Competing Failure Modes: Difference between revisions

From ReliaWiki
Jump to navigation Jump to search
No edit summary
No edit summary
Line 6: Line 6:
{{Reference_Example_Heading1}}
{{Reference_Example_Heading1}}


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.   




Line 76: Line 76:
| F||266||w
| F||266||w
|-
|-
|
|}
|}


Line 82: Line 81:
{{Reference_Example_Heading3}}
{{Reference_Example_Heading3}}


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:


* For failure mode s, the log-likelihood value is -101.36.
* For failure mode s, the log-likelihood value is -101.36.
Line 94: Line 93:
{{Reference_Example_Heading4}}
{{Reference_Example_Heading4}}


* 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.




Line 100: Line 99:




* 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.





Revision as of 16:36, 9 June 2014

Weibull Reference Examples Banner.png


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.




Competing Failure Modes

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.


CFM results.png


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


CFM bounds.png


  • 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.