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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:
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| ::<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>
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| :where:
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| ::<math>M=\left\{ \begin{matrix}
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| N\text{ if the test is time terminated} \\
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| N-1\text{ if the test is failure terminated} \\
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| \end{matrix} \right\}</math>
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| The failure times, <math>{{T}_{i}}</math> , must be ordered so that <math>{{T}_{1}}<{{T}_{2}}<\ldots <{{T}_{M}}</math> .
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| 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> .
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