Kenneth W. Fertig

Life-Test Sampling Plans for Two-Parameter Weibull Populations

In this paper, we give variables sampling plans for items whose failure times are distributed as either extreme-value variates or Weibull variates (the logarithms of which are from an extreme-value distribution). Tables applying to acceptance regions and operating characteristics for sample size n, ranging from 3 to 18, are given. The tables allow for Type II censoring, with censoring number r ranging from 3 to n. In order to fix the maximum time on test, the sampling plan also allows for Type I censoring. Acceptance/rejection is based upon a statistic incorporating best linear invariant estimates, or, alternatively, maximum likelihood estimates of the location and scale parameters of the underlying extreme value distribution. The operating characteristics are computed using an approximation discussed by Fertig and Mann (1980).

Tables for Obtaining Weibull Confidence Bounds and Tolerance Bounds Based on Best Linear Invariant Estimates of Parameters of the Extreme-Value Distribution

Tables are given for obtaining confidence bounds for the two parameters and the 90th, 95th, and 99th percentiles of the two-parameter Weibull or extreme-value distribut.ions. The tables are based on best linear invariant estimlators of extreme-value location and scale parameters and apply to samples of size n, n = 3(1)16, which may be censored at the mth smallest sample observation, m = 3(1)%. More extensive tables for samples of size n, n = 3(1)25, with censoring from m = 3(1)n, may be found in [12]. Discussion is given concerning other methods of obtaining confidence and tolerance bounds for these distributions, properties of the estimators on which the bounds are based and computational procedures used.

Interval Influence Diagrams

We describe a mechanism for performing probabilistic reasoning in influence diagrams using interval rather than point valued probabilities. We derive the procedures for node removal (corresponding to conditional expectation) and arc reversal (corresponding to Bayesian conditioning) in influence diagrams where lower bounds on probabilities are stored at each node. The resulting bounds for the transformed diagram are shown to be optimal within the class of constraints on probability distributions that can be expressed exclusively as lower bounds on the component probabilities of the diagram. Sequences of these operations can be performed to answer probabilistic queries with indeterminacies in the input and for performing sensitivity analysis on an influence diagram. The storage requirements and computational complexity of this approach are comparable to those for point-valued probabilistic inference mechanisms, making the approach attractive for performing sensitivity analysis and where probability information is not available. Limited empirical data on an implementation of the methodology are provided.

Simplified Efficient Point and Interval Estimators for Weibull Parameters

In this paper are tabulated values which allow one to obtain, without lengthy tables of weights, simplified linear estimates of Weibull or extreme-value distribution parameters which approximate best linear invariant, best linear unbiased, or maximumlikelihood estimates. Asymptotic distributional results proved herein, together with these tabulated values, make it possible to obtain confidence bounds for both parameters from censored or uncensored samples of size n, with n = 20(5)60. Other tabulations and related results are given by Bain (1972) and Engelhardt, and Bain (1973, 1974).

One-Sided Prediction Intervals for at Least p Out of m Future Observations from a Normal Population

Tables of factors are provided for constructing prediction intervals to contain at least p = m – l future observations from a normal distribution; the simplified new procedure for obtaining these factors is described. The tables pertain to selected values of the given sample size, n. the future sample size, m, and the number of items, M – l. in a future sample to be included in the prediction interval. Factors are given for n = 2(1)15(5)30(10)5O,∞, m = 20,25,30(10)80, and l = 1(1)8. An example demonstrates the use of the tables.

A Decision-Theoretic Approach to Defining Variables Sampling Plans for Finite Lots: Single Sampling for Exponential and Gaussian Processes

Abstract A decision-theoretic approach is used to derive a variables sampling plan applicable to finite lots. A sampling distribution is derived for a general manufacturing process, and one-sided acceptance regions are determined for the particular cases in which the manufacturing process is described by either a normal distribution or an exponential distribution. Comparisons show that for a given risk level, a substantial savings in sample sizes can be effected over those required by previously available procedures (hypergeometric plans or variables plans with an infinite lot assumption).

On Constructing Prediction Intervals for Samples from a Weibull or Extreme Value Distribution

Assume that a (possibly censored) sample is available from a population having a Weibull or extreme value distribution, and that a single future observation is to be obtained from the same population. This paper provides Monte Carlo estimates of percentiles of the distribution of a statistic S, that may be used to construct prediction intervals to contain the future observation. Such intervals are of practical importance to reliability engineers and to those investigating treatment of undesirable habitual behavior. An approximation is given for the distribution of S that is in very close agreement with the Monte Carlo results that are presented. The approximation applies also when the single future observation of interest is the smallest in a lot (or sample) of size m > 1.

A Goodness-of-Fit Test for the Two Parameter vs. Three Parameter Weibull; Confidence Bounds for ThreshoId

An investigation is described herein of modifications of the two-parameter Weibullgoodness-of-fit test of Mann, Scheuer and Fertig (1973). It is assumed that the sole alternative of interest is any three-parameter Weibull distribution. The power of candidate test statistics is investigated, therefore, only under various three-parameter Weibull alternative hypotheses. It is found that for a fixed selection of gaps (differences of adjacent order statistics) used in the numerator and in the denominator of the approximately F dist ributed test statistic, nothing is gained by weighting the gaps in order to minimize the variances and thus to maximize the numbers of degrees of freedom. A test statistic which is a modified version of that of Mann, Scheuer and Fertig is shown to have higher power under three-parameter Weibull alternatives, and a simple method for approximating critical values of the test statistic is described. The test statistic is shown to be a monotone function of an unknown threshold (location...

An Accurate Approximation to the Sampling Distribution of the Studentized Extreme Value Statistic

The principal result of this paper is the discovery of an excellent approximation to the sampling distribution of the statistic used in the construction of Weibull tolerance bounds. The approximation is compared over a wide range of cases to Monte Carlo results and to two other recently derived approximations. In no case has it been found to be deficient.

Efficient Unbiased Quantile Estimators for Moderate-Size Complete Samples from Extreme-Value and Weibull Distributions; Confidence Bounds and Tolerance and Prediction Intervals

Tables of factors are given for complete samples of size n (n = 20(1)40) for correcting small-sample bias in Hassaneins [4] asymptotically unbiased quantile estimators of extreme-value location and scale parameters. Hassaneins k-order-statistic estimators of the two parameters are based on the same set of spacings for each k (k = 2. … 10) and are asymptotically best of this type of linear estimator (with asymptotic efficiencies of .977 and .937, respectively, for k = 10). The tabulated values not only allow one to obtain estimates based on the specified set of ordered observations that are best linear unbiased or best linear invariant (for the specified set of weights), but they also enable one to use procedures described in Mann. Schafer, and Singpurwalla [13] to compute approximate confidence bounds and tolerance and prediction intervals. Also tabulated are efficiencies of unbiased versions of the estimators relative to Cramer-Rao bounds for regular unbiased estimators and to best linear unbiased esti...