Template:Parameter bounds duane

Parameter Bounds

Apply least squares analysis on the Duane model:

[math]\displaystyle{ \ln ({{\hat{m}}_{c}})=\ln (b)+\alpha \ln (t) }[/math]

The unbiased estimator of can be obtained from:

[math]\displaystyle{ {{\sigma }^{2}}=Var\left[ \ln {{m}_{c}}(t) \right]=\frac{SSE}{(n-2)} }[/math]

where:

[math]\displaystyle{ SSE=\underset{i=1}{\overset{n}{\mathop \sum }}\,{{\left[ \ln {{{\hat{m}}}_{c}}({{t}_{i}})-\ln {{m}_{c}}({{t}_{i}}) \right]}^{2}} }[/math]

Thus, the confidence bounds on [math]\displaystyle{ \alpha }[/math] and [math]\displaystyle{ b }[/math] are:

[math]\displaystyle{ C{{B}_{\alpha }}=\hat{\alpha }\pm {{t}_{n-2,\alpha /2}}SE(\hat{\alpha }) }[/math]

[math]\displaystyle{ C{{B}_{b}}=\hat{b}{{e}^{\pm {{t}_{n-2,\alpha /2}}SE\left[ \ln (\hat{b}) \right]}} }[/math]

where [math]\displaystyle{ {{t}_{n-2,\alpha /2}} }[/math] denotes the percentage point of the [math]\displaystyle{ t }[/math] distribution with [math]\displaystyle{ n-2 }[/math] degrees of freedom such that [math]\displaystyle{ P\{{{t}_{n-2}}\ge {{t}_{\alpha /2,n-2}}\}=\alpha /2 }[/math] and:

[math]\displaystyle{ SE(\hat{\alpha })=\frac{\sigma }{\sqrt{{{S}_{xx}}}} }[/math]

[math]\displaystyle{ 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]\displaystyle{ {{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]