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.     

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 class of subset selection procedures for Weibull populations

Among distributions that represent component and system life, the Weibull family is presented. Goodness of a Weibull population is defined in terms of the shape parameter. A class of subset selection-procedures based on the U-statistic (derived via the concept of convex ordering) is proposed for two cases: (1) selecting a subset from k Weibull populations which contains the population with the largest shape parameter; and (2) selecting a subset containing all the populations whose shape parameters are at least that of the control population; unknown and known control populations are considered. The approximate implementation of the selection procedures with the help of the existing tables is discussed, and a numerical example is given. Statistical simulation is used to investigate the optimal members of the proposed class in case 1. The simulation shows that a common sample-size of 11 or more can be used to implement the asymptotic results     

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.     

Bayes interval for the Weibull parameters utilizing guessed estimates

In this paper an attempt is made to provide a method of obtaining the HPD-intervals for the scale and shape parameters of the Weibull distribution, when a prior distribution of the parameter places a weight (1 − b) on the guess value of the parameter and distributes the rest probability mass b according to some specified distribution. The equal tail credible intervals have been obtained and it is proposed that these limits be used as initial points for obtaining the HPD-intervals.     

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     

Bayes estimation of reliability in stress-strength model of Weibull distribution with equal scale parameters

The paper provides a Bayesian approach to inference about the reliability in a multicomponent stress-strength system. We consider Bayes' estimator of the system reliability from data consisting of a random sample from the stress distribution and one from the strength distribution when the two distributions are Weibull with equal and known scale parameters. The estimator of λ, ratio of two shape parameters, is also considered. The proposed estimators can be compared with the maximum likelihood estimators (mles). However, the comparison is carried out for single component stress-strength system and the Monte Carlo efficiencies are obtained. It is found that the proposed estimators are better than the corresponding mles.     

Accelerated Life Test Planning With Independent Weibull Competing Risks With Known Shape Parameter

This article presents methodology for accelerated life test (ALT) planning when there are two or more failure modes, or competing risks which are dependent on one accelerating factor. It is assumed that the failure modes have respective latent (unobservable) failure times, and the minimum of these times corresponds to the product lifetime. The latent failure times are assumed to be s-independently distributed Weibull with known, common shape parameter. Expressions for the Fisher information matrix, and test plan criteria are presented. The methodology is applied to the ALT of Class-H insulation for motorettes, where temperature is the accelerating factor. Two-level, and 4:2:1 allocation test plans based on determinants, and on estimating quantiles or hazard functions, are presented. Sensitivity of optimal test plans to the specified Weibull shape parameter is also studied     

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.     

A cautionary tale about Weibull analysis [reliability estimation]

This paper makes three points about possible perils of unguarded fitting of Weibull distributions to data: (1) bias is introduced by incomplete data, which may have counter-intuitive effects; (2) bias is introduced into percentile estimates by using regression on log-transformed variables to fit the Weibull parameters, particularly if the percentile to be predicted lies outside the range of the data; and (3) the amount of variation associated with such estimates can be very substantial. A partial solution to the incomplete data problem using simulation is presented, and the maximum likelihood approach to parameter estimation and its advantages relative to regression estimation are explained. The problem arose in predicting life expectancy of long-lived components subject to natural aging which cannot be investigated using accelerated testing and for which the collection of data provides an incomplete life record     

Accelerated life tests for products of unequal size

This paper considers `estimation of the lifetime distribution' and `optimal design of constant-stress accelerated life test plans' for products of unequal size. The distribution is Weibull with a scale parameter that is a `log-linear function of stress' and a `power function of product size with a size-effect parameter'. Maximum likelihood estimators (MLE) of model parameters are obtained, and their properties are studied. Two stress-level optimal test plans are obtained for products that come in two sizes, and a table useful for finding optimal test plans is given. The sum of asymptotic variances of MLE of a specified quantile of the distributions for products of both sizes is used as the optimality criterion. Optimum plans can be used when the ratio of two sizes is not too large. When the ratio is very large, the preestimate of size effect parameter should be carefully chosen     

Selecting, under type-II censoring, Weibull populations that aremore reliable

It is pointed out that the problem of selecting Weibull populations that are more reliable is complex; the main result is that there is no simple selection rule. Under type-II censoring, the use of a locally optimal selection rule when the shape parameters are known and the use of a modified selection rule when the unknown shape parameters have some prior distributions are proposed. The performance of this modified rule was tested extensively by simulation; this rule was shown to be quite robust for a variety of beta prior distributions     

Bayesian Analysis of the Weibull Process With Unknown Scale and Shape Parameters

Previously, the Weibull process with an unknown scale parameter was examined as a model for Bayesian decision making. The analysis is extended by treating both the shape and scale parameters as unknown. It is not possible to find a family of continuous joint prior distributions on the two parameters that is closed under sampling, so a family of prior distributions is used that places continuous distributions on the scale parameter and discrete distributions on the shape parameter. Prior and posterior analyses are examined and seen to be no more difficult than for the case in which only the scale parameter is treated as unknown, but preposterior analysis and determination of optimal sampling plans are considerably more complicated in this case. To illustrate the use of the present model, an example is presented in which it is necessary to make probability statements about the mean life and reliability of a long-life component both before and after life testing.     

