Competing Failure Modes

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Competing Failure Modes

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


Reference Case

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.


Data

State F/S Time to F/S Failure Mode
F 275 w
F 13 s
F 147 w
F 23 s
F 181 w
F 30 s
F 65 s
F 10 s
S 300
F 173 s
F 106 s
S 300
S 300
F 212 w
S 300
S 300
S 300
F 2 s
F 261 s
F 293 w
F 88 s
F 247 s
F 28 s
F 143 s
S 300
F 23 s
S 300
F 80 s
F 245 w
F 266 w


Result

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

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


Results in Weibull++

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


CFM results.png


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


CFM bounds.png


  • In terms of [math]\mu\,\![/math] and [math]\sigma\,\![/math], the results are:
  • For failure mode s, [math]\mu_{s} = ln(\eta_{s})\,\![/math] = 6.11 and its approximated 95% confidence interval are [5.27, 6.95].
  • For failure mode s, [math]\sigma_{s} = \frac{1}{\beta_{s}}\,\![/math] = 1.49 and its approximated 95% confidence interval are [0.94, 2.36].
  • For failure mode w, [math]\mu_{w} = ln(\eta_{w})\,\![/math] = 5.83 and its approximated 95% confidence interval are [5.62, 6.04].
  • For failure mode w, [math]\sigma_{s} = \frac{1}{\beta_{s}}\,\![/math] = 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.