Competing Failure Modes

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Competing Failure Modes

This example compares the competing failure mode calculations.


Reference Case

The data is from the 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, parameter [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 ML estimates and the variance covariance matrix for each failure mode are:


CFM results.png


  • The 95% confidence intervals for the parameters of each failure mode are:


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.