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.     

Solution of the Three-Parameter Weibull Equations by Constrained Modified Quasilinearization (Progressively Censored Samples)

The location, shape, and scale parameters of the Weibull distribution are estimated from Type I progressively censored samples by the method of maximum likelihood. Nonlinear logarithmic likelihood estimating equations are derived, and the approximate asymptotic variance-covariance matrix for the maximum likelihood parameter estimates is given. The iterative procedure to solve the likelihood equations is a stable and rapidly convergent constrained modified quasilinearization algorithm which is applicable to the general case in which all three parameters are unknown. The numerical results indicate that, in terms of the number of iterations required for convergence and in the accuracy of the solution, the proposed algorithm is a very effective technique for solving systems of logarithmic likelihood equations for which all iterative approximations to the solution vector must satisfy certain intrinsic constraints on the parameters. A FORTRAN IV program implementing the maximum likelihood estimation procedure is included.     

Life Tests with Periodic Change in the Scale Parameter of a Weibull Distribution

Two life testing procedures, namely, the progressively censored samples and Bartholomew's experiment are discussed under the assumption that the life of an item follows a specialized Weibull distribution. The scale parameter is different under two different conditions of usage of the item at regular intervals of time, the shape parameter remains unchanged throughout the experiment. The maximum likelihood estimates of the two scale parameters have been derived along with their variances. A numerical example illustrates the type of data and relevant calculations for the experiment involving progressively censored samples.     

Bayesian Estimation of Parameters of Life Distributions and Reliability from Type II Censored Samples

The Bayesian approach to reliability estimation from Type II censored samples is discussed here with emphasis on obtaining natural conjugate prior distributions. The underlying sampling distribution from which the censored samples are drawn follows a generalized life model (GLM) which includes a model proposed by Epstein and Sobel, Weibull, exponential, and Rayleigh distributions as special cases. Results are given for the Type II asymptotic distribution of largest values, Pareto, and Limited distribution. The natural conjugate prior, Bayes estimate for the generalized scale parameter, posterior risk, Bayes risk and Bayes estimate of the reliability function were derived for the distributions studied. In every case the natural conjugate prior is a 2-parameter family which provides a wide range of possible prior knowledge. Conjugate diffuse priors were derived. A diffuse prior, also called a quasi-pdf, is not a pdf because its integral is not unity. It represents roughly an informationless prior state of knowledge. The proper choice of the parameter for the diffuse prior leads to maximum likelihood, classical uniform minimum-variance unbiased estimator, and an admissible biased estimator with minimum mean square error as the generalized Bayes estimate. A feature of the GLM is the increasing function g(·) with possible applications in accelerated testing. KG(·) is a s-complete s-sufficient statistic for ?, and KG(·)/m is a maximum likelihood estimate for ?. Similar results were obtained for the Pareto, Type II asymptotic distribution of extremes, Pareto (associated with Pearl-Reed growth distribution) and others.     

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.     

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.     

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.     

Maximum likelihood estimation of Burr XII distribution parameters under Type II censoring

The mathematical theory for the point estimation of the parameters of the Burr Type XII distribution by maximum likelihood (ML) is developed for Type II censored samples. Also derived are necessary and sufficient conditions on the sample data that guarantee the existence, uniqueness and finiteness of the ML parameter estimates for all possible permissible parameter combinations. The asymptotic theory of ML is invoked to obtain approximate confidence intervals for the ML parameter estimates. An application to reliability data arising in a life test experiment is discussed.     

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.     

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.     

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     

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     

Inference Based on Type-II Hybrid Censored Data From a Weibull Distribution

A hybrid censoring scheme is a mixture of type-I and type-II censoring schemes. This article presents the statistical inferences on Weibull parameters when the data are type-II hybrid censored. The maximum likelihood estimators, and the approximate maximum likelihood estimators are developed for estimating the unknown parameters. Asymptotic distributions of the maximum likelihood estimators are used to construct approximate confidence intervals. Bayes estimates, and the corresponding highest posterior density credible intervals of the unknown parameters, are obtained using suitable priors on the unknown parameters, and by using Markov chain Monte Carlo techniques. The method of obtaining the optimum censoring scheme based on the maximum information measure is also developed. We perform Monte Carlo simulations to compare the performances of the different methods, and we analyse one data set for illustrative purposes.     

Estimation of the Parameters of the Weibull Distribution from Multicensored Samples

Monte Carlo simulations are used to investigate the maximum likelihood estimators for the parameters of the Weibull distribution using multicensored samples. For the case of unequal number of items censored, a single situation is considered. When the same number of items are removed at each test, several different estimators are compared for several values of the parameters.     

Best Linear Unbiased Estimator of the Parameter of the Rayleigh Distribution?Part I: Small Sample Theory for Censored Order Statistics

The best linear unbiased estimator of the parameter of the Rayleigh distribution using order statistics in a Type II censored sample from a potential sample of size N is considered. The coefficients for this estimator are tabled to five decimal places for N = 2(1)15 and censoring values of r1, (the number of observations censored from the left) and r2 (the number of observations censored from the right) such that r1 + r2 ? N - 2 for N = 2(1)10, r1 + r2 ? N - 3 for N = 11(1)15.     

Analysis of a simple step-stress life test with a random stress-change time

This paper studies a variation of a simple step-stress life testing in which the stress change time is random, and the test is subject to type II censoring. We assume that only two order statistics from the test are observed. The first observed order statistic is the stress change time from a low level stress to a high level stress during the testing, and the second observed order statistic is the final failure time when the test is censored. We first present the joint probability distribution of the two order statistics observed from the simple step-stress accelerated life test. Maximum likelihood estimates, and the method of moment estimates for model parameters based on the joint distribution are considered. We also present the exact confidence interval estimates for the model parameters based on various pivotal quantities, and demonstrate the estimation procedure by a simulated example.     

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     

Best linear unbiased estimator of the Rayleigh scale parameterbased on fairly large censored samples

This paper tabulates: coefficients and relative s-efficiency of the best linear unbiased estimator (BLUE) of the scale parameter of the Rayleigh distribution for type II censored samples of size N=20(5)40, r=0(1)4 (number of observations censored from the left) and s=0(1)4 (number of observations censored from the right); and ranks, coefficients, variances, and relative s-efficiencies of the BLUE of a based on a selected few order statistics (k) for sample size N=20(1)40 and k=2(1)4. These estimators have the minimum variance among the BLUE of a based on the same number of order statistics. As compared to maximum likelihood estimators (MLE) and approximate MLE, the s-efficiency of BLUE of ρ is very high. When estimating the parameter using only a few observations, the k-optimum BLUE of ρ is the only choice, as the MLE of ρ is not available. Therefore, these tables for coefficients of BLUE of ρ based on censored samples and few observations for moderately large samples, have many applications     

Bayes estimation of the parameters and reliability function of the3-parameter Weibull distribution

The authors obtain Bayes estimates of the parameters and reliability function of a 3-parameter Weibull distribution and compare posterior standard-deviation estimates with the corresponding asymptotic standard-deviation estimates of their maximum likelihood counterparts. Numerical examples are given     

Inference for a Multivariate Exponential Distribution with a Censored Sample

Maximum likelihood estimators for the parameters of a multivariate exponential Cdf are easily obtained from partial information about a random sample, censored or not. The partial information consists of the minimum from each multivariate observation and the counts of how often each r.v. was equal to the minimum in an observation. The censoring might cause only the smallest r out of n minima to be observed along with the counts. The estimators depend on the total time-on-test statistic familiar in univariate exponential life testing. A likelihood ratio test for s-independence is derived which has s-significance ? = 0 and easily calculated power function.