Competing Failure Modes

This example validates the competing failure mode calculations in Weibull++ standard folios.

The data set is from Table 15.1 on page 383 in the book Statistical Methods for Reliability Data by Dr. Meeker and Dr. Escobar, John Wiley & Sons, 1998.

In the book, parameters $$\mu\,\!$$ and $$\sigma\,\!$$ are used for the Weibull distribution. They are defined by $$\mu = ln(\eta)\,\!$$ and $$\sigma = \frac{1}{\beta}\,\!$$. The results are:


 * For failure mode s, the log-likelihood value is -101.36.
 * For failure mode s, $$\mu_{s}\,\!$$ = 6.11 and its approximated 95% confidence interval are [5.27, 6.95].
 * For failure mode s, $$\sigma_{s}\,\!$$ = 1.49 and its approximated 95% confidence interval are [0.94, 2.36].
 * For failure mode w, the log-likelihood value is -47.16.
 * For failure mode w, $$\mu_{w}\,\!$$ = 5.83 and its approximated 95% confidence interval are [5.62, 6.04].
 * For failure mode w, $$\sigma_{s}\,\!$$ = 0.23 and its approximated 95% confidence interval are [0.12, 0.44].


 * The following picture shows the ML estimates and the variance covariance matrix for each failure mode.




 * The following picture shows the 95% confidence intervals for the parameters of each failure mode.




 * In terms of $$\mu\,\!$$ and $$\sigma\,\!$$, the results are:


 * For failure mode s, $$\mu_{s} = ln(\eta_{s})\,\!$$ = 6.11 and its approximated 95% confidence interval are [5.27, 6.95].
 * For failure mode s, $$\sigma_{s} = \frac{1}{\beta_{s}}\,\!$$ = 1.49 and its approximated 95% confidence interval are [0.94, 2.36].
 * For failure mode w, $$\mu_{w} = ln(\eta_{w})\,\!$$ = 5.83 and its approximated 95% confidence interval are [5.62, 6.04].
 * For failure mode w, $$\sigma_{s} = \frac{1}{\beta_{s}}\,\!$$ = 0.23 and its approximated 95% confidence interval are [0.12, 0.44].

The above results are exactly the same as the results in the book.