Template:Cramer-con mises test for individual failure times: Difference between revisions

Latest revision as of 00:48, 24 August 2012

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-===Cramér-von Mises Test for Individual Failure Times===
+#REDIRECT [[Crow-AMSAA - NHPP]]
-If the individual failure times are known, a Cramér-von Mises statistic is used to test the null hypothesis that a non-homogeneous Poisson process with failure intensity function  <math>\rho \left( t \right)=\lambda \,\beta \,{{t}^{\beta -1}}\left( \lambda >0,\beta >0,t>0 \right)</math>  properly describes the reliability growth of a system. The Cramér-von Mises goodness-of-fit statistic is then given by the following expression:
-::<math>C_{M}^{2}=\frac{1}{12M}+\underset{i=1}{\overset{M}{\mathop \sum }}\,{{\left[ {{\left( \frac{{{T}_{i}}}{T} \right)}^{{\hat{\beta }}}}-\frac{2i-1}{2M} \right]}^{2}}</math>
-:where:
-::<math>M=\left\{ \begin{matrix}
-   N\text{ if the test is time terminated}  \\
-   N-1\text{ if the test is failure terminated}  \\
-\end{matrix} \right\}</math>
-The failure times,  <math>{{T}_{i}}</math> , must be ordered so that  <math>{{T}_{1}}<{{T}_{2}}<\ldots <{{T}_{M}}</math> .
-If the statistic  <math>C_{M}^{2}</math>  exceeds the critical value corresponding to  <math>M</math>  for a chosen significance level, then the null hypothesis that the Crow-AMSAA model adequately fits the data shall be rejected. Otherwise, the model shall be accepted. Critical values of  <math>C_{M}^{2}</math>  are shown in Appendix B, Table B.2, where the table is indexed by the total number of observed failures,  <math>M</math> .