Template:Eyring-ex cb on mean life - Revision history

Richard House: Redirected page to Eyring Relationship#Eyring Confidence Bounds

2012-08-17T01:01:17Z

Redirected page to Eyring Relationship#Eyring Confidence Bounds

← Older revision		Revision as of 01:01, 17 August 2012
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	~~===Confidence Bounds on Mean Life===~~		#REDIRECT [[Eyring_Relationship#Eyring_Confidence_Bounds]]
	~~<br>~~

	The mean life for the Eyring relationship is given by setting <math>m=L(V)</math> . The upper <math>({{m}_{U}})</math> and lower <math>({{m}_{L}})</math> bounds on the mean life (ML estimate of the mean life) are estimated by:

	~~<br>~~
	~~::<math>{{m}_{U}}=\widehat{m}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{m})}}{\widehat{m}}}}</math>~~

	~~<br>~~
	~~::<math>{{m}_{L}}=\widehat{m}\cdot {{e}^{-\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{m})}}{\widehat{m}}}}</math>~~

	~~<br>~~
	~~where <math>{{K}_{\alpha }}</math> is defined by:~~

	~~<br>~~
	~~::<math>\alpha =\frac{1}{\sqrt{2\pi }}\int_{{{K}_{\alpha }}}^{\infty }{{e}^{-\tfrac{{{t}^{2}}}{2}}}dt=1-\Phi ({{K}_{\alpha }})</math>~~

	~~<br>~~
	If <math>\delta </math> is the confidence level, then <math>\alpha =\tfrac{1-\delta }{2}</math> for the two-sided bounds, and <math>\alpha =1-\delta </math> for the one-sided bounds. The variance of <math>\widehat{m}</math> is given by:

	~~<br>~~
	~~::<math>\begin{align}~~
	& Var(\widehat{m})= & {{\left( \frac{\partial m}{\partial A} \right)}^{2}}Var(\widehat{A})+{{\left( \frac{\partial m}{\partial B} \right)}^{2}}Var(\widehat{B}) +2\left( \frac{\partial m}{\partial A} \right)\left( \frac{\partial m}{\partial B} \right)Cov(\widehat{A},\widehat{B})
	~~\end{align}</math>~~

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

	~~<br>~~
	~~::<math>Var(\widehat{m})=\frac{1}{{{V}^{2}}}{{e}^{-2\left( \widehat{A}-\tfrac{\widehat{B}}{V} \right)}}\left~~[ ~~Var(\widehat{A})+\frac{1}{{{V}^{2}}}Var(\widehat{B})-\frac{1}{V}Cov(\widehat{A},\widehat{B}) \right]</math>~~

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

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

	~~<br>~~

Athanasios Gerokostopoulos: /* Confidence Bounds on Mean Life */

2012-02-27T23:58:21Z

Confidence Bounds on Mean Life

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

	<br>		<br>
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	-\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial A} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{B}^{2}}} \\		-\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: /* Confidence Bounds on Mean Life */

2012-02-15T00:25:26Z

Confidence Bounds on Mean Life

← Older revision		Revision as of 00:25, 15 February 2012
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	<br>		<br>
	::<math>\alpha =\frac{1}{\sqrt{2\pi }}\~~mathop{}_~~{{{K}_{\alpha }}}^{\infty }{{e}^{-\tfrac{{{t}^{2}}}{2}}}dt=1-\Phi ({{K}_{\alpha }})</math>		::<math>\alpha =\frac{1}{\sqrt{2\pi }}\int_{{{K}_{\alpha }}}^{\infty }{{e}^{-\tfrac{{{t}^{2}}}{2}}}dt=1-\Phi ({{K}_{\alpha }})</math>

