Template:Cramer-con mises test for individual failure times

Cramér-von Mises Test for Individual Failure Times
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 $$\rho \left( t \right)=\lambda \,\beta \,{{t}^{\beta -1}}\left( \lambda >0,\beta >0,t>0 \right)$$  properly describes the reliability growth of a system. The Cramér-von Mises goodness-of-fit statistic is then given by the following expression:


 * $$C_{M}^{2}=\frac{1}{12M}+\underset{i=1}{\overset{M}{\mathop \sum }}\,{{\left[ {{\left( \frac{{{T}_{i}}}{T} \right)}^{{\hat{\beta }}}}-\frac{2i-1}{2M} \right]}^{2}}$$


 * where:


 * $$M=\left\{ \begin{matrix}

N\text{ if the test is time terminated} \\ N-1\text{ if the test is failure terminated} \\ \end{matrix} \right\}$$

The failure times, $${{T}_{i}}$$, must be ordered so that  $${{T}_{1}}<{{T}_{2}}<\ldots <{{T}_{M}}$$. If the statistic $$C_{M}^{2}$$  exceeds the critical value corresponding to  $$M$$  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 $$C_{M}^{2}$$  are shown in Appendix B, Table B.2, where the table is indexed by the total number of observed failures,  $$M$$.