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