Template:App cb 4 the eyring-ex - Revision history

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

2012-08-17T01:00:20Z

Redirected page to Eyring Relationship#Eyring Confidence Bounds

← Older revision		Revision as of 01:00, 17 August 2012
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	~~==Approximate Confidence Bounds for the Eyring-Exponential==~~		#REDIRECT [[Eyring_Relationship#Eyring_Confidence_Bounds]]
	~~<br>~~
	~~{{eyring-ex cb on mean life}}~~

	~~{{eyring-ex cb on reliability}}~~

	~~{{eyring-ex cb on time}}~~

Nicolette Young: /* Confidence Bounds on Time */

2012-01-12T23:36:29Z

Confidence Bounds on Time

← Older revision		Revision as of 23:36, 12 January 2012
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	{{eyring-ex cb on reliability}}		{{eyring-ex cb on reliability}}

	~~===Confidence Bounds on Time===~~		{{eyring-ex cb on time}}
	~~<br>~~
	~~The bounds on time (ML estimate of time) for a given reliability are estimated by first solving the reliability function with respect to time:~~

	~~<br>~~
	~~::<math>\widehat~~{~~T}=-\widehat~~{~~m}\cdot \ln (R)</math>~~

	~~<br>~~
	~~The corresponding confidence bounds are estimated from:~~

	~~<br>~~
	~~::<math>\begin{align}~~
	~~& {{T}_{U}}= &~~ -~~{{m}_{U}}\cdot \ln (R) \\~~
	~~& & \\~~
	~~& {{T}_{L}}= & -{{m}_{L}}\cdot \ln (R)~~
	~~\end{align}</math>~~

	~~<br>~~
	~~where <math>{{m}_{U}}</math> and <math>{{m}_{L~~}}~~</math> are estimated using Eqns. (EyrxpMeanUpper) and (EyrxpMeanLower).~~

Nicolette Young: /* Confidence Bounds on Reliability */

2012-01-12T23:36:00Z

Confidence Bounds on Reliability

← Older revision		Revision as of 23:36, 12 January 2012
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	{{eyring-ex cb on mean life}}		{{eyring-ex cb on mean life}}

	~~===Confidence Bounds~~ on ~~Reliability===~~		{{eyring-ex cb on reliability}}

	~~The bounds on reliability at a given time, <math>T</math> , are estimated by:~~

	~~<br>~~
	~~::<math>\begin{align}~~
	~~& {{R}_{U}}= & {{e}^{-\tfrac{T}{{{m}_{U}}}}} \\~~
	~~& & \\~~
	~~& {{R}_{L}}= & {{e}^{-\tfrac{T}{{{m}_{L}}}}}~~
	~~\end{align}</math>~~

	~~<br>~~
	~~where <math>{{m}_{U}}</math> and <math>{{m}_{L}}</math> are estimated using Eqns. (EyrxpMeanUpper) and (EyrxpMeanLower).~~
	~~<br>~~
	===Confidence Bounds on Time===		===Confidence Bounds on Time===
	<br>		<br>

Nicolette Young: /* Confidence Bounds on Mean Life */

2012-01-12T23:35:34Z

Confidence Bounds on Mean Life

@@ Line 1: / Line 1: @@
 ==Approximate Confidence Bounds for the Eyring-Exponential==
 <br>
-===Confidence Bounds on Mean Life===
+{{eyring-ex cb on mean life}}
 ===Confidence Bounds on Reliability===

Nicolette Young: Created page with '==Approximate Confidence Bounds for the Eyring-Exponential==
===Confidence Bounds on Mean Life===
The mean life for the Eyring relationship l is given by Eqn. (eyring)…'

2012-01-12T23:34:38Z

Created page with '==Approximate Confidence Bounds for the Eyring-Exponential== ===Confidence Bounds on Mean Life=== The mean life for the Eyring relationship l is given by Eqn. (eyring)…'

New page

==Approximate Confidence Bounds for the Eyring-Exponential==
 
===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>

===Confidence Bounds on Reliability===

The bounds on reliability at a given time, <math>T</math> , are estimated by:

 
::<math>\begin{align}
& {{R}_{U}}= & {{e}^{-\tfrac{T}{{{m}_{U}}}}} \\
& & \\
& {{R}_{L}}= & {{e}^{-\tfrac{T}{{{m}_{L}}}}}
\end{align}</math>

 
where <math>{{m}_{U}}</math> and <math>{{m}_{L}}</math> are estimated using Eqns. (EyrxpMeanUpper) and (EyrxpMeanLower).
 
===Confidence Bounds on Time===
 
The bounds on time (ML estimate of time) for a given reliability are estimated by first solving the reliability function with respect to time:

 
::<math>\widehat{T}=-\widehat{m}\cdot \ln (R)</math>

 
The corresponding confidence bounds are estimated from:

 
::<math>\begin{align}
& {{T}_{U}}= & -{{m}_{U}}\cdot \ln (R) \\
& & \\
& {{T}_{L}}= & -{{m}_{L}}\cdot \ln (R)
\end{align}</math>

 
where <math>{{m}_{U}}</math> and <math>{{m}_{L}}</math> are estimated using Eqns. (EyrxpMeanUpper) and (EyrxpMeanLower).

Template:App cb 4 the eyring-ex - Revision history

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

Nicolette Young: /* Confidence Bounds on Time */

Nicolette Young: /* Confidence Bounds on Reliability */

Nicolette Young: /* Confidence Bounds on Mean Life */

Nicolette Young: Created page with '==Approximate Confidence Bounds for the Eyring-Exponential== ===Confidence Bounds on Mean Life=== The mean life for the Eyring relationship l is given by Eqn. (eyring)…'

Nicolette Young: Created page with '==Approximate Confidence Bounds for the Eyring-Exponential==
===Confidence Bounds on Mean Life===
The mean life for the Eyring relationship l is given by Eqn. (eyring)…'