Template:Erying-weib cb on reliability - Revision history

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

2012-08-17T01:08:59Z

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

← Older revision		Revision as of 01:08, 17 August 2012
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	~~===Confidence Bounds on Reliability===~~		#REDIRECT [[Eyring_Relationship#Approximate_Confidence_Bounds_for_the_Eyring-Weibull]]

	~~<br>~~
	~~The reliability function for the Eyring-Weibull model (ML estimate) is given by:~~

	~~<br>~~
	~~::<math>\widehat{R}(T,V)={{e}^{-{{\left( T\cdot V\cdot {{e}^{\left( \widehat{A}-\tfrac{\widehat{B}}{V} \right)}} \right)}^{\widehat{\beta }}}}}</math>~~

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

	~~<br>~~
	~~::<math>\widehat{R}(T,V)={{e}^{-{{e}^{\ln \left~~[ ~~{{\left( T\cdot V\cdot {{e}^{\left( \widehat{A}-\tfrac{\widehat{B}}{V} \right)}} \right)}^{\widehat{\beta }}} \right]}}}}</math>~~

	~~<br>~~
	~~Setting:~~

	~~<br>~~
	~~::<math>\widehat{u}=\ln \left~~[ ~~{{\left( T\cdot V\cdot {{e}^{\left( \widehat{A}~~-~~\tfrac{\widehat{B}}{V} \right)}} \right)}^{\widehat{\beta }}} \right~~]~~</math>~~

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

	~~<br>~~
	~~::<math>\widehat{u}=\widehat{\beta }\left[ \ln (T)+\ln (V)+\widehat{A}-\frac{\widehat{B}}{V} \right~~]~~</math>~~

	~~<br>~~
	~~The reliability function now becomes:~~

	~~<br>~~
	~~::<math>\widehat{R}(T,V)={{e}^{-e\widehat{^{u}}}}</math>~~

	~~<br>~~
	~~The next step is to find the upper and lower bounds on <math>\widehat{u}</math> :~~

	~~<br>~~
	~~::<math>{{u}_{U}}=\widehat{u}+{{K}_{\alpha }}\sqrt{Var(\widehat{u})}</math>~~

	~~<br>~~
	~~::<math>{{u}_{L}}=\widehat{u}-{{K}_{\alpha }}\sqrt{Var(\widehat{u})}</math>~~

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

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

	~~<br>~~
	~~or:~~
	~~<br>~~
	~~<br>~~
	~~::<math>\begin{align}~~
	Var(\widehat{u})= & {{\left( \frac{\widehat{u}}{\widehat{\beta }} \right)}^{2}}Var(\widehat{\beta })+{{\widehat{\beta }}^{2}}Var(\widehat{A}) +{{\left( \frac{\widehat{\beta }}{V} \right)}^{2}}Var(\widehat{B}) +2\widehat{u}\cdot Cov(\widehat{\beta },\widehat{A})-\frac{2\widehat{u}}{V}Cov(\widehat{\beta },\widehat{B}) -\frac{2{{\widehat{\beta }}^{2}}}{V}Cov(\widehat{A},\widehat{B})
	~~\end{align}</math>~~

	~~<br>~~
	~~The upper and lower bounds on reliability are:~~

	~~<br>~~
	~~::<math>\begin{align}~~
	~~& {{R}_{U}}= & {{e}^{-{{e}^{\left( {{u}_{L}} \right)}}}} \\~~
	~~& {{R}_{L}}= & {{e}^{-{{e}^{\left( {{u}_{U}} \right)}}}}~~
	~~\end{align}</math>~~

	~~<br>~~

Athanasios Gerokostopoulos at 00:30, 15 February 2012

2012-02-15T00:30:35Z

← Older revision		Revision as of 00:30, 15 February 2012
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	<br>		<br>
	::<math>\begin{align}		::<math>\begin{align}
	Var(\widehat{u})= & {{\left( \frac{\widehat{u}}{\widehat{\beta }} \right)}^{2}}Var(\widehat{\beta })+{{\widehat{\beta }}^{2}}Var(\widehat{A}) +{{\left( \frac{\widehat{\beta }}{V} \right)}^{2}}Var(\widehat{B}) \\		Var(\widehat{u})= & {{\left( \frac{\widehat{u}}{\widehat{\beta }} \right)}^{2}}Var(\widehat{\beta })+{{\widehat{\beta }}^{2}}Var(\widehat{A}) +{{\left( \frac{\widehat{\beta }}{V} \right)}^{2}}Var(\widehat{B}) +2\widehat{u}\cdot Cov(\widehat{\beta },\widehat{A})-\frac{2\widehat{u}}{V}Cov(\widehat{\beta },\widehat{B}) -\frac{2{{\widehat{\beta }}^{2}}}{V}Cov(\widehat{A},\widehat{B})
	& +2\widehat{u}\cdot Cov(\widehat{\beta },\widehat{A})-\frac{2\widehat{u}}{V}Cov(\widehat{\beta },\widehat{B}) -\frac{2{{\widehat{\beta }}^{2}}}{V}Cov(\widehat{A},\widehat{B})
	\end{align}</math>		\end{align}</math>

Athanasios Gerokostopoulos: /* Confidence Bounds on Reliability */

2012-02-15T00:29:48Z

Confidence Bounds on Reliability

← Older revision		Revision as of 00:29, 15 February 2012
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	<br>		<br>
	:or:		or:

	<br>		<br>
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	<br>		<br>
	:Setting:		Setting:

