Template:Cramer-con mises test rsa

Cramér-von Mises Test
To illustrate the application of the Cramér-von Mises statistic for multiple system data, suppose that $$K$$  like systems are under study and you wish to test the hypothesis  $${{H}_{1}}$$  that their failure times follow a non-homogeneous Poisson process. Suppose information is available for the $${{q}^{th}}$$  system over the interval  $$[0,{{T}_{q}}]$$ , with successive failure times    ,  $$(q=1,2,\ldots ,\,K)$$. The Cramér-von Mises test can be performed with the following steps: Step 1: If $${{x}_{{{N}_{q}}q}}={{T}_{q}}$$  (failure terminated) let  $${{M}_{q}}={{N}_{q}}-1$$, and if  $${{x}_{{{N}_{q}}q}}<T$$  (time terminated) let  $${{M}_{q}}={{N}_{q}}$$. Then:


 * $$M=\underset{q=1}{\overset{K}{\mathop \sum }}\,{{M}_{q}}$$

Step 2: For each system divide each successive failure time by the corresponding end time $${{T}_{q}}$$, $$\,i=1,2,...,{{M}_{q}}.$$  Calculate the  $$M$$  values:


 * $${{Y}_{iq}}=\frac,i=1,2,\ldots ,{{M}_{q}},\text{ }q=1,2,\ldots ,K$$

Step 3: Next calculate $$\overline{\beta }$$, the unbiased estimate of  $$\beta $$ , from:


 * $$\overline{\beta }=\frac{M-1}{\underset{q=1}{\overset{K}{\mathop{\sum }}}\,\underset{i=1}{\overset{Mq}{\mathop{\sum }}}\,\ln \left( \tfrac \right)}$$

Step 4: Treat the $${{Y}_{iq}}$$  values as one group and order them from smallest to largest. Name these ordered values $${{z}_{1}},\,{{z}_{2}},\ldots ,{{z}_{M}}$$, such that  $${{z}_{1}}<\ \ {{z}_{2}}<\ldots <{{z}_{M}}$$. Step 5: Calculate the parametric Cramér-von Mises statistic.


 * $$C_{M}^{2}=\frac{1}{12M}+\underset{j=1}{\overset{M}{\mathop \sum }}\,{{(Z_{j}^{\overline{\beta }}-\frac{2j-1}{2M})}^{2}}$$

Critical values for the Cramér-von Mises test are presented in Table B.2 of Appendix B. Step 6: If the calculated $$C_{M}^{2}$$  is less than the critical value then accept the hypothesis that the failure times for the  $$K$$  systems follow the non-homogeneous Poisson process with intensity function  $$u(t)=\lambda \beta {{t}^{\beta -1}}$$.

Example 2
For the data from Example 1, use the Cramér-von Mises test to examine the compatibility of the model at a significance level $$\alpha =0.10$$ Solution Step 1:


 * $$\begin{align}

& {{X}_{9,1}}= & 1913.5<2000,\,\ {{M}_{1}}=9 \\ & {{X}_{11,2}}= & 1867<2000,\,\ {{M}_{2}}=11 \\ & {{X}_{14,3}}= & 1604.8<2000,\,\ {{M}_{3}}=14 \\ & M= & \underset{q=1}{\overset{3}{\mathop \sum }}\,{{M}_{q}}=34 \end{align}$$

Step 2: Calculate $${{Y}_{iq}},$$  treat the  $${{Y}_{iq}}$$  values as one group and order them from smallest to largest. Name these ordered values $${{z}_{1}},\,{{z}_{2}},\ldots ,{{z}_{M}}$$. Step 3: Calculate $$\overline{\beta }=\tfrac{M-1}{\underset{q=1}{\overset{K}{\mathop{\sum }}}\,\underset{i=1}{\overset{Mq}{\mathop{\sum }}}\,\ln \left( \tfrac \right)}=0.4397$$ Step 4: Calculate $$C_{M}^{2}=\tfrac{1}{12M}+\underset{j=1}{\overset{M}{\mathop{\sum }}}\,{{(Z_{j}^{\overline{\beta }}-\tfrac{2j-1}{2M})}^{2}}=0.0611$$ Step 5: Find the critical value (CV) from Table B.2 for $$M=34$$  at a significance level  $$\alpha =0.10$$. $$CV=0.172$$. Step 6: Since $$C_{M}^{2}<CV$$, accept the hypothesis that the failure times for the  $$K=3$$  repairable systems follow the non-homogeneous Poisson process with intensity function  $$u(t)=\lambda \beta {{t}^{\beta -1}}$$.