Template:Parameter bounds duane

Parameter Bounds
Apply least squares analysis on the Duane model:


 * $$\ln ({{\hat{m}}_{c}})=\ln (b)+\alpha \ln (t)$$

The unbiased estimator of    can be obtained from:


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


 * where:


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

Thus, the confidence bounds on $$\alpha $$  and  $$b$$  are:


 * $$C{{B}_{\alpha }}=\hat{\alpha }\pm {{t}_{n-2,\alpha /2}}SE(\hat{\alpha })$$


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

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


 * $$SE(\hat{\alpha })=\frac{\sigma }{\sqrt}$$


 * $$SE\left[ \ln (\hat{b}) \right]=\sigma \cdot \sqrt{\frac{\underset{i=1}{\overset{n}{\mathop{\sum }}}\,{{(\ln {{t}_{i}})}^{2}}}{n\cdot {{S}_{xx}}}}$$


 * $${{S}_{xx}}=\left[ \underset{i=1}{\overset{n}{\mathop \sum }}\,{{(\ln {{t}_{i}})}^{2}} \right]-\frac{1}{n}{{\left( \underset{i=1}{\overset{n}{\mathop \sum }}\,\ln ({{t}_{i}}) \right)}^{2}}$$