Albert H. Moore

Maximum-Likelihood Estimation of the Parameters of Gamma and Weibull Populations from Complete and from Censored Samples

Iterative procedures are given for joint maximum-likelihood estimation, from complete and censored samples, of the three parameters of Gamma and of Weibull populations. For each of these populations, the likelihood function is written down, and the three maximum-likelihood equations are obtained. In each case, simultaneous solution of these three equations would yield joint maximum-likelihood estimators for the three parameters. The iterative procedures proposed to solve the equations are applicable to the most general case, in which all three parameters are unknown, and also to special cases in which any one or any two of the parameters are known. Numerical examples are worked out in which the parameters are estimated from the first m failure times in simulated life tests of n items (m ≤ n), using data drawn from Gamma and Weibull populations, each with two different values of the shape parameter.

Maximum-Likelihood Estimation, from Doubly Censored Samples, of the Parameters of the First Asymptotic Distribution of Extreme Values

Let X be a random variable having the first asymptotic distribution of smallest (largest) values, with location parameter u and scale parameter b, b > 0. The natural logarithm of the likelihood function of a sample of size n from such a distribution, the lowest r1 and the highest r2 sample values having been censored, is written down and its first and second partial derivatives with respect to the parameters are worked out. The likelihood equations, obtained by equating to zero the first partial derivatives, do not have explicit solutions, but an iterative procedure for solving them on an electronic computer is described. The asymptotic variances and covariances of the maximum-likelihood estimators of the parameters are obtained by inverting the information matrix, whose elements are the negatives of the limits, as n → ∞, of the expected values of the second partial derivatives, and tabulated for censoring proportions q1 = 0.0(0.1)0.9 from below and q2 = 0.0(0.1) (0.9 – q1) from above. The asymptotic vari...

Point and Interval Estimators, Based on m Order Statistics, for the Scale Parameter of a Weibull Population with Known Shape Parameter

A derivation is given of the maximum likelihood estimator , based on the first m out of n ordered observations, of the scale parameter θ of a Weibull population with known shape parameter K. It is shown that 2m k/θK has a chi-square distribution with 2m degrees of freedom (independent of n). Use is made of this fact to set upper confidence bounds with confidence level 1 – P (lower confidence bounds with confidence level P) on the scale parameter θ. Formulas are given for the mean squared deviations of the upper and lower confidence bounds from the true value of the parameter. From these one can obtain expressions for the efficiency of confidence bounds and confidence intervals. The expected value of is also determined, and from it the unbiasing factor / by which must be multiplied to obtain an unbiased estimator . An expression for the variance of the unbiased estimator is found. Values of the unbiasing factor and of the variance of the unbiased estimator, both of which are independent of n, are tabled fo...

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.

A Monte Carlo Technique for Obtaining System Reliability Confidence Limits from Component Test Data

A digital computer technique is developed, using a Monte Carlo simulation based on common probability models, with which component test data may be translated into approximate system reliability limits at any confidence level. The probability distributions from which the component failures are assumed to come are the exponential, Weibull (shape parameter K known), gamma (shape parameter ? known), normal, and lognormal. The components can be arranged in any system configuration, series, parallel, or both. Since reliability prediction is meaningful only when expressed with an associated confidence leve, this method provides a valuable and economical tool for the reliability analyst.

Maximum-Likelihood Estimation, from Censored Samples, of the Parameters of a Logistic Distribution

Abstract Consider an ordered random sample of size n from a logistic distribution with location parameter μ and scale parameter σ. Let the r 1 smallest and the r 2 largest sample values be censored, where r 1 ≥0,r 2 ≥0, and r 1 +r 2 ≤n −2. An iterative procedure for solution (on a computer) of the likelihood equations is outlined and illustrated by a numerical example involving Strontium-90 concentrations in milk. The information matrix is inverted numerically to obtain the asymptotic variances and covariances of the maximum-likelihood estimators of the parameters for censoring proportions, q 1 = r 1/n and q 2 = r 2 /n, at intervals of one-tenth. The means and variances of , and , together with the covariances of and , are reported for a Monte Carlo study of 1000 samples each of size n = 10. The mean square errors of the estimates are compared with the variances of the best linear unbiased estimators and with the variances of the maximum-likelihood estimators as given by the asymptotic formulas.

Series-system reliability-estimation using very small binomial samples

This investigation explored the effect of incorporating prior information into series-system reliability estimates, where the inferences are made using very small sets (less than 10 observations) of binomial test-data. To capture this effect, the performance of a set of Bayes interval estimators was compared to that of a set of classical estimators over a wide range of subsystem beta prior-distribution parameters. During a Monte Carlo simulation, the Bayes estimators tended to provide shorter interval estimators when the mean of the prior system-reliability differed from the true reliability by 20 percent of less, but the classical estimators dominated when the difference was greater. Based on these results, the authors conclude that there is no clear advantage to using Bayes interval estimation for sample sizes less than 10 unless the poor mean system reliability is believed to be within 20 percent of the true system reliability. Otherwise, the Lindstrom-Madden estimator, a useful classical alternative for very small samples, should be used.

A Modified Kolmogorov-Smirnov Test for Weibull Distributions with Unknown Location and Scale Parameters

When the parameters in a continuous distribution are unknown and must be estimated, the standard Kolmogorov-Smirnov (K-S) goodness-of-fit tables do not represent the true distribution of the test statistics. This paper uses Monte Carlo techniques to create tables of critical values for a K-S type test for Weibull distributions with unknown location and scale parameters, but known shape parameter. The power of the proposed test is investigated, as is the relationship between critical values and the shape parameters. The results indicate that the modified K-S test appears to be a reasonable goodness-of-fit test for the Weibull family with unknown scale and location parameters.

A Monte-Carlo Technique for Estimating Lower Confidence Limits on System Reliability Using Pass-Fail Data

Several methods for estimating lower s-confidence limits (LCLs) for system reliability were examined using pass-fail data on the components. A new technique was used to obtain limits for selected systems. These limits were compared to those obtained by other methods. The new method was tested to verify its accuracy. Results indicate that the proposed technique is simple to understand, easy to implement, and accurate. One need only supply the component reliabilities, the number of component tests, and the desired level of s-confidence, to obtain, not only an estimated LCL of the system reliability, but also an idea of the accuracy of the estimate. Most of the other techniques are not valid in the case of zero-failures, whereas this method easily accommodates such a situation. This method is not restricted to series systems; it can easily handle parallel configurations.

Modified goodness-of-fit tests for the inverse Gaussian distribution

Abstract Modified Kolmogorov-Smirnov (KS), Kuiper (V), Cramer-von Mises (CV), Watson (W), Anderson-Darling (AD) and sequential goodness-of-fit tests are developed for the inverse Gaussian distribution with unknown parameters. A Monte Carlo procedure is employed to generate critical values for a wide range of sample sizes and shape parameters. Power studies indicate that the W test is most effective against alternate distributions that are very similar in shape to the null inverse Gaussian distribution. Otherwise, the modified AD test generally demonstrates the highest power among single tests. To eliminate the need for extensive critical value tables, functional relationships between critical values, sample sizes, and shape parameters are reported.