Maximum Likelihood Estimation For The 2-Parameter Weibull Distribution Based On Interval-Data

MLE techniques are presented for estimating time-to-failure distributions from interval-data. Interval-data consist of adjacent inspection times that surround an unknown failure time. Censored interval-data bound the unknown failure time with only a lower time. The 2-parameter Weibull distribution is examined as the failure distribution. Parameter estimates from interval-data and from the midpoints of the intervals are compared for 6 shapes of the Weibull distribution. The results from Monte Carlo simulation runs are used to examine the s-bias and S-variability of the parameter estimates.     

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     

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.     

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     

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     

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.     

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.     

Goodness-of-fit tests for type-I extreme-value and 2-parameterWeibull distributions

This study investigates the properties of the Kolmogorov-Smirnov (K-S), Cramer-von Mises (C-M) and Anderson-Darling (A-D) statistics for goodness-of-fit tests for type-I extreme-value and for 2-parameter Weibull distributions, when the population parameters are estimated from a complete sample by graphical plotting techniques (GPT). Three GPT-median ranks, mean ranks, symmetrical sample cumulative distribution (symmetrical ranks)-are combined with the least-squares method (LSM) on extreme-value and Weibull probability paper to estimate the population parameters. The critical values of the K-S, C-M, A-D statistics are calculated by Monte Carlo simulation, in which 10⁶ sets of samples for each sample size of 3(1)20, 25(5)50, and 60(10)100 are generated. The power of the K-S, C-M, A-D statistics are investigated for 3 graphical plotting techniques and for maximum likelihood estimators (MLE). A Monte Carlo simulation provided the power results using 10⁴ repetitions for each sample size of 5, 10, 25, 40. The power comparison showed that: Among 3 GPT, the symmetrical ranks give more powerful results than the median and mean ranks for the K-S, C-M, A-D statistics; Among 3 GPT and the MLE, the symmetrical ranks provide more powerful results than the MLE for the K-S and A-D statistics; for the C-M statistic, the MLE provide more powerful results than 3 GPT; Generally, the A-D statistic coupled with the symmetrical ranks and LSM is most powerful among the competitors in this study and is recommended for practical use     

Minimum Distance Estimation of the Three Parameters of the Gamma Distribution

Procedures have been investigated to establish robust, adaptive estimating techniques for the 3-parameter Gamma distribution, The procedures incorporate minimum distance statistics for determining the location parameter for a range of sample sizes and shape parameters. Seven new estimators were developed of which six incorporate minimum distance estimation for determining the location parameter or guaranteed life with the remaining parameters estimated by maximum likelihood. All the estimators were compared with maximum likelihood estimators (MLEs) using 1000 Monte Carlo repetitions. The criteria of comparison was the ratio of the mean square errors of the parameter estimates. All the new estimators give better results than the MLE. The minimum distance estimation of the location parameter using the Anderson Darling goodness of fit statistic provided the overall best estimates of the parameters. As the sample size increased the relative position of MLEs improved but were still very inefficient with respect to best of the new estimators at sample size 20.     

A multiplicative damage model for strength of fibrous compositematerials

Knowledge of the tensile strength properties of a fibrous composite material is essential in the design of reliable structures from that material. Determination of statistical models for the tensile strength of a composite material which provide good fits to experimental data from tensile tests on material specimens is therefore important for engineering design. Perhaps the most commonly used statistical model is the Weibull distribution, based on `weakest link of a chain' arguments. However, in many cases the usual Weibull distribution does not adequately fit experimental data on tensile strength for composite materials made from brittle fibers such as carbon. Here, an alternative model is developed for tensile strength of carbon composites, which is based on a multiplicative cumulative-damage approach. This approach results in a 3-parameter extension of the Birnbaum-Saunders fatigue model and incorporates the material specimen size (size effect) as a known variable. This new distribution can also be written as an inverse Gaussian-type distribution, which can be interpreted as the first passage of the accumulated damage past a damage threshold, resulting in material failure. The new model fits experimental tensile-strength data, for carbon micro-composites better than existing models, providing more accurate estimates of material strength     

Inspection Intervals for Fail-Safe Structure

The Fail-Safe principle as applied to aircraft structural design implies that there is insufficient knowledge of the life capability of the design. Control of inspection intervals is not supported by risk calculations, yet only a sample of aircraft is inspected, at intervals whose duration is rapidly increased. This paper provides risk estimates based on a simple mathematical model. Catastrophic failure is treated in two stages modeled respectively by 2-parameter and 3-parameter Weibull distributions. Bayes inferences are made about the scale parameter using in-service survivor times. Only those cases are treated for which no failures have occurred. This results in a suggested form of inspection policy. A separate non-Bayes analysis confirms the Bayes risk estimate; thus the assumed improper prior is interesting. This prior, the only simple one which is tractable for the case of no failures, transforms, for the exponential distribution, to the uniform prior, in contrast to the hyperbolic one usually used. The analysis is simplistic but provides a ball-park estimate which would otherwise be unavailable. It can be used with caution as a check on inspection programs already derived by other means. It can also serve in tutorial demonstration of the statistical effects of the various parameters, to airworthiness managers. Possibly it might form the basis of a more sophisticated analysis.     

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.     

