Template:Acb4weib on time - Revision history

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

2012-08-16T06:02:20Z

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

← Older revision		Revision as of 06:02, 16 August 2012
Line 1:		Line 1:
	~~====Confidence Bounds on Time====~~		#REDIRECT [[Arrhenius_Relationship#Approximate_Confidence_Bounds_for_the_Arrhenius-Weibull]]
	~~<br>~~
	~~The bounds on time for a given reliability are estimated by first solving the reliability function with respect to time:~~

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

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

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

	~~where <math>\widehat{u}=\ln \widehat{T}</math> .~~

	~~<br>~~
	~~The upper and lower bounds on <math>u</math> are estimated from:~~


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

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

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

	~~<br>~~

Athanasios Gerokostopoulos at 01:12, 14 February 2012

2012-02-14T01:12:30Z

← Older revision		Revision as of 01:12, 14 February 2012
Line 32:		Line 32:
	<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>

Line 42:		Line 40:
	<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 })+\frac{1}{{{V}^{2}}}Var(\widehat{B})+\frac{1}{{{\widehat{C}}^{2}}}Var(\widehat{C}) \\		& Var(\widehat{u})= & \frac{1}{{{\widehat{\beta }}^{4}}}{{\left[ \ln (-\ln (R)) \right]}^{2}}Var(\widehat{\beta })+\frac{1}{{{V}^{2}}}Var(\widehat{B})+\frac{1}{{{\widehat{C}}^{2}}}Var(\widehat{C})-\frac{2\ln (-\ln (R))}{{{\widehat{\beta }}^{2}}V}Cov(\widehat{\beta },\widehat{B})-\frac{2\ln (-\ln (R))}{{{\widehat{\beta }}^{2}}\widehat{C}}Cov(\widehat{\beta },\widehat{C}) +\frac{2}{V\widehat{C}}Cov(\widehat{B},\widehat{C})
	~~& &~~ -\frac{2\ln (-\ln (R))}{{{\widehat{\beta }}^{2}}V}Cov(\widehat{\beta },\widehat{B})-\frac{2\ln (-\ln (R))}{{{\widehat{\beta }}^{2}}\widehat{C}}Cov(\widehat{\beta },\widehat{C}) \\
	~~& &~~ +\frac{2}{V\widehat{C}}Cov(\widehat{B},\widehat{C})
	\end{align}</math>		\end{align}</math>

Line 57:		Line 53:

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

Athanasios Gerokostopoulos at 01:11, 14 February 2012

2012-02-14T01:11:13Z

← Older revision		Revision as of 01:11, 14 February 2012
Line 5:		Line 5:
	<br>		<br>
	::<math>\begin{align}		::<math>\begin{align}
	& \ln (R)= & -{{\left( \frac{\widehat{T}}{\widehat{C}\cdot {{e}^{\tfrac{\widehat{B}}{V}}}} \right)}^{\widehat{\beta