Template:Eyring-weib bounds on parameters - Revision history

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

2012-08-17T01:07:34Z

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

← Older revision		Revision as of 01:07, 17 August 2012
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	~~===Bounds on the Parameters===~~		#REDIRECT [[Eyring_Relationship#Approximate_Confidence_Bounds_for_the_Eyring-Weibull]]
	~~<br>~~

	From the asymptotically normal property of the maximum likelihood estimators, and since <math>\widehat{\beta }</math> is a positive parameter, <math>\ln (\widehat{\beta })</math> can then be treated as normally distributed. After performing this transformation, the bounds on the parameters are 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}~~
	~~& {{A}_{U}}= & \widehat{A}+{{K}_{\alpha }}\sqrt{Var(\widehat{A})} \\~~
	~~& {{A}_{L}}= & \widehat{A}-{{K}_{\alpha }}\sqrt{Var(\widehat{A})}~~
	~~\end{align}</math>~~

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

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

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

	~~<br>~~

Athanasios Gerokostopoulos: /* Bounds on the Parameters */

2012-02-27T23:59:09Z

Bounds on the Parameters

← Older revision		Revision as of 23:59, 27 February 2012
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	-\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial \beta } & -\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial A} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{B}^{2}}} \\		-\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial \beta } & -\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial A} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{B}^{2}}} \\
	\end{matrix} \right]}^{-1}}</math>		\end{matrix} \right]}^{-1}}</math>

			<br>

Athanasios Gerokostopoulos: /* Bounds on the Parameters */

2012-02-27T23:58:50Z

Bounds on the Parameters

← Older revision		Revision as of 23:58, 27 February 2012
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	<br>		<br>
	:also:		also:

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

	<br>		<br>

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

2012-01-12T23:43:02Z

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

New page

===Bounds on the Parameters===
 

From the asymptotically normal property of the maximum likelihood estimators, and since <math>\widehat{\beta }</math> is a positive parameter, <math>\ln (\widehat{\beta })</math> can then be treated as normally distributed. After performing this transformation, the bounds on the parameters are 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}
& {{A}_{U}}= & \widehat{A}+{{K}_{\alpha }}\sqrt{Var(\widehat{A})} \\
& {{A}_{L}}= & \widehat{A}-{{K}_{\alpha }}\sqrt{Var(\widehat{A})}
\end{align}</math>

 
:and:

 
::<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>

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

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

Template:Eyring-weib bounds on parameters - Revision history

Richard House: Redirected page to Eyring Relationship#Approximate Confidence Bounds for the Eyring-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 } is a positive paramet…'

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