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| This example compares the competing failure mode calculations. | | This example validates the competing failure mode calculations in Weibull++ standard folios. |
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| {{Reference_Example_Heading1}} | | {{Reference_Example_Heading1}} |
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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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| [[Image:CFM_results.png|center|550px|]] | | [[Image:CFM_results.png|center]] |
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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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| [[Image:CFM_bounds.png|center|550px|]] | | [[Image:CFM_bounds.png|center]] |
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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]\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.
