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| ==Confidence Bounds==
| | #REDIRECT [[Duane_Model#Confidence_Bounds]] |
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| Least squares confidence bounds can be computed for both the model parameters and metrics of interest for the Duane model.
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| ===Parameter Bounds===
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| Apply least squares analysis on the Duane model:
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| ::<math>\ln ({{\hat{m}}_{c}})=\ln (b)+\alpha \ln (t)</math>
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| The unbiased estimator of can be obtained from:
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| ::<math>{{\sigma }^{2}}=Var\left[ \ln {{m}_{c}}(t) \right]=\frac{SSE}{(n-2)}</math>
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| :where:
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| ::<math>SSE=\underset{i=1}{\overset{n}{\mathop \sum }}\,{{\left[ \ln {{{\hat{m}}}_{c}}({{t}_{i}})-\ln {{m}_{c}}({{t}_{i}}) \right]}^{2}}</math>
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| <br>
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| Thus, the confidence bounds on <math>\alpha </math> and <math>b</math> are:
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| ::<math>C{{B}_{\alpha }}=\hat{\alpha }\pm {{t}_{n-2,\alpha /2}}SE(\hat{\alpha })</math>
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| ::<math>C{{B}_{b}}=\hat{b}{{e}^{\pm {{t}_{n-2,\alpha /2}}SE\left[ \ln (\hat{b}) \right]}}</math>
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| where <math>{{t}_{n-2,\alpha /2}}</math> denotes the percentage point of the <math>t</math> distribution with <math>n-2</math> degrees of freedom such that <math>P\{{{t}_{n-2}}\ge {{t}_{\alpha /2,n-2}}\}=\alpha /2</math> and:
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| ::<math>SE(\hat{\alpha })=\frac{\sigma }{\sqrt{{{S}_{xx}}}}</math>
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| ::<math>SE\left[ \ln (\hat{b}) \right]=\sigma \cdot \sqrt{\frac{\underset{i=1}{\overset{n}{\mathop{\sum }}}\,{{(\ln {{t}_{i}})}^{2}}}{n\cdot {{S}_{xx}}}}</math>
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| ::<math>{{S}_{xx}}=\left[ \underset{i=1}{\overset{n}{\mathop \sum }}\,{{(\ln {{t}_{i}})}^{2}} \right]-\frac{1}{n}{{\left( \underset{i=1}{\overset{n}{\mathop \sum }}\,\ln ({{t}_{i}}) \right)}^{2}}</math>
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| ===Other Bounds===
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| Confidence bounds also can be obtained on the cumulative MTBF and the cumulative failure intensity:
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| ::<math>C{{B}_{{{m}_{c}}}}={{\hat{m}}_{c}}(t){{e}^{\pm {{z}_{\alpha }}\sqrt{Var\left[ \ln ({{{\hat{m}}}_{c}}) \right]}}}</math>
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| ::<math>\begin{align}
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| & {{[{{\lambda }_{c}}(t)]}_{L}}= & \frac{1}{{{[{{m}_{c}}(t)]}_{u}}} \\
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| & {{[{{\lambda }_{c}}(t)]}_{U}}= & \frac{1}{{{[{{m}_{c}}(t)]}_{l}}}
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| \end{align}</math>
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| When <math>n</math> is large, the approximate <math>100(1-\alpha )%</math> confidence bounds for instantaneous MTBF are given by:
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| ::<math>\begin{align}
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| & {{m}_{i}}{{(t)}_{L}}= & \frac{{{[{{m}_{c}}(t)]}_{L}}}{{\hat{\beta }}} \\
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| & {{m}_{i}}{{(t)}_{U}}= & \frac{{{[{{m}_{c}}(t)]}_{U}}}{{\hat{\beta }}}
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| \end{align}</math>
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| and therefore, the confidence bounds on the instantaneous failure intensity are:
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| ::<math>\begin{align}
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| & {{[{{\lambda }_{i}}(t)]}_{L}}= & \frac{1}{{{[{{m}_{i}}(t)]}_{U}}} \\
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| & {{[{{\lambda }_{c}}(t)]}_{U}}= & \frac{1}{{{[{{m}_{i}}(t)]}_{L}}}
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| \end{align}</math>
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| '''Example 5<math>{{\lambda }_{i}}(t)=\tfrac{1}{{{m}_{i}}(t)}</math>'''
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| For the data given in Table 4.3, calculate the 90% confidence bounds for:
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| #The parameters <math>\alpha\text{and} b</math>. | |
| #The cumulative and instantaneous failure intensity.
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| #The cumulative and instantaneous MTBF.
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| <br>
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| '''Solution'''
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| :1. Using the values of <math>\widehat{b}</math> and <math>\widehat{\alpha }</math> estimated from the least squares analysis in Example 3:
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| ::<math>\widehat{b}=1.9453</math>
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| ::<math>\widehat{\alpha}=0.6133
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| Eqn. (duanec9) is:
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| ::<math>\begin{align}
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| & {{S}_{xx}}= & 1400.9084-1301.4545 \\
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| & = & 99.4539
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| \end{align}</math>
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| Eqn. (duanec7) is:
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| ::<math>\begin{align}
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| & SE(\hat{\alpha })= & \frac{\sigma }{\sqrt{{{S}_{xx}}}} \\
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| & = & \frac{0.08428}{9.9727} \\
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| & = & 0.008452
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| \end{align}</math>
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| Eqn. (duanec8) is:
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| ::<math>\begin{align}
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| & SE(\ln \hat{b})= & \sigma \cdot \sqrt{\frac{\underset{i=1}{\overset{n}{\mathop{\sum }}}\,{{(\ln {{T}_{i}})}^{2}}}{n\cdot {{S}_{xx}}}} \\
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| & = & 0.065960
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| \end{align}</math>
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| Thus, 90% confidence bounds on parameter <math>\alpha </math> using Eqn. (duanec1) are:
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| <br>
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| ::<math>\begin{align}
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| & {{\alpha }_{L}}= & 0.602050 \\
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| & {{\alpha }_{U}}= & 0.624417
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| \end{align}</math>
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| And 90% confidence bounds on parameter <math>b</math> using Eqn. (duanec2) are:
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| ::<math>\begin{align}
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| & {{b}_{L}}= & 1.7831 \\
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| & {{b}_{U}}= & 2.1231
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| \end{align}</math>
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| <br>
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| :2. The cumulative failure intensity is:
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| ::<math>\begin{align}
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| & {{\lambda }_{c}}= & \frac{1}{1.9453}\cdot {{22000}^{-0.6133}} \\
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| & = & 0.00111689
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| \end{align}</math>
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| And the instantaneous failure intensity is equal to:
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| ::<math>\begin{align}
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| & {{\lambda }_{i}}= & \frac{1}{1.9453}\cdot (1-0.6133)\cdot {{22000}^{-0.6133}} \\
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| & = & 0.00043198
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| \end{align}</math>
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| So, at the 90% confidence level and for <math>T=22,000</math> hr, the confidence bounds on cumulative failure intensity are:
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| ::<math>\begin{align}
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| & {{[{{\lambda }_{c}}(t)]}_{L}}= & 0.00100254 \\
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| & {{[{{\lambda }_{c}}(t)]}_{U}}= & 0.00124429
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| \end{align}</math>
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| For the instantaneous failure intensity:
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| ::<math>\begin{align}
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| & {{[{{\lambda }_{i}}(t)]}_{L}}= & 0.00038775 \\
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| & {{[{{\lambda }_{c}}(t)]}_{U}}= & 0.00048125
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| \end{align}</math>
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| Figures figure75 and figure76 show the graphs of the cumulative and instantaneous failure intensity. Both are plotted with confidence bounds.
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| [[Image:rga4.7.png|thumb|center|400px|Cumulative Failure Intensity plot with 2-sided 90% confidence bounds.]]
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| [[Image:rga4.8.png|thumb|center|400px|Instantaneous Failure Intensity plot with 2-sided 90% confidence bounds.]]
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| :3. The cumulative MTBF is:
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| ::<math>\begin{align}
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| & {{m}_{c}}(T)= & 1.9453\cdot {{22000}^{0.6133}} \\
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| & = & 895.3395
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| \end{align}</math>
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| And the instantaneous MTBF is:
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| ::<math>\begin{align}
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| & {{m}_{i}}(T)= & \frac{1.9453}{1-0.6133}\cdot {{22000}^{0.6133}} \\
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| & = & 2314.9369
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| \end{align}</math>
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| <br>
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| So, at 90% confidence level and for <math>T=22,000</math> hr, the confidence bounds on the cumulative MTBF are:
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| ::<math>\begin{align}
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| & {{m}_{c}}{{(t)}_{l}}= & 803.6695 \\
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| & {{m}_{c}}{{(t)}_{u}}= & 997.4658
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| \end{align}</math>
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| The confidence bounds for the instantaneous MTBF are:
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| ::<math>\begin{align}
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| & {{m}_{i}}{{(t)}_{l}}= & 2077.9204 \\
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| & {{m}_{i}}{{(t)}_{u}}= & 2578.9886
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| \end{align}</math>
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| <br>
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| Figure CumMTBFCB displays the cumulative MTBF while Figure InstMTBFCB displays the instantaneous MTBF. Both are plotted with confidence bounds.
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| [[Image:rga4.9.png|thumb|center|400px|Cumulative MTBF plot with 2-sided 90% condfidence bounds.]]
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| [[Image:rga4.10.png|thumb|center|400px|Instantaneous MTBF plot with 2-sided 90% confidence bounds.]]
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