Template:Eyring-weib cb on time - Revision history

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

2012-08-17T01:10:09Z

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

← Older revision		Revision as of 01:10, 17 August 2012
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	~~===Confidence Bounds on Time===~~		#REDIRECT [[Eyring_Relationship#Approximate_Confidence_Bounds_for_the_Eyring-Weibull]]

	~~<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>\begin{align}~~
	~~\ln (R)&=\ -{{\left( \widehat{T}\cdot V\cdot {{e}^{\left( \widehat{A}-\tfrac{\widehat{B}}{V} \right)}} \right)}^{\widehat{\beta }}} \\~~
	~~\ln (-\ln (R))&=\ \widehat{\beta }\left( \ln \widehat{T}+\ln V+\widehat{A}-\frac{\widehat{B}}{V} \right)~~
	~~\end{align}</math>~~

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

	~~<br>~~
	~~::<math>\widehat{u}=\frac{1}{\widehat{\beta }}\ln (-\ln (R))-\ln V-\widehat{A}+\frac{\widehat{B}}{V}</math>~~

	~~<br>~~
	~~where <math>\widehat{u}=ln(\widehat{T})</math>. The upper and lower bounds on <math>\widehat{u}</math> are then estimated from:~~

	~~<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>~~
	~~::<math>\begin{align}~~
	~~& Var(\widehat{u})= \frac{1}{{{\widehat{\beta }}^{4}}}{{\left~~[ ~~\ln (~~-~~\ln (R)) \right~~]}^{2}}Var(\widehat{\beta }) +Var(\widehat{A})+\frac{1}{{{V}^{2}}}Var(\widehat{B}) +\frac{2\ln (-\ln (R))}{{{\widehat{\beta }}^{2}}}Cov(\widehat{\beta },\widehat{A})-\frac{2\ln (-\ln (R))}{{{\widehat{\beta }}^{2}}V}Cov(\widehat{\beta },\widehat{B}) -\frac{2}{V}Cov(\widehat{A},\widehat{B})
	~~\end{align}</math>~~

	~~<br>~~
	~~The upper and lower bounds on time are then found by:~~

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

	~~<br>~~

Athanasios Gerokostopoulos: /* Confidence Bounds on Time */

2012-02-28T00:03:23Z

Confidence Bounds on Time

← Older revision		Revision as of 00:03, 28 February 2012
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	<br>		<br>
	::<math>\widehat{u}=\frac{1}{\widehat{\beta }}\ln (-\ln (R))-\ln V-\widehat{A}+\frac{\widehat{B}}{V}</math>		::<math>\widehat{u}=\frac{1}{\widehat{\beta }}\ln (-\ln (R))-\ln V-\widehat{A}+\frac{\widehat{B}}{V}</math>

	<br>		<br>
	where		where <math>\widehat{u}=ln(\widehat{T})</math>. The upper and lower bounds on <math>\widehat{u}</math> are then estimated from:
	~~<br>~~
	::<math></math>
	~~<br>~~
	The upper and lower bounds on <math>\widehat{u}</math> are then estimated from:

	<br>		<br>

Athanasios Gerokostopoulos: /* Confidence Bounds on Time */

2012-02-27T23:59:35Z

Confidence Bounds on Time

← Older revision		Revision as of 23:59, 27 February 2012
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	<br>		<br>
	:or:		or:

	<br>		<br>

Athanasios Gerokostopoulos: /* Confidence Bounds on Time */

2012-02-15T00:34:47Z

Confidence Bounds on Time

← Older revision		Revision as of 00:34, 15 February 2012
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	<br>		<br>
	::<math>\begin{align}		::<math>\begin{align}
	& \ln (R)= & -{{\left( \widehat{T}\cdot V\cdot {{e}^{\left( \widehat{A}-\tfrac{\widehat{B}}{V} \right)}} \right)}^{\widehat{\beta }}} \\		\ln (R)&=\ -{{\left( \widehat{T}\cdot V\cdot {{e}^{\left( \widehat{A}-\tfrac{\widehat{B}}{V} \right)}} \right)}^{\widehat{\beta }}} \\
	& \ln (-\ln (R))= & \widehat{\beta }\left( \ln \widehat{T}+\ln V+\widehat{A}-\frac{\widehat{B}}{V} \right)		\ln (-\ln (R))&=\ \widehat{\beta }\left( \ln \widehat{T}+\ln V+\widehat{A}-\frac{\widehat{B}}{V} \right)
	\end{align}</math>		\end{align}</math>

