Template:Eyring-log cb on parameters - Revision history

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

2012-08-17T01:14:21Z

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

← Older revision		Revision as of 01:14, 17 August 2012
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	~~===Bounds on the Parameters===~~		#REDIRECT [[Eyring_Relationship#Approximate_Confidence_Bounds_for_the_Eyring-Lognormal]]
	~~<br>~~
	~~The lower and upper bounds on <math>A</math> and <math>B</math> are estimated from:~~

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

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

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

	~~<br>~~
	Since the standard deviation, <math>{{\widehat{\sigma }}_{{T}',}}</math> is a positive parameter, <math>\ln ({{\widehat{\sigma }}_{{{T}'}}})</math> is treated as normally distributed, and the bounds are estimated from:

	~~<br>~~
	~~::<math>\begin{align}~~
	~~{{\sigma }_{U}} &=\ {{\widehat{\sigma }}_{{{T}'}}}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var({{\widehat{\sigma }}_{{{T}'}}})}}{{{\widehat{\sigma }}_{{{T}'}}}}}}\text{ (Upper bound)} \\~~
	~~{{\sigma }_{L}} &=\ \frac{{{\widehat{\sigma }}_{{{T}'}}}}{{{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var({{\widehat{\sigma }}_{{{T}'}}})}}{{{\widehat{\sigma }}_{{{T}'}}}}}}}\text{ (Lower bound)}~~
	~~\end{align}</math>~~

	~~<br>~~
	The variances and covariances of <math>A,</math> <math>B,</math> and <math>{{\sigma }_{{{T}'}}}</math> are estimated from the local Fisher matrix (evaluated at <math>\widehat{A},</math> <math>\widehat{B}</math> , <math>{{\widehat{\sigma }}_{{{T}'}}})</math> as follows:

	~~<br>~~
	~~::<math>\left( \begin{matrix}~~
	~~Var\left( {{\widehat{\sigma }}_{{{T}'}}} \right) & Cov\left( \widehat{A},{{\widehat{\sigma }}_{{{T}'}}} \right) & Cov\left( \widehat{B},{{\widehat{\sigma }}_{{{T}'}}} \right) \\~~
	~~Cov\left( {{\widehat{\sigma }}_{{{T}'}}},\widehat{A} \right) & Var\left( \widehat{A} \right) & Cov\left( \widehat{A},\widehat{B} \right) \\~~
	~~Cov\left( {{\widehat{\sigma }}_{{{T}'}}},\widehat{B} \right) & Cov\left( \widehat{B},\widehat{A} \right) & Var\left( \widehat{B} \right) \\~~
	~~\end{matrix} \right)={{[F~~]~~}^{-1}}</math>~~

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

	~~<br>~~
	~~::<math>F=\left( \begin{matrix}~~
	-\tfrac{{{\partial }^{2}}\Lambda }{\partial \sigma _{{{T}'}}^{2}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\sigma }_{{{T}'}}}\partial A} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\sigma }_{{{T}'}}}\partial B} \\
	~~-\tfrac{{{\partial }^{2}}\Lambda }{\partial A\partial {{\sigma }_{{{T}'}}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{A}^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial A\partial B} \\~~
	~~-\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial {{\sigma }_{{{T}'}}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial A} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{B}^{2}}} \\~~
	~~\end{matrix} \right)</math>~~

	~~<br>~~

Athanasios Gerokostopoulos: /* Bounds on the Parameters */

2012-02-28T00:03:58Z

Bounds on the Parameters

← Older revision		Revision as of 00:03, 28 February 2012
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	<br>		<br>
	:where:		where:

	<br>		<br>

Athanasios Gerokostopoulos: /* Bounds on the Parameters */

2012-02-15T00:35:57Z

Bounds on the Parameters

← Older revision		Revision as of 00:35, 15 February 2012
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	<br>		<br>
	:and:		and:

