Template:Acb4weib on parameters - Revision history

Richard House: Redirected page to Arrhenius Relationship#Approximate Confidence Bounds for the Arrhenius-Weibull

2012-08-16T05:59:44Z

Redirected page to Arrhenius Relationship#Approximate Confidence Bounds for the Arrhenius-Weibull

← Older revision		Revision as of 05:59, 16 August 2012
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	~~====Bounds on the Parameters====~~		#REDIRECT [[Arrhenius_Relationship#Approximate_Confidence_Bounds_for_the_Arrhenius-Weibull]]
	~~<br>~~
	From the asymptotically normal property of the maximum likelihood estimators, and since <math>\widehat{\beta },</math> and <math>\widehat{C}</math> are positive parameters, <math>\ln (\widehat{\beta }),</math> and <math>\ln (\widehat{C})</math> can then be treated as normally distributed. After performing this transformation, the bounds on the parameters can be estimated from:

	~~<br>~~
	~~::<math>\begin{align}~~
	~~& {{\beta }_{U}}= \widehat{\beta }\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{\beta })}}{\widehat{\beta }}}} \\~~
	~~& {{\beta }_{L}}= \widehat{\beta }\cdot {{e}^{-\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{\beta })}}{\widehat{\beta }}}}~~
	~~\end{align}</math>~~

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

	~~<br>~~
	~~::<math>\begin{align}~~
	~~& {{B}_{U}}= \widehat{B}+{{K}_{\alpha }}\sqrt{Var(\widehat{B})} \\~~
	~~& {{B}_{L}}= \widehat{B}-{{K}_{\alpha }}\sqrt{Var(\widehat{B})}~~
	~~\end{align}</math>~~

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

	~~<br>~~
	~~::<math>\begin{align}~~
	~~& {{C}_{U}}= \widehat{C}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{C})}}{\widehat{C}}}} \\~~
	~~& {{C}_{L}}= \widehat{C}\cdot {{e}^{-\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{C})}}{\widehat{C}}}}~~
	~~\end{align}</math>~~

	~~<br>~~
	The variances and covariances of <math>\beta ,</math> <math>B,</math> and <math>C</math> are estimated from the local Fisher matrix (evaluated at <math>\widehat{\beta },</math> <math>\widehat{B},</math> <math>\widehat{C})</math> , as follows:

	~~<br>~~
	~~::<math>\left~~[ ~~\begin{matrix}~~
	~~Var(\widehat{\beta }) & Cov(\widehat{\beta },\widehat{B}) & Cov(\widehat{\beta },\widehat{C}) \\~~
	~~Cov(\widehat{B},\widehat{\beta }) & Var(\widehat{B}) & Cov(\widehat{B},\widehat{C}) \\~~
	~~Cov(\widehat{C},\widehat{\beta }) & Cov(\widehat{C},\widehat{B}) & Var(\widehat{C}) \\~~
	~~\end{matrix} \right]={{\left~~[ ~~\begin{matrix}~~
	-~~\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\beta }^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial \beta \partial B} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial \beta \partial C} \\~~
	~~-\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial \beta } & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{B}^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial C} \\~~
	~~-\tfrac{{{\partial }^{2}}\Lambda }{\partial C\partial \beta } & -\tfrac{{{\partial }^{2}}\Lambda }{\partial C\partial B} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{C}^{2}}} \\~~
	~~\end{matrix} \right~~]~~}^{-1}}</math>~~

Athanasios Gerokostopoulos: /* Bounds on the Parameters */

2012-02-14T01:03:57Z

Bounds on the Parameters

← Older revision		Revision as of 01:03, 14 February 2012
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	<br>		<br>
	::<math>\begin{align}		::<math>\begin{align}
	& {{\beta }_{U}}= & \widehat{\beta }\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{\beta })}}{\widehat{\beta }}}} \\		& {{\beta }_{U}}= \widehat{\beta }\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{\beta })}}{\widehat{\beta }}}} \\
	& {{\beta }_{L}}= & \widehat{\beta }\cdot {{e}^{-\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{\beta })}}{\widehat{\beta }}}}		& {{\beta }_{L}}= \widehat{\beta }\cdot {{e}^{-\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{\beta })}}{\widehat{\beta }}}}
	\end{align}</math>		\end{align}</math>

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	<br>		<br>
	::<math>\begin{align}		::<math>\begin{align}
	& {{B}_{U}}= & \widehat{B}+{{K}_{\alpha }}\sqrt{Var(\widehat{B})} \\		& {{B}_{U}}= \widehat{B}+{{K}_{\alpha }}\sqrt{Var(\widehat{B})} \\
	& {{B}_{L}}= & \widehat{B}-{{K}_{\alpha }}\sqrt{Var(\widehat{B})}		& {{B}_{L}}= \widehat{B}-{{K}_{\alpha }}\sqrt{Var(\widehat{B})}
	\end{align}</math>		\end{align}</math>

