Template:Acb4weib on reliability - Revision history

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

2012-08-16T06:01:14Z

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

← Older revision		Revision as of 06:01, 16 August 2012
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	~~====Confidence Bounds on Reliability====~~		#REDIRECT [[Arrhenius_Relationship#Approximate_Confidence_Bounds_for_the_Arrhenius-Weibull]]

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

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

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

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

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

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

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

	~~<br>~~
	~~::<math>\widehat{u}=\widehat{\beta }\left[ \ln (T)-\ln (\widehat{C})-\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 B} \right)}^{2}}Var(\widehat{B})+{{\left( \frac{\partial \widehat{u}}{\partial C} \right)}^{2}}Var(\widehat{C}) +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 \beta } \right)\left( \frac{\partial \widehat{u}}{\partial C} \right)Cov(\widehat{\beta },\widehat{C}) +2\left( \frac{\partial \widehat{u}}{\partial B} \right)\left( \frac{\partial \widehat{u}}{\partial C} \right)Cov(\widehat{B},\widehat{C})
	~~\end{align}</math>~~

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

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

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

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

	~~<br>~~

Athanasios Gerokostopoulos: /* Confidence Bounds on Reliability */

2012-02-14T01:08:35Z

Confidence Bounds on Reliability

← Older revision		Revision as of 01:08, 14 February 2012
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	<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 B} \right)}^{2}}Var(\widehat{B})+{{\left( \frac{\partial \widehat{u}}{\partial C} \right)}^{2}}Var(\widehat{C}) \\		& Var(\widehat{u})= {{\left( \frac{\partial \widehat{u}}{\partial \beta } \right)}^{2}}Var(\widehat{\beta })+{{\left( \frac{\partial \widehat{u}}{\partial B} \right)}^{2}}Var(\widehat{B})+{{\left( \frac{\partial \widehat{u}}{\partial C} \right)}^{2}}Var(\widehat{C}) +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 \beta } \right)\left( \frac{\partial \widehat{u}}{\partial C} \right)Cov(\widehat{\beta },\widehat{C}) +2\left( \frac{\partial \widehat{u}}{\partial B} \right)\left( \frac{\partial \widehat{u}}{\partial C} \right)Cov(\widehat{B},\widehat{C})
	~~& &~~ +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 \beta } \right)\left( \frac{\partial \widehat{u}}{\partial C} \right)Cov(\widehat{\beta },\widehat{C}) \\
	~~& &~~ +2\left( \frac{\partial \widehat{u}}{\partial B} \right)\left( \frac{\partial \widehat{u}}{\partial C} \right)Cov(\widehat{B},\widehat{C})
	\end{align}</math>		\end{align}</math>

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

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


	~~where <math>{{u}_{U}}</math> and <math>{{u}_{L}}</math> are estimated from Eqns. (ArreibRupper) and (ArreibRlower).~~
	<br>		<br>

Athanasios Gerokostopoulos: /* Confidence Bounds on Reliability */

2012-02-14T00:02:43Z

Confidence Bounds on Reliability

← Older revision		Revision as of 00:02, 14 February 2012
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	===Confidence Bounds on Reliability===		====Confidence Bounds on Reliability====

	<br>		<br>

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

2012-01-12T22:26:04Z

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

New page

===Confidence Bounds on Reliability===

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

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

 
or:

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

 
Setting:

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

 
or:

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

 
or:

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

 
The upper and lower bounds on reliability are:

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

where <math>{{u}_{U}}</math> and <math>{{u}_{L}}</math> are estimated from Eqns. (ArreibRupper) and (ArreibRlower).

Template:Acb4weib on reliability - Revision history

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

Athanasios Gerokostopoulos: /* Confidence Bounds on Reliability */

Athanasios Gerokostopoulos: /* Confidence Bounds on Reliability */

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

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