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===Goodness-of-Fit Tests for Repairable System Analysis===
#REDIRECT [[RGA Models for Repairable Systems Analysis]]
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It is generally desirable to test the compatibility of a model and data by a statistical goodness-of-fit test. A parametric Cramér-von Mises goodness-of-fit test is used for the multiple system and repairable system Power Law model, as proposed by Crow in [17]. This goodness-of-fit test is appropriate whenever the start time for each system is 0 and the failure data is complete over the continuous interval  <math>[0,{{T}_{q}}]</math>  with no gaps in the data. The Chi-Squared test is a goodness-of-fit test that can be applied under more general circumstances. In addition, the Common Beta Hypothesis test also can be used to compare the intensity functions of the individual systems by comparing the  <math>{{\beta }_{q}}</math>  values of each system. Lastly, the Laplace Trend test checks for trends within the data. Due to their general applicatoin, the Common Beta Hypothesis test and the Laplace Trend test are both presented in Appendix B. The Cramér-von Mises and Chi-Squared goodness-of-fit tests are illustrated next.
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{{cramer-con mises test rsa}}
 
====Chi-Squared Test====
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The parametric Cramér-von Mises test described above requires that the starting time,  <math>{{S}_{q}}</math> , be equal to 0 for each of the  <math>K</math>  systems. Although not as powerful as the Cramér-von Mises test, the Chi-Squared test can be applied regardless of the starting times. The expected number of failures for a system over its age  <math>(a,b)</math>  for the Chi-Squared test is estimated by  <math>\widehat{\lambda }{{b}^{\widehat{\beta }}}-\widehat{\lambda }{{a}^{\widehat{\beta }}}=\widehat{\theta }</math> , where  <math>\widehat{\lambda }</math>  and  <math>\widehat{\beta }</math>  are the maximum likelihood estimates.
The computed  <math>{{\chi }^{2}}</math>  statistic is:
 
::<math>{{\chi }^{2}}=\underset{j=1}{\overset{d}{\mathop \sum }}\,{{\frac{\left[ N(j)-\theta (j) \right]}{\widehat{\theta }(j)}}^{2}}</math>
 
where  <math>d</math>  is the total number of intervals. The random variable  <math>{{\chi }^{2}}</math>  is approximately Chi-Square distributed with  <math>df=d-2</math>  degrees of freedom. There must be at least three intervals and the length of the intervals do not have to be equal. It is common practice to require that the expected number of failures for each interval,  <math>\theta (j)</math> , be at least five. If  <math>\chi _{0}^{2}>\chi _{\alpha /2,d-2}^{2}</math>  or if  <math>\chi _{0}^{2}<\chi _{1-(\alpha /2),d-2}^{2}</math> , reject the null hypothesis.

Latest revision as of 06:18, 23 August 2012

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