	<br>		<br>
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	<br>		<br>
	::<math>\begin{align}		::<math>\begin{align}
	& {{\sigma }_{U}}= & {{\widehat{\sigma }}_{{{T}'}}}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var({{\widehat{\sigma }}_{{{T}'}}})}}{{{\widehat{\sigma }}_{{{T}'}}}}}}\text{ (Upper bound)} \\		{{\sigma }_{U}} &=\ {{\widehat{\sigma }}_{{{T}'}}}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var({{\widehat{\sigma }}_{{{T}'}}})}}{{{\widehat{\sigma }}_{{{T}'}}}}}}\text{ (Upper bound)} \\
	& {{\sigma }_{L}}= & \frac{{{\widehat{\sigma }}_{{{T}'}}}}{{{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var({{\widehat{\sigma }}_{{{T}'}}})}}{{{\widehat{\sigma }}_{{{T}'}}}}}}}\text{ (Lower bound)}		{{\sigma }_{L}} &=\ \frac{{{\widehat{\sigma }}_{{{T}'}}}}{{{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var({{\widehat{\sigma }}_{{{T}'}}})}}{{{\widehat{\sigma }}_{{{T}'}}}}}}}\text{ (Lower bound)}
	\end{align}</math>		\end{align}</math>

Nicolette Young: Created page with '===Bounds on the Parameters===
The lower and upper bounds on $A$ and $B$ are estimated from:
:: $\begin{align} & {{A}_{U}}= & \widehat{…'$

2012-01-12T23:46:05Z

Created page with '===Bounds on the Parameters=== The lower and upper bounds on <math>A</math> and <math>B</math> are estimated from: ::<math>\begin{align} & {{A}_{U}}= & \widehat{…'

New page

===Bounds on the Parameters===
 
The lower and upper bounds on <math>A</math> and <math>B</math> are estimated from:

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

 
:and:

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

 
Since the standard deviation, <math>{{\widehat{\sigma }}_{{T}',}}</math> is a positive parameter, <math>\ln ({{\widehat{\sigma }}_{{{T}'}}})</math> is treated as normally distributed, and the bounds are estimated from:

 
::<math>\begin{align}
& {{\sigma }_{U}}= & {{\widehat{\sigma }}_{{{T}'}}}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var({{\widehat{\sigma }}_{{{T}'}}})}}{{{\widehat{\sigma }}_{{{T}'}}}}}}\text{ (Upper bound)} \\
& {{\sigma }_{L}}= & \frac{{{\widehat{\sigma }}_{{{T}'}}}}{{{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var({{\widehat{\sigma }}_{{{T}'}}})}}{{{\widehat{\sigma }}_{{{T}'}}}}}}}\text{ (Lower bound)}
\end{align}</math>

 
The variances and covariances of <math>A,</math> <math>B,</math> and <math>{{\sigma }_{{{T}'}}}</math> are estimated from the local Fisher matrix (evaluated at <math>\widehat{A},</math> <math>\widehat{B}</math> , <math>{{\widehat{\sigma }}_{{{T}'}}})</math> as follows:

 
::<math>\left( \begin{matrix}
Var\left( {{\widehat{\sigma }}_{{{T}'}}} \right) & Cov\left( \widehat{A},{{\widehat{\sigma }}_{{{T}'}}} \right) & Cov\left( \widehat{B},{{\widehat{\sigma }}_{{{T}'}}} \right) \\
Cov\left( {{\widehat{\sigma }}_{{{T}'}}},\widehat{A} \right) & Var\left( \widehat{A} \right) & Cov\left( \widehat{A},\widehat{B} \right) \\
Cov\left( {{\widehat{\sigma }}_{{{T}'}}},\widehat{B} \right) & Cov\left( \widehat{B},\widehat{A} \right) & Var\left( \widehat{B} \right) \\
\end{matrix} \right)={{[F]}^{-1}}</math>

 
:where:

 
::<math>F=\left( \begin{matrix}
-\tfrac{{{\partial }^{2}}\Lambda }{\partial \sigma _{{{T}'}}^{2}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\sigma }_{{{T}'}}}\partial A} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\sigma }_{{{T}'}}}\partial B} \\
-\tfrac{{{\partial }^{2}}\Lambda }{\partial A\partial {{\sigma }_{{{T}'}}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{A}^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial A\partial B} \\
-\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial {{\sigma }_{{{T}'}}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial A} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{B}^{2}}} \\
\end{matrix} \right)</math>

Template:Eyring-log cb on parameters - Revision history

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

Athanasios Gerokostopoulos: /* Bounds on the Parameters */

Athanasios Gerokostopoulos: /* Bounds on the Parameters */

Nicolette Young: Created page with '===Bounds on the Parameters=== The lower and upper bounds on A and B are estimated from: ::\begin{align} & {{A}_{U}}= & \widehat{…'

Nicolette Young: Created page with '===Bounds on the Parameters===
The lower and upper bounds on $A$ and $B$ are estimated from:
:: $\begin{align} & {{A}_{U}}= & \widehat{…'$