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	<br>		<br>
	::<math>\begin{align}		::<math>\begin{align}
	& {{C}_{U}}= & \widehat{C}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{C})}}{\widehat{C}}}} \\		& {{C}_{U}}= \widehat{C}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{C})}}{\widehat{C}}}} \\
	& {{C}_{L}}= & \widehat{C}\cdot {{e}^{-\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{C})}}{\widehat{C}}}}		& {{C}_{L}}= \widehat{C}\cdot {{e}^{-\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{C})}}{\widehat{C}}}}
	\end{align}</math>		\end{align}</math>

Athanasios Gerokostopoulos: /* Bounds on the Parameters */

2012-02-14T00:02:36Z

Bounds on the Parameters

← Older revision		Revision as of 00:02, 14 February 2012
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	===Bounds on the Parameters===		====Bounds on the Parameters====
	<br>		<br>
	From the asymptotically normal property of the maximum likelihood estimators, and since <math>\widehat{\beta },</math> and <math>\widehat{C}</math> are positive parameters, <math>\ln (\widehat{\beta }),</math> and <math>\ln (\widehat{C})</math> can then be treated as normally distributed. After performing this transformation, the bounds on the parameters can be estimated from:		From the asymptotically normal property of the maximum likelihood estimators, and since <math>\widehat{\beta },</math> and <math>\widehat{C}</math> are positive parameters, <math>\ln (\widehat{\beta }),</math> and <math>\ln (\widehat{C})</math> can then be treated as normally distributed. After performing this transformation, the bounds on the parameters can be estimated from:

Nicolette Young: Created page with '===Bounds on the Parameters===
From the asymptotically normal property of the maximum likelihood estimators, and since $\widehat{\beta },$ and $\widehat{C…'$

2012-01-12T22:25:39Z

Created page with '===Bounds on the Parameters=== From the asymptotically normal property of the maximum likelihood estimators, and since <math>\widehat{\beta },</math> and <math>\widehat{C…'

New page

===Bounds on the Parameters===
 
From the asymptotically normal property of the maximum likelihood estimators, and since <math>\widehat{\beta },</math> and <math>\widehat{C}</math> are positive parameters, <math>\ln (\widehat{\beta }),</math> and <math>\ln (\widehat{C})</math> can then be treated as normally distributed. After performing this transformation, the bounds on the parameters can be estimated from:

 
::<math>\begin{align}
& {{\beta }_{U}}= & \widehat{\beta }\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{\beta })}}{\widehat{\beta }}}} \\
& {{\beta }_{L}}= & \widehat{\beta }\cdot {{e}^{-\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{\beta })}}{\widehat{\beta }}}}
\end{align}</math>

 
also:

 
::<math>\begin{align}
& {{B}_{U}}= & \widehat{B}+{{K}_{\alpha }}\sqrt{Var(\widehat{B})} \\
& {{B}_{L}}= & \widehat{B}-{{K}_{\alpha }}\sqrt{Var(\widehat{B})}
\end{align}</math>

 
and:

 
::<math>\begin{align}
& {{C}_{U}}= & \widehat{C}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{C})}}{\widehat{C}}}} \\
& {{C}_{L}}= & \widehat{C}\cdot {{e}^{-\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{C})}}{\widehat{C}}}}
\end{align}</math>

 
The variances and covariances of <math>\beta ,</math> <math>B,</math> and <math>C</math> are estimated from the local Fisher matrix (evaluated at <math>\widehat{\beta },</math> <math>\widehat{B},</math> <math>\widehat{C})</math> , as follows:

 
::<math>\left[ \begin{matrix}
Var(\widehat{\beta }) & Cov(\widehat{\beta },\widehat{B}) & Cov(\widehat{\beta },\widehat{C}) \\
Cov(\widehat{B},\widehat{\beta }) & Var(\widehat{B}) & Cov(\widehat{B},\widehat{C}) \\
Cov(\widehat{C},\widehat{\beta }) & Cov(\widehat{C},\widehat{B}) & Var(\widehat{C}) \\
\end{matrix} \right]={{\left[ \begin{matrix}
-\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\beta }^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial \beta \partial B} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial \beta \partial C} \\
-\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial \beta } & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{B}^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial C} \\
-\tfrac{{{\partial }^{2}}\Lambda }{\partial C\partial \beta } & -\tfrac{{{\partial }^{2}}\Lambda }{\partial C\partial B} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{C}^{2}}} \\
\end{matrix} \right]}^{-1}}</math>

Template:Acb4weib on parameters - Revision history

Richard House: Redirected page to Arrhenius Relationship#Approximate Confidence Bounds for the Arrhenius-Weibull

Athanasios Gerokostopoulos: /* Bounds on the Parameters */

Athanasios Gerokostopoulos: /* Bounds on the Parameters */

Nicolette Young: Created page with '===Bounds on the Parameters=== From the asymptotically normal property of the maximum likelihood estimators, and since \widehat{\beta }, and \widehat{C…'

Nicolette Young: Created page with '===Bounds on the Parameters===
From the asymptotically normal property of the maximum likelihood estimators, and since $\widehat{\beta },$ and $\widehat{C…'$