	<br>		<br>
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	<br>		<br>
	::<math>\begin{align}		::<math>\begin{align}
	& Var(\widehat{m})= & {{\left( \frac{\partial m}{\partial A} \right)}^{2}}Var(\widehat{A})+{{\left( \frac{\partial m}{\partial B} \right)}^{2}}Var(\widehat{B}) \\		& Var(\widehat{m})= & {{\left( \frac{\partial m}{\partial A} \right)}^{2}}Var(\widehat{A})+{{\left( \frac{\partial m}{\partial B} \right)}^{2}}Var(\widehat{B}) +2\left( \frac{\partial m}{\partial A} \right)\left( \frac{\partial m}{\partial B} \right)Cov(\widehat{A},\widehat{B})
	~~& &~~ +2\left( \frac{\partial m}{\partial A} \right)\left( \frac{\partial m}{\partial B} \right)Cov(\widehat{A},\widehat{B})
	\end{align}</math>		\end{align}</math>

Athanasios Gerokostopoulos: /* Confidence Bounds on Mean Life */

2012-02-15T00:24:17Z

Confidence Bounds on Mean Life

← Older revision		Revision as of 00:24, 15 February 2012
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	<br>		<br>

	The mean life for the Eyring relationship l is given ~~by Eqn. (eyring)~~ by setting <math>m=L(V)</math> . The upper <math>({{m}_{U}})</math> and lower <math>({{m}_{L}})</math> bounds on the mean life (ML estimate of the mean life) are estimated by:		The mean life for the Eyring relationship is given by setting <math>m=L(V)</math> . The upper <math>({{m}_{U}})</math> and lower <math>({{m}_{L}})</math> bounds on the mean life (ML estimate of the mean life) are estimated by:

	<br>		<br>

Nicolette Young: Created page with '===Confidence Bounds on Mean Life===
The mean life for the Eyring relationship l is given by Eqn. (eyring) by setting $m=L(V)$ . The upper $({{m}_{U}})$
2012-01-12T23:35:39Z

Created page with '===Confidence Bounds on Mean Life=== The mean life for the Eyring relationship l is given by Eqn. (eyring) by setting <math>m=L(V)</math> . The upper <math>({{m}_{U}})</m…'

New page
===Confidence Bounds on Mean Life===
 

The mean life for the Eyring relationship l is given by Eqn. (eyring) by setting <math>m=L(V)</math> . The upper <math>({{m}_{U}})</math> and lower <math>({{m}_{L}})</math> bounds on the mean life (ML estimate of the mean life) are estimated by:

 
::<math>{{m}_{U}}=\widehat{m}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{m})}}{\widehat{m}}}}</math>

 
::<math>{{m}_{L}}=\widehat{m}\cdot {{e}^{-\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{m})}}{\widehat{m}}}}</math>

 
where <math>{{K}_{\alpha }}</math> is defined by:

 
::<math>\alpha =\frac{1}{\sqrt{2\pi }}\mathop{}_{{{K}_{\alpha }}}^{\infty }{{e}^{-\tfrac{{{t}^{2}}}{2}}}dt=1-\Phi ({{K}_{\alpha }})</math>

 
If <math>\delta </math> is the confidence level, then <math>\alpha =\tfrac{1-\delta }{2}</math> for the two-sided bounds, and <math>\alpha =1-\delta </math> for the one-sided bounds. The variance of <math>\widehat{m}</math> is given by:

 
::<math>\begin{align}
& Var(\widehat{m})= & {{\left( \frac{\partial m}{\partial A} \right)}^{2}}Var(\widehat{A})+{{\left( \frac{\partial m}{\partial B} \right)}^{2}}Var(\widehat{B}) \\
& & +2\left( \frac{\partial m}{\partial A} \right)\left( \frac{\partial m}{\partial B} \right)Cov(\widehat{A},\widehat{B})
\end{align}</math>

 
:or:

 
::<math>Var(\widehat{m})=\frac{1}{{{V}^{2}}}{{e}^{-2\left( \widehat{A}-\tfrac{\widehat{B}}{V} \right)}}\left[ Var(\widehat{A})+\frac{1}{{{V}^{2}}}Var(\widehat{B})-\frac{1}{V}Cov(\widehat{A},\widehat{B}) \right]</math>

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

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