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 [math]\displaystyle{ K }[/math] like systems are under study and you wish to test the hypothesis [math]\displaystyle{ {{H}_{1}} }[/math] that their failure times follow a non-homogeneous Poisson process. Suppose information is available for the [math]\displaystyle{ {{q}^{th}} }[/math] system over the interval [math]\displaystyle{ [0,{{T}_{q}}] }[/math] , with successive failure times , [math]\displaystyle{ (q=1,2,\ldots ,\,K) }[/math] . The Cramér-von Mises test can be performed with the following steps:

Step 1: If [math]\displaystyle{ {{x}_{{{N}_{q}}q}}={{T}_{q}} }[/math] (failure terminated) let [math]\displaystyle{ {{M}_{q}}={{N}_{q}}-1 }[/math] , and if [math]\displaystyle{ {{x}_{{{N}_{q}}q}}\lt T }[/math] (time terminated) let [math]\displaystyle{ {{M}_{q}}={{N}_{q}} }[/math] . Then:

[math]\displaystyle{ M=\underset{q=1}{\overset{K}{\mathop \sum }}\,{{M}_{q}} }[/math]

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

[math]\displaystyle{ {{Y}_{iq}}=\frac{{{X}_{iq}}}{{{T}_{q}}},i=1,2,\ldots ,{{M}_{q}},\text{ }q=1,2,\ldots ,K }[/math]

Step 3: Next calculate [math]\displaystyle{ \overline{\beta } }[/math] , the unbiased estimate of [math]\displaystyle{ \beta }[/math] , from:

[math]\displaystyle{ \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]

Step 4: Treat the [math]\displaystyle{ {{Y}_{iq}} }[/math] values as one group and order them from smallest to largest. Name these ordered values [math]\displaystyle{ {{z}_{1}},\,{{z}_{2}},\ldots ,{{z}_{M}} }[/math] , such that [math]\displaystyle{ {{z}_{1}}\lt \ \ {{z}_{2}}\lt \ldots \lt {{z}_{M}} }[/math] .

Step 5: Calculate the parametric Cramér-von Mises statistic.

[math]\displaystyle{ C_{M}^{2}=\frac{1}{12M}+\underset{j=1}{\overset{M}{\mathop \sum }}\,{{(Z_{j}^{\overline{\beta }}-\frac{2j-1}{2M})}^{2}} }[/math]

Critical values for the Cramér-von Mises test are presented in Table B.2 of Appendix B.

Step 6: If the calculated [math]\displaystyle{ C_{M}^{2} }[/math] is less than the critical value then accept the hypothesis that the failure times for the [math]\displaystyle{ K }[/math] systems follow the non-homogeneous Poisson process with intensity function [math]\displaystyle{ u(t)=\lambda \beta {{t}^{\beta -1}} }[/math] .

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 [math]\displaystyle{ \alpha =0.10 }[/math]

Solution
Step 1:

[math]\displaystyle{ \begin{align} & {{X}_{9,1}}= & 1913.5\lt 2000,\,\ {{M}_{1}}=9 \\ & {{X}_{11,2}}= & 1867\lt 2000,\,\ {{M}_{2}}=11 \\ & {{X}_{14,3}}= & 1604.8\lt 2000,\,\ {{M}_{3}}=14 \\ & M= & \underset{q=1}{\overset{3}{\mathop \sum }}\,{{M}_{q}}=34 \end{align} }[/math]

Step 2: Calculate [math]\displaystyle{ {{Y}_{iq}}, }[/math] treat the [math]\displaystyle{ {{Y}_{iq}} }[/math] values as one group and order them from smallest to largest. Name these ordered values [math]\displaystyle{ {{z}_{1}},\,{{z}_{2}},\ldots ,{{z}_{M}} }[/math] .

Step 3: Calculate [math]\displaystyle{ \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]

Step 4: Calculate [math]\displaystyle{ 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]

Step 5: Find the critical value (CV) from Table B.2 for [math]\displaystyle{ M=34 }[/math] at a significance level [math]\displaystyle{ \alpha =0.10 }[/math] . [math]\displaystyle{ CV=0.172 }[/math] .

Step 6: Since [math]\displaystyle{ C_{M}^{2}\lt CV }[/math] , accept the hypothesis that the failure times for the [math]\displaystyle{ K=3 }[/math] repairable systems follow the non-homogeneous Poisson process with intensity function [math]\displaystyle{ u(t)=\lambda \beta {{t}^{\beta -1}} }[/math] .