Bayesian Weibull Estimation


Lamberson, Leonard R.; DeSouza Jr., Daniel I.   (1987, ASQC)   Wayne State University, Detroit, MI; Fluminese Federal University, Rio De Janeiro, Brazil

41st Annual Quality Congress, May 1987, Minneapolis, MN    Vol. 41    No. 0
QICID: 3339    May 1987    pp. 497-506
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Article Abstract

The Weibull distribution is a very popular life testing model used for mechanical components. In laboratory testing of mechanical components the sample size is often relatively small and the tests tend to be done in a similar fashion from one design evolution to the next. In this product development environment the engineer has considerable past experience on what constitutes good results from the life test and hence attaches more significance to the test data than would be warranted by application of traditional statistical estimation procedures. This suggests an ideal situation for the application of the Bayesian estimation approach.The philosophy of the Bayesian estimation approach is that one attempts to quantify prior knowledge about parameters of the life testing distribution through the use of a prior distribution for the unknown parameters. In the case of the two-parameter Weibull distribution we have two parameters which must be quantified by prior distributions. These parameters are the Weibull slope (shape parameter) and characteristic life (location parameter). We have modeled the typical life testing situation and provided a working Bayesian estimation procedure.Bayesian estimation has been previously applied to the Weibull distribution. In this previous work, the characteristic life parameter was assumed to follow an inverse gamma in one instance and an inverse Weibull in a second while the Weibull slope parameter was taken as discretely distributed in one instance and as uniformly distributed in a second. We have used the inverse Weibull for the prior distribution of the characteristic life and the Beta distribution as the prior for the Weibull slope parameter. The use of these two prior distributions provides a flexible and reasonable approach for quantifying prior information on the Weibull parameters. We have then developed an estimation procedure that provides a basis for a software package.The problem in applying Weibull Bayesian estimation is that the numerical calculations are very complex. With the proliferation of microcomputers we can now solve this problem. We have developed the basis for a software package that guides the practicioner through the application of the Bayesian estimation procedure. The software should run on most microcomputers with sufficient memory capacity. This software will be placed in the public domain and will be made freely available to those that want to experiment with it.

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