Unbiased Maximum-Likelihood Estimation of a Weibull Percentile when the Shape Parameter is Known

The maximum-likelihood (ML) estimator for a percentile of a Weibull distribution with a known shape parameter is considered. Multiplicative correction factors are listed for rendering the ML estimator mean or median unbiased in the cases where the samples are type II censored with or without replacement. The correction factors depend upon the number of failures and the shape parameter but are independent of the sample size and the percentile being estimated.     

Inference on Weibull Percentiles From Sudden Death Tests Using Maximum Likelihood

A sudden death test is a special case of a multiply censored life test wherein an equal number of randomly selected surviving items are removed from the test following the occurrence of each failure. Confidence limits for the Weibull-shape parameter and a Weibull percentile may be set with a sudden death sample, using the method of maximum likelihood. An expression is found for the medium ratio of the upper to lower 100 (1 - ?) percent confidence limits for a Weibul percentile under either sudden death or conventional type-II censored testing. It is proposed that this ratio be used as a criterion for determining whether a given sudden death test is more precise than a given conventional test.     

k-Sample Maximum Likelihood Ratio Test for Change of Weibull Shape Parameter

A k-sample maximum likelihood ratio (MLR) test is derived to test equality of shape parameters for 2-parameter Weibull populations. The test is independent of the scale parameters, and the power depends on ratios of the shape parameters. Critical points and power calculations were obtained by Monte Carlo techniques for k = 2. The MLR test is equivalent to the MLR test of scale parameters for the extreme value distribution.     

Some Graphical Techniques for Estimating Weibull Confidence Intervals

Many methods for estimating the parameter and percentile statistical confidence intervals for the Weibull and Gumbel (extreme value) distributions have been described in the literature. Most of these methods depend on extensive computer programs, require reference to tables which do not cover all sample sizes of interest and/or are not widely available. This paper describes a semi-empirical technique which permits rapid estimation of the 2-sided 90% statistical confidence intervals for the Weibull or Gumbel distribution parameters, as well as for the 1, 5, 10 percentiles. The estimates can be obtained for type II censoring and sample sizes to 25. The statistical confidence intervals calculated using this method are not exact, but are very good approximations and are useful to engineers who do not have ready access to programs or lengthy tables, or who require quick estimates. If more accurate statistical confidence intervals are required, then the more complicated methods described in the references should be used.     

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     

Complete Sample Estimation Techniques for Reparameterized Weibull Distributions

The Weibull distribution, frequently employed to assign probabilities to the lifetimes of components and systems operating under stress, is habitually characterized by a pair of positive parameters, termed the scale and shape parameters. Two fundamental reparameterizations of the Weibull probability density function are proposed. The first reparameterization replaces the shape parameter by its inverse, the resulting positive parameter thereafter termed the shaping parameter. This permits a more facile exposition of the properties of parameter estimates, derived in the event that a complete random sample from the Weibull distribution is available. The characteristics of these parameter estimation techniques are then reviewed and compared, and their variances and distributional properties are delineated whenever possible. A second reparameterization extends the parameter space so as to include nonpositive values of the shape parameter. This extension augments the utility and applicability of the Weibull distribution without requiring radical alteration of the standard parameter estimation procedures applicable to the original parameter space.     

Modified KS, AD, and C-vM tests for the Pareto distribution withunknown location and scale parameters

Standard goodness-of-fit tests based on the empirical CdF (Edf) require continuous underlying distributions with all parameters specified. Three modified Edf-type tests, the Kolmogorov-Smirnov (K-S), Anderson-Darling (A-D), and Cramer-von Mises (C-vM), are developed for the Pareto distribution with unknown parameters of location and scale and known shape parameter. The unknown parameters are estimated using best linear unbiased estimators. For each test, Monte Carlo techniques are used to generate critical values for sample sizes 5(5)30 and Pareto shape parameters 0.5(0.5)4.0. The powers of the modified tests are investigated under eight alterative distributions. In most cases, the powers of the modified K-S, A-D, C-vM tests are considerably higher than the chi-square test. Finally, a functional relationship is identified between the modified K-S and C-vM test statistics and the Pareto shape parameter. Powerful goodness-of-fit tests that supplement the best linear unbiased estimates are provided