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 [math]\displaystyle{ \rho \left( t \right)=\lambda \,\beta \,{{t}^{\beta -1}}\left( \lambda \gt 0,\beta \gt 0,t\gt 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]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ {{T}_{i}} }[/math] , must be ordered so that [math]\displaystyle{ {{T}_{1}}\lt {{T}_{2}}\lt \ldots \lt {{T}_{M}} }[/math] . If the statistic [math]\displaystyle{ C_{M}^{2} }[/math] exceeds the critical value corresponding to [math]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ M }[/math] .