Template:Parameter bounds duane: Difference between revisions

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(Created page with '===Parameter Bounds=== <br> Apply least squares analysis on the Duane model: <br> ::<math>\ln ({{\hat{m}}_{c}})=\ln (b)+\alpha \ln (t)</math> <br> The unbiased estimator of …')
 
(Redirected page to Duane Model#Parameter Bounds)
 
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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:
 
 
::<math>C{{B}_{\alpha }}=\hat{\alpha }\pm {{t}_{n-2,\alpha /2}}SE(\hat{\alpha })</math>
 
 
 
::<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>
 
 
::<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>
 
 
::<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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Latest revision as of 02:00, 24 August 2012

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