The Weibull process with unknown scale parameter is taken as a model for Bayesian decision making. The family of natural conjugate prior distributions for the scale parameter is exhibited and used in prior and posterior analysis. Preposterior analysis and several sampling schemes are then discussed. Preposterior analysis is given for an acceptance sampling problem with utility linear in the unknown mean of the Weibull process, in which the sampling scheme yields the first r failures in a life test of n items. An example is included.

Bayesian Analysis of the Weibull Process with Unknown Scale Parameter and Its Application to Acceptance Sampling

This paper presents a simple method for obtaining exact lower confidence bounds for reliabilities (tail probabilities) for items whose life times follow a Weibull distribution where both the “shape” and “scale” parameters are unknown. These confidence bounds are obtained both for the censored and non-censored cases and are asymptotically efficient. They are exact even for small sample sizes in that they attain the desired confidence level precisely. The case of an additional unknown “location” or “shift” parameter is also discussed in the large sample case. Tables are given of exact and asymptotic lower confidence bounds for the reliability for sample sizes of 10, 15, 20, 30, 50 and 100 for various censoring fractions.

An Exact Asymptotically Efficient Confidence Bound for Reliability in the Case of the Weibull Distribution

A test based on maximum likelihood estimators is given for testing the equality of the shape parameters in two Weibull distributions with the scale parameters unknown. Tests for the equality of the scale parameters are also presented along with a procedure for selecting the Weibull process with the larger mean life.

Two Sample Tests in the Weibull Distribution

Some adaptive tests are proposed (a) for the general two-sample scale problem and (b) for the restricted two-sample scale problem, in which the random variables are known to be right-skewed and also positive valued. These are based on either the ratio of the adaptively trimmed sample variances or adaptive rank tests of scale, whose quantiles are estimated by means of the bootstrap. Simulations show that, for practical sample sizes, these tests tend to perform better than their competitors. The methodology is illustrated by an application to a problem considered by Nair regarding the breakdown times of an insulating fluid under elevated voltage stress.

Adaptive inference for the two-sample scale problem

A method for construction of cumulative sum control charts for controlling the mean of a Weibull distribution is described As a special case, charts appropriate to exponentially distributed variables can be constructed There is some investigation of the results of using such charts when a non-exponential Weibull distribution would be more appropriate The paper concludes with a discussion of certain formulas in the related analysis of sequential probability ratio tests for Weibull distributions

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Cumulative Sum Control Charts and the Weibull Distribution

Given two production processes, the units from which fail in accordance with the Weibull distribution, the problem of selecting the particular process with the larger mean life is considered. Three techniques are considered and three procedures are constructed to show their advantages. An alternative sequential procedure has also been proposed and shown to be valuable in some situations. This paper may be regarded as a generalization of Sobel's earlier work [21] on the exponential distribution.

A. S. Qureishi

Papers

The Discximination Between Two Weibull Processes