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

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

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

	<br>		<br>
	:or:		or:
	<br>		<br>
	<br>		<br>
	::<math>\begin{align}		::<math>\begin{align}
	& Var(\widehat{u})= & {{\left( \frac{\widehat{u}}{\widehat{\beta }} \right)}^{2}}Var(\widehat{\beta })+{{\widehat{\beta }}^{2}}Var(\widehat{A}) \\		Var(\widehat{u})= & {{\left( \frac{\widehat{u}}{\widehat{\beta }} \right)}^{2}}Var(\widehat{\beta })+{{\widehat{\beta }}^{2}}Var(\widehat{A}) +{{\left( \frac{\widehat{\beta }}{V} \right)}^{2}}Var(\widehat{B}) \\
	~~& &~~ +{{\left( \frac{\widehat{\beta }}{V} \right)}^{2}}Var(\widehat{B}) \\		& +2\widehat{u}\cdot Cov(\widehat{\beta },\widehat{A})-\frac{2\widehat{u}}{V}Cov(\widehat{\beta },\widehat{B}) -\frac{2{{\widehat{\beta }}^{2}}}{V}Cov(\widehat{A},\widehat{B})
	& & +2\widehat{u}\cdot Cov(\widehat{\beta },\widehat{A})-\frac{2\widehat{u}}{V}Cov(\widehat{\beta },\widehat{B}) \\
	~~& &~~ -\frac{2{{\widehat{\beta }}^{2}}}{V}Cov(\widehat{A},\widehat{B})
	\end{align}</math>		\end{align}</math>

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	<br>		<br>
	~~where <math>{{u}_{U}}</math> and <math>{{u}_{L}}</math> are estimated using Eqns (EyrExpu) and (EyrExpl).~~

Nicolette Young: Created page with '===Confidence Bounds on Reliability===
The reliability function for the Eyring-Weibull model (ML estimate) is given by:
:: $\widehat{R}(T,V)={{e}^{-{{\left( T\cdo…'$

2012-01-12T23:43:53Z

Created page with '===Confidence Bounds on Reliability=== The reliability function for the Eyring-Weibull model (ML estimate) is given by: ::<math>\widehat{R}(T,V)={{e}^{-{{\left( T\cdo…'

New page

===Confidence Bounds on Reliability===

 
The reliability function for the Eyring-Weibull model (ML estimate) is given by:

 
::<math>\widehat{R}(T,V)={{e}^{-{{\left( T\cdot V\cdot {{e}^{\left( \widehat{A}-\tfrac{\widehat{B}}{V} \right)}} \right)}^{\widehat{\beta }}}}}</math>

 
:or:

 
::<math>\widehat{R}(T,V)={{e}^{-{{e}^{\ln \left[ {{\left( T\cdot V\cdot {{e}^{\left( \widehat{A}-\tfrac{\widehat{B}}{V} \right)}} \right)}^{\widehat{\beta }}} \right]}}}}</math>

 
:Setting:

 
::<math>\widehat{u}=\ln \left[ {{\left( T\cdot V\cdot {{e}^{\left( \widehat{A}-\tfrac{\widehat{B}}{V} \right)}} \right)}^{\widehat{\beta }}} \right]</math>

 
:or:

 
::<math>\widehat{u}=\widehat{\beta }\left[ \ln (T)+\ln (V)+\widehat{A}-\frac{\widehat{B}}{V} \right]</math>

 
The reliability function now becomes:

 
::<math>\widehat{R}(T,V)={{e}^{-e\widehat{^{u}}}}</math>

 
The next step is to find the upper and lower bounds on <math>\widehat{u}</math> :

 
::<math>{{u}_{U}}=\widehat{u}+{{K}_{\alpha }}\sqrt{Var(\widehat{u})}</math>

 
::<math>{{u}_{L}}=\widehat{u}-{{K}_{\alpha }}\sqrt{Var(\widehat{u})}</math>

 
:where:

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

 
:or:
 
 
::<math>\begin{align}
& Var(\widehat{u})= & {{\left( \frac{\widehat{u}}{\widehat{\beta }} \right)}^{2}}Var(\widehat{\beta })+{{\widehat{\beta }}^{2}}Var(\widehat{A}) \\
& & +{{\left( \frac{\widehat{\beta }}{V} \right)}^{2}}Var(\widehat{B}) \\
& & +2\widehat{u}\cdot Cov(\widehat{\beta },\widehat{A})-\frac{2\widehat{u}}{V}Cov(\widehat{\beta },\widehat{B}) \\
& & -\frac{2{{\widehat{\beta }}^{2}}}{V}Cov(\widehat{A},\widehat{B})
\end{align}</math>

 
The upper and lower bounds on reliability are:

 
::<math>\begin{align}
& {{R}_{U}}= & {{e}^{-{{e}^{\left( {{u}_{L}} \right)}}}} \\
& {{R}_{L}}= & {{e}^{-{{e}^{\left( {{u}_{U}} \right)}}}}
\end{align}</math>

 
where <math>{{u}_{U}}</math> and <math>{{u}_{L}}</math> are estimated using Eqns (EyrExpu) and (EyrExpl).

Template:Erying-weib cb on reliability - Revision history

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

Athanasios Gerokostopoulos at 00:30, 15 February 2012

Athanasios Gerokostopoulos: /* Confidence Bounds on Reliability */

Nicolette Young: Created page with '===Confidence Bounds on Reliability=== The reliability function for the Eyring-Weibull model (ML estimate) is given by: ::\widehat{R}(T,V)={{e}^{-{{\left( T\cdo…'

Nicolette Young: Created page with '===Confidence Bounds on Reliability===
The reliability function for the Eyring-Weibull model (ML estimate) is given by:
:: $\widehat{R}(T,V)={{e}^{-{{\left( T\cdo…'$