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| 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. | | 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. |
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| 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: | | 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: |
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| * For failure mode s, the log-likelihood value is -101.36. | | * For failure mode s, the log-likelihood value is -101.36. |
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| * The ML estimates and the variance covariance matrix for each failure mode are: | | * The following picture shows the ML estimates and the variance covariance matrix for each failure mode. |
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| * The 95% confidence intervals for the parameters of each failure mode are: | | * The following picture shows the 95% confidence intervals for the parameters of each failure mode. |
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This example compares the competing failure mode calculations.
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]\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 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]\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.