Point and Interval Estimation, From One-Order Statistic, of the Location Parameter of an Extreme-Value Distribution with Known Scale Parameter and of the Scale Parameter of a Weibull Distribution with Known Shape Parameter

This paper derives a one-order statistic estimator ?mn b for the location parameter of the (first) extreme-value distribution of smallest values with cumulative distribution function F(x;u,b) = 1 - exp {-exp[(x-u)/b]} using the minimum-variance unbiased one-order statistic estimator for the scale parameter of an exponential distribution, as was done in an earlier paper for the scale parameter of a Weibull distribution. It is shown that exact confidence bounds, based on one-order statistic, can be easily derived for the location parameter of the extreme-value distribution and for the scale parameter of the Weibull distribution, using exact confidence bounds for the scale parameter of the exponential distribution. The estimator for u is shown to be b ln cmn + xmn, where xmn is the mth order statistic from an ordered sample of size n from the extreme-value distribution with scale parameter b and Cmn is the coefficient for a one-order statistic estimator of the scale parameter of an exponential distribution. Values of the factor cmn, which have previously viously been tabulated for n = 1(1)20, are given for n = 21(1)40. The ratios of the mean-square-errors of the maximum-likelihood estimators based on m order statistics to those of the one-order statistic estimators for the location parameter of the extreme-value distribution and the scale parameter of the Weibull distribution are investigated by Monte Carlo methods. The use of the table and related tables is discussed and illustrated by numerical examples.     

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.     

Solving ML equations for 2-parameter Poisson-process models forungrouped software-failure data

Existence conditions are given for maximum likelihood (ML) parameter estimates for several families of 2-parameter software-reliability Poisson-process models. For each such model, the ML equations can be expressed in terms of one equation in one unknown. Bounds are given on solutions to these one equation problems to serve as initial intervals for search algorithms like bisection. Uniqueness of the solutions is established in some cases. Solutions are also tabulated for certain simple cases. Results are given for ungrouped failure data (exact times are available for all failures). ML estimation problems for such a situation are treated as limiting cases of problems based on failure times grouped into intervals of decreasing mesh     

Sequential tests for integrated-circuit failures

We propose a sequential probability ratio test (SPRT) based on a 2-parameter Weibull distribution for integrated-circuit (IC) failure analysis. The shape parameter of the Weibull distribution characterizes the decreasing, constant, or increasing failure rate regions in the bath tub model for IC. The algorithm (SD) detects the operating region of the IC based on the observed failure times. Unlike the fixed-length tests, the SD, due to its sequential nature, uses the minimum average number of devices for the test for fixed error tolerances in the detection procedure. We find that SD is, on average, 96% more statistically efficient than the fixed-length test. SD is highly robust to the variations in the model parameters, unlike other existing sequential tests. Since the accuracy of the tests and the test length are conflicting requirements, we also propose a truncated SD which allows a better control of this tradeoff. It has both the sequential nature of examining measurements and the fixed-length property of guaranteeing that the tolerances be met approximately with a specified number of available measurements     

Inference for a Simple Step-Stress Model With Type-II Censoring, and Weibull Distributed Lifetimes

The simple step-stress model under Type-II censoring based on Weibull lifetimes, which provides a more flexible model than the exponential model, is considered in this paper. For this model, the maximum likelihood estimates (MLE) of its parameters, as well as the corresponding observed Fisher Information Matrix, are derived. The likelihood equations do not lead to closed-form expressions for the MLE, and they need to be solved by using an iterative procedure, such as the Newton-Raphson method. We also present a simplified estimator, which is easier to compute, and hence is suitable to use as an initial estimate in the iterative process for the determination of the MLE. We then evaluate the bias, and mean square error of these estimates; and provide asymptotic, and bootstrap confidence intervals for the parameters of the Weibull simple step-stress model. Finally, the results are illustrated with some examples.      

A new model for repairable systems with bounded failure intensity

This paper proposes a new model, called the 2-parameter Engelhardt-Bain process (2-EBP) model, to describe the failure pattern of complex repairable systems subjected to reliability deterioration with the operating time, and showing a finite bound for the intensity function. The characteristics of the 2-EBP model are discussed, and the physical meaning of its parameters is derived. The 2-EBP model can be viewed as a dynamic power law process, whose shape parameter ranges from 2 to 1 as the system age increases, converging asymptotically to the homogeneous Poisson process. Maximum likelihood estimates of model parameters & other quantities of interest, as well as a testing procedure (based on the likelihood ratio statistic) for time trend, are provided. Numerical applications are given to illustrate the 2-EBP model & the related inferential procedures, and to emphasize on the caution to use in assuming the (very often used) power law process when the presence of a finite bound for the failure intensity is conjecturable.     

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     

Distinguishing between lognormal and Weibull distributions[time-to-failure data]

A previous method for deciding if a set of time-to-fail data follows a lognormal distribution or a Weibull distribution is expanded upon. Pearson's s-correlation coefficient is calculated for lognormal and Weibull probability plots of the time-to-fail data. The test statistic is the ratio of the two s-correlation coefficients. When "standardized", the lognormal and Weibull variables map into 1 of 2 gamma distributions with no dependence on the shape or scaling factors, confirming earlier observations. Using a set of Monte Carlo simulations, the test statistic was found to be s-normally distributed to good approximation. Formulas for estimating the mean and standard deviation of the test statistic were derived, allowing for an estimate of the probability of hypothesis test errors. As anticipated, the test capability increases with increasing sample size, but only if a substantial fraction of the parts actually fail. If less than 10% of the parts are stressed to failure, then it is almost impossible to distinguish between lognormal and Weibull distributions. If all parts are stressed to failure, the probability of making a correct choice is fair for sample sizes as small as 10, and becomes quite good if the sample size is at least 50. The statistical technique for distinguishing lognormal from Weibull distributions is presented. Its theoretical foundation is given at a qualitative level, and the range of useful application is explored. An approximate form for the distribution of the test statistic is inferred from Monte Carlo simulation