Richard House: Redirected page to Duane Model#Parameter Bounds

2012-08-24T02:00:57Z

Redirected page to Duane Model#Parameter Bounds

← Older revision		Revision as of 02:00, 24 August 2012
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	~~===Parameter Bounds===~~		#REDIRECT [[Duane_Model#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 can be obtained from:~~

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

	~~<br>~~
	~~:where:~~

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

	~~<br>~~
	~~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>~~

	~~<br>~~
	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:

	~~<br>~~
	~~::<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>~~

	~~<br>~~

Nicolette Young: Created page with '===Parameter Bounds===
Apply least squares analysis on the Duane model:
:: $\ln ({{\hat{m}}_{c}})=\ln (b)+\alpha \ln (t)$
The unbiased estimator of …'

2012-01-05T20:46:15Z

Created page with '===Parameter Bounds=== Apply least squares analysis on the Duane model: ::<math>\ln ({{\hat{m}}_{c}})=\ln (b)+\alpha \ln (t)</math> The unbiased estimator of …'

New page

===Parameter Bounds===
 
Apply least squares analysis on the Duane model:

 
::<math>\ln ({{\hat{m}}_{c}})=\ln (b)+\alpha \ln (t)</math>

 
The unbiased estimator of can be obtained from:

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

 
:where:

 
::<math>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>\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>

 
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:

 
::<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>

Template:Parameter bounds duane - Revision history

Richard House: Redirected page to Duane Model#Parameter Bounds

Nicolette Young: Created page with '===Parameter Bounds=== Apply least squares analysis on the Duane model: ::\ln ({{\hat{m}}_{c}})=\ln (b)+\alpha \ln (t) The unbiased estimator of …'

Nicolette Young: Created page with '===Parameter Bounds===
Apply least squares analysis on the Duane model:
:: $\ln ({{\hat{m}}_{c}})=\ln (b)+\alpha \ln (t)$
The unbiased estimator of …'