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Appendix A Further Details Concerning the Bearing Cage Example _ ' n Chapter 3, an overview of a frequentist analysis of the reliability of a bearing cage was provided, starting with the analysis when the two ~ ~ Weibull parameters, 77 and A, are both estimated, and then demon- strating the benefits from using other sources of information to fix p. This was followed by a Bayesian analysis in which other sources of information were used to provide a prior for p. With respect to the frequentist analysis, this appendix provides addi- tional probability plots for the bearing cage data, fixing ~ at the values 1.5, 2, and 3. These plots show the lack of support for the assertion that B 10 is 8,000 hours. With respect to the Bayesian analysis, this appendix provides two plots, the first of the prior distribution for B10 and A, and the second for the posterior distribution for B10 and p. Both plots have the data likelihood superimposed. It is clear that the data likelihood has moved the posterior to be more consistent with it than the prior. Figures A-1, A-2, and A-3 show probability plots with the Weibull shape parameter ~ fixed at 1.5, 2, and 3, respectively. Although Figure A-1 may allow for a little optimism, the overall conclusion suggested by these figures is that the bearing cage is, most likely, not meeting its reliability goal. 81
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82 .2 1 - 05 - .02 - .01 - .005 - .003 .001 - .0005 - APPENDIXA _,, . ~ - 77= 28982 ,0= 1.5 200 500 1,000 2,000 5,000 10,000 Hours of Service FIGURE A-1 Weibull probability plot of the bearing cage failure data showing the maximum likelihood estimate of fraction failing with fixed ,0 = 1.5, the reliability target, and approximate confidence limits. SOURCE: Meeker and Escobar (1998~. .98 - 9 .7 .5 . _ ct. .05 ~ .02 .O .01 cd .005 IL .003 .001 .0005 0002 .0001 .00005 .00003 ., ., by= 12320 ~=2 1 1 1 · 1 1 1 200 500 1,000 2,000 Hours of Service 5,000 1 0,000 FIGURE A-2 Weibull probability plot of the bearing cage failure data showing the maximum likelihood estimate of fraction failing with fixed ,0 = 2, the reliability target, and approximate confidence limits. SOURCE: Meeker and Escobar (1998~.
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THE BEARING CA GE EXAMPLE .98 9- .7 5 - . _ IL .02 ~ a` o .u1 C.) .005 ~ .003 IL 83 77= 5409 p=3 I ~ I ~ I ~ ~ T ~ ~ 200 500 1,000 2,000 Hours of Service 5,000 1 0,000 FIGURE A-3 Weibull probability plot of the bearing cage failure data showing the maximum likelihood estimate of the fraction failing with fixed ,0 = 3, the reliability target, and approximate confidence limits. SOURCE: Meeker and Escobar (1998~. BAYES ANALYSIS Figure A-4 shows a sample of points from the joint prior distribution of ,0 and B 10 superimposed on a graph of the relative likelihood contours. The truth is expected to lie at the intersection of the prior and the likeli- hood. A sample from the posterior can be obtained by filtering the sample from the prior distribution. This can be done by selecting points in the prior distribution with a probability equal to the likelihood contour going through the point. Points are run through this filter until a sufficient num- ber of posterior points are obtained to estimate the posterior distribution (for two parameters, something on the order of 6,000 to 10,000 is suffi- cient). Figure A-5 shows a sample of points from the joint posterior distribu- tion, again with the likelihood contours.
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84 APPENDIXA 4.0 - 3.5 - 3.0 , 2.5] 2.0 - 1.5 - 1.0 - 0.5 ' 1~\ · . . · ~ · :,,.; ';~1,~,,:~, , . . . ~1 ~1 500 1,000 2,000 5,000 10,000 20,000 B10 FIGURE A-4 Sample points from the jointpriordistribution of B10 of bearing cage life and the Weibull ~ with the data likelihood superimposed. SOURCE: Meeker and Escobar (1998~. 4.0 3.5 3.0 2.5 - 1 1 ~ 1 ~~ ,,, Jo:" , 2.0 - 1.5 - 1.0 - 0.5 · 1 1 1 1 1 1 \ it\ . ~~\ ~ U. 1 ~1 500 1,000 2,000 5,000 B10 1 o,ooo 20,000 FIGURE A-5 Sample points from the jointposteriordistribution of B10 of bearing cage life and the Weibull ~ with the data likelihood superimposed. SOURCE: Meeker and Escobar (1998~.
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