方程复杂,导致软件编程繁琐,不易掌握,而且也不一定能得到参数估计。鉴于此,文章针对恒加试验提出一种简便地求解三参数Weibull分布参数估计的方法,该方法不涉及超越方程的求解问题,软件编程相当简单,且统计思想清晰。通过LED照明灯恒加试验下的几个案例数据说明方法的应用,并与已有的方法做了对比分析。     

Confidence Limits for Weibull Regression With Censored Data

The response variable in an experiment follows a 2-parameter Weibull distribution having a scale parameter that varies inversely with a power of a deterministic, externally controlled, variable generically termed a stress. The shape parameter is invariant with stress. A numerical scheme is given for solving a pair of nonlinear simultaneous equations for the maximum likelihood (ML) estimates of the common shape parameter and the stress-life exponent. Interval and median unbiased point estimates for the shape parameter, stress-life exponent and a specified percentile at any stress, are expressed in terms of percentage points of the sampling distributions of pivotal functions of the ML estimates. A numerical example is given.     

A Lifetime Distribution With an Upside-Down Bathtub-Shaped Hazard Function

A three-parameter lifetime distribution with increasing, decreasing, bathtub, and upside down bathtub shaped failure rates is introduced. The new model includes the Weibull distribution as a special case. A motivation is given using a competing risks interpretation when restricting its parametric space. Various statistical properties, and reliability aspects are explored; and the estimation of parameters is studied using the standard maximum likelihood procedures. Applications of the model to real data are also included     

A general linear-regression analysis applied to the 3-parameterWeibull distribution

The conventional techniques of linear regression analysis (linear least squares) applied to the 3-parameter Weibull distribution are extended (not modified), and new techniques are developed for the 3-parameter Weibull distribution. The three pragmatic estimation methods in this paper are simple, accurate, flexible, and powerful in dealing with difficult problems such as estimates of the 3 parameters becoming nonpositive. In addition, the inherent disadvantages of the 3-parameter Weibull distribution are revealed; the advantages of a new 3-parameter Weibull-like distribution over the original Weibull distribution are explored; and the potential of a 4-parameter Weibull-like distribution is briefly mentioned. This paper demonstrates how a general linear regression analysis or linear least-squares breaks away from the classical or modern nonlinear regression analysis or nonlinear least-squares. By adding a parameter to the simplest 2-parameter linear regression model (AB-model), two kinds of ABC models (elementary 3-parameter nonlinear regression models) are found, and then a 4-parameter AABC model is built as an example of multi-parameter nonlinear regression models. Although some other techniques are still necessary, additional applications of the ABC models are strongly implied     

Multiple Comparison for Weibull Parameters

Two problems are considered: 1) testing the hypothesis that the shape parameters of k 2-parameter Weibull populations are equal, given a sample of n observations censored (Type II) at r failures, from each population; and 2) Under the assumption of equal shape parameters, the problem of testing the equality of the p-th percentiles. Test statistics (for these hypotheses), which are simple functions of the maximum likelihood estimates, follow distributions that depend only upon r,n,k,p and not upon the Weibull parameters. Critical values of the test statistics found by Monte Carlo sampling are given for selected values of r,n,k,p. An expression is found and evaluated numerically for the exact distribution of the ratio of the largest to smallest maximum likelihood estimates of the Weibull shape parameter in k samples of size n, Type II censored at r = 2. The asymptotic behavior of this distribution for large n is also found.     

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.     

Estimation of threshold stress in accelerated life-testing

The author presents a method that uses accelerated life-test data to estimate the mean life at the service stress and the threshold stress below which a failure is unlikely to occur. The relation between stress and mean-life at that stress is assumed to follow an inverse power law that includes a threshold stress. The failure times at a given stress are assumed to follow a Weibull distribution in which the shape parameter varies with the stress. This model extends the well-known Weibull inverse power law model. If only the mean life but not a specific percentile point at a service stress is sought, the maximum likelihood method is useful for parameter estimation. This is a tradeoff in the parametric approach. For adoption of an appropriate probability model, the likelihood ratio test as well as the Akaike Information Criterion are used. Type I right censored data are considered. Extensions of the method are discussed     

Threshold optimization of decentralized CFAR detection in weibull clutter using genetic algorithms

The use of genetic algorithms (GAs) tool for the solution of distributed constant false alarm rate (CFAR) detection for Weibull clutter statistics is considered. An approximate expression of the probability of detection (P _D) of the ordered statistics CFAR (OS-CFAR) detector in Weibull clutter is derived. Optimal threshold values of distributed maximum likelihood CFAR (ML-CFAR) detectors and distributed OS-CFAR detectors with a known shape parameter of the background statistics are obtained using GA tool. For the distributed ML-CFAR detection, we consider also the case when the shape parameter is unknown of the Weibull distribution. A performance assessment is carried out, and the results are compared and given as a function of the shape parameter and of system parameters.     

Estimation of P[Y

Kundu  D. Gupta  R.D. 《Reliability, IEEE Transactions on》2006,55(2):270-280

Improved percentile estimation for the two-parameter Weibull distribution

This paper presents an improvement of a technique recently published to estimate the parameters of the two-parameter Weibull distribution. A simple percentile method is used to estimate the two parameters. Computer simulation is employed to compare the proposed method with the maximum likelihood estimation and graphical methods results. A set of frequently-used and newer expressions for estimating the cumulative density are examined. Comparisons are made with both complete and censored data. The primary advantage of the method is its computational simplicity. Results indicate that with respect to Mean Square Error and estimation of the characteristic value with complete data, the percentile method cannot outperform the maximum likelihood method, although differences are minor in many instances. However, with censored data, improvements over the maximum likelihood are observed. When the shape parameter is estimated, the percentile method is quite competitive with that of maximum likelihood for both complete and censored data under a variety of conditions.     

One method for interval estimation of Weibull shape parameter

For three appropriated values of the random variable t, the corresponding values of the confidence limits for the cumulative Weibull distribution F(t), with parameters: β, γ and η, were calculated. On the basis of the appropriated and calculated values, the interval estimation of the shape parameter β was computed. The practical application of this method is illustrated by two examples.     

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     

Analysis of step-stress accelerated-life-test data: a new approach

A linear cumulative exposure model (LCEM) is used to analyze data from a step-stress accelerated-life-test, in particular, those with failure-free life (FFL). FFL is characterized by a location parameter in the distribution. For the 2-parameter Weibull distribution, the Nelson cumulative exposure model is a special case of LCEM. Under LCEM a general expression is derived for computing the maximum likelihood estimator (MLE) of stress-dependent distribution parameters under multiple censoring. The estimation procedure is simple and is illustrated by a set of experimental data using the 3-parameter Weibull distribution