New format available! This reference is now available in a new format that offers faster page load, improved display for calculations and images and more targeted search.
As of January 2024, this Reliawiki page will not continue to be updated. Please update all links and bookmarks to the latest references at Weibull examples and Weibull reference examples.
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]\displaystyle{ \mu\,\! }[/math] and [math]\displaystyle{ \sigma\,\! }[/math] are used for the Weibull distribution. They are defined by [math]\displaystyle{ \mu = ln(\eta)\,\! }[/math] and [math]\displaystyle{ \sigma = \frac{1}{\beta}\,\! }[/math]. The results are:
- For failure mode s, the log-likelihood value is -101.36.
- For failure mode s, [math]\displaystyle{ \mu_{s}\,\! }[/math] = 6.11 and its approximated 95% confidence interval are [5.27, 6.95].
- For failure mode s, [math]\displaystyle{ \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]\displaystyle{ \mu_{w}\,\! }[/math] = 5.83 and its approximated 95% confidence interval are [5.62, 6.04].
- For failure mode w, [math]\displaystyle{ \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:
- The 95% confidence intervals for the parameters of each failure mode are:
- In terms of [math]\displaystyle{ \mu\,\! }[/math] and [math]\displaystyle{ \sigma\,\! }[/math], the results are:
- For failure mode s, [math]\displaystyle{ \mu_{s} = ln(\eta_{s})\,\! }[/math] = 6.11 and its approximated 95% confidence interval are [5.27, 6.95].
- For failure mode s, [math]\displaystyle{ \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]\displaystyle{ \mu_{w} = ln(\eta_{w})\,\! }[/math] = 5.83 and its approximated 95% confidence interval are [5.62, 6.04].
- For failure mode w, [math]\displaystyle{ \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.