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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>
	::<math>\begin{align}		::<math>\begin{align}
	& Var(\widehat{u})= & \frac{1}{{{\widehat{\beta }}^{4}}}{{\left[ \ln (-\ln (R)) \right]}^{2}}Var(\widehat{\beta }) \\		& Var(\widehat{u})= \frac{1}{{{\widehat{\beta }}^{4}}}{{\left[ \ln (-\ln (R)) \right]}^{2}}Var(\widehat{\beta }) +Var(\widehat{A})+\frac{1}{{{V}^{2}}}Var(\widehat{B}) +\frac{2\ln (-\ln (R))}{{{\widehat{\beta }}^{2}}}Cov(\widehat{\beta },\widehat{A})-\frac{2\ln (-\ln (R))}{{{\widehat{\beta }}^{2}}V}Cov(\widehat{\beta },\widehat{B}) -\frac{2}{V}Cov(\widehat{A},\widehat{B})
	~~& &~~ +Var(\widehat{A})+\frac{1}{{{V}^{2}}}Var(\widehat{B}) \\
	~~& &~~ +\frac{2\ln (-\ln (R))}{{{\widehat{\beta }}^{2}}}Cov(\widehat{\beta },\widehat{A})-\frac{2\ln (-\ln (R))}{{{\widehat{\beta }}^{2}}V}Cov(\widehat{\beta },\widehat{B}) \\
	~~& &~~ -\frac{2}{V}Cov(\widehat{A},\widehat{B})
	\end{align}</math>		\end{align}</math>

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	\end{align}</math>		\end{align}</math>

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

Nicolette Young: Created page with '===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 tim…'

2012-01-12T23:44:46Z

Created page with '===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 tim…'

New page

===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>\begin{align}
& \ln (R)= & -{{\left( \widehat{T}\cdot V\cdot {{e}^{\left( \widehat{A}-\tfrac{\widehat{B}}{V} \right)}} \right)}^{\widehat{\beta }}} \\
& \ln (-\ln (R))= & \widehat{\beta }\left( \ln \widehat{T}+\ln V+\widehat{A}-\frac{\widehat{B}}{V} \right)
\end{align}</math>

 
:or:

 
::<math>\widehat{u}=\frac{1}{\widehat{\beta }}\ln (-\ln (R))-\ln V-\widehat{A}+\frac{\widehat{B}}{V}</math>
 
where
 
::<math></math>
 
The upper and lower bounds on <math>\widehat{u}</math> are then estimated from:

 
::<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})= & \frac{1}{{{\widehat{\beta }}^{4}}}{{\left[ \ln (-\ln (R)) \right]}^{2}}Var(\widehat{\beta }) \\
& & +Var(\widehat{A})+\frac{1}{{{V}^{2}}}Var(\widehat{B}) \\
& & +\frac{2\ln (-\ln (R))}{{{\widehat{\beta }}^{2}}}Cov(\widehat{\beta },\widehat{A})-\frac{2\ln (-\ln (R))}{{{\widehat{\beta }}^{2}}V}Cov(\widehat{\beta },\widehat{B}) \\
& & -\frac{2}{V}Cov(\widehat{A},\widehat{B})
\end{align}</math>

 
The upper and lower bounds on time are then found by:

 
::<math>\begin{align}
& {{T}_{U}}= & {{e}^{{{u}_{U}}}} \\
& {{T}_{L}}= & {{e}^{{{u}_{L}}}}
\end{align}</math>

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

Template:Eyring-weib cb on time - Revision history

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

Athanasios Gerokostopoulos: /* Confidence Bounds on Time */

Athanasios Gerokostopoulos: /* Confidence Bounds on Time */

Athanasios Gerokostopoulos: /* Confidence Bounds on Time */

Nicolette Young: Created page with '===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 tim…'

Nicolette Young: Created page with '===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 tim…'