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| ====Cramér-von Mises Test====
| | #REDIRECT [[RGA_Models_for_Repairable_Systems_Analysis#Cram.C3.A9r-von_Mises_Test]] |
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| To illustrate the application of the Cramér-von Mises statistic for multiple system data, suppose that <math>K</math> like systems are under study and you wish to test the hypothesis <math>{{H}_{1}}</math> that their failure times follow a non-homogeneous Poisson process. Suppose information is available for the <math>{{q}^{th}}</math> system over the interval <math>[0,{{T}_{q}}]</math> , with successive failure times , <math>(q=1,2,\ldots ,\,K)</math> . The Cramér-von Mises test can be performed with the following steps:
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| Step 1: If <math>{{x}_{{{N}_{q}}q}}={{T}_{q}}</math> (failure terminated) let <math>{{M}_{q}}={{N}_{q}}-1</math> , and if <math>{{x}_{{{N}_{q}}q}}<T</math> (time terminated) let <math>{{M}_{q}}={{N}_{q}}</math> . Then:
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| ::<math>M=\underset{q=1}{\overset{K}{\mathop \sum }}\,{{M}_{q}}</math>
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| Step 2: For each system divide each successive failure time by the corresponding end time <math>{{T}_{q}}</math> , <math>\,i=1,2,...,{{M}_{q}}.</math> Calculate the <math>M</math> values:
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| ::<math>{{Y}_{iq}}=\frac{{{X}_{iq}}}{{{T}_{q}}},i=1,2,\ldots ,{{M}_{q}},\text{ }q=1,2,\ldots ,K</math>
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| Step 3: Next calculate <math>\overline{\beta }</math> , the unbiased estimate of <math>\beta </math> , from:
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| ::<math>\overline{\beta }=\frac{M-1}{\underset{q=1}{\overset{K}{\mathop{\sum }}}\,\underset{i=1}{\overset{Mq}{\mathop{\sum }}}\,\ln \left( \tfrac{{{T}_{q}}}{{{X}_{i}}{{}_{q}}} \right)}</math>
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| Step 4: Treat the <math>{{Y}_{iq}}</math> values as one group and order them from smallest to largest. Name these ordered values <math>{{z}_{1}},\,{{z}_{2}},\ldots ,{{z}_{M}}</math> , such that <math>{{z}_{1}}<\ \ {{z}_{2}}<\ldots <{{z}_{M}}</math> .
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| Step 5: Calculate the parametric Cramér-von Mises statistic.
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| ::<math>C_{M}^{2}=\frac{1}{12M}+\underset{j=1}{\overset{M}{\mathop \sum }}\,{{(Z_{j}^{\overline{\beta }}-\frac{2j-1}{2M})}^{2}}</math>
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| Critical values for the Cramér-von Mises test are presented in Table B.2 of Appendix B.
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| Step 6: If the calculated <math>C_{M}^{2}</math> is less than the critical value then accept the hypothesis that the failure times for the <math>K</math> systems follow the non-homogeneous Poisson process with intensity function <math>u(t)=\lambda \beta {{t}^{\beta -1}}</math> .
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| =====Example 2=====
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| For the data from Example 1, use the Cramér-von Mises test to examine the compatibility of the model at a significance level <math>\alpha =0.10</math>
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| ''Solution''
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| Step 1:
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| ::<math>\begin{align}
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| & {{X}_{9,1}}= & 1913.5<2000,\,\ {{M}_{1}}=9 \\
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| & {{X}_{11,2}}= & 1867<2000,\,\ {{M}_{2}}=11 \\
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| & {{X}_{14,3}}= & 1604.8<2000,\,\ {{M}_{3}}=14 \\
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| & M= & \underset{q=1}{\overset{3}{\mathop \sum }}\,{{M}_{q}}=34
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| \end{align}</math>
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| Step 2: Calculate <math>{{Y}_{iq}},</math> treat the <math>{{Y}_{iq}}</math> values as one group and order them from smallest to largest. Name these ordered values <math>{{z}_{1}},\,{{z}_{2}},\ldots ,{{z}_{M}}</math> .
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| Step 3: Calculate <math>\overline{\beta }=\tfrac{M-1}{\underset{q=1}{\overset{K}{\mathop{\sum }}}\,\underset{i=1}{\overset{Mq}{\mathop{\sum }}}\,\ln \left( \tfrac{{{T}_{q}}}{{{X}_{i}}{{}_{q}}} \right)}=0.4397</math>
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| Step 4: Calculate <math>C_{M}^{2}=\tfrac{1}{12M}+\underset{j=1}{\overset{M}{\mathop{\sum }}}\,{{(Z_{j}^{\overline{\beta }}-\tfrac{2j-1}{2M})}^{2}}=0.0611</math>
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| Step 5: Find the critical value (CV) from Table B.2 for <math>M=34</math> at a significance level <math>\alpha =0.10</math> . <math>CV=0.172</math> .
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| Step 6: Since <math>C_{M}^{2}<CV</math> , accept the hypothesis that the failure times for the <math>K=3</math> repairable systems follow the non-homogeneous Poisson process with intensity function <math>u(t)=\lambda \beta {{t}^{\beta -1}}</math> .
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