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| ===Parameter Bounds===
| | #REDIRECT [[Duane_Model#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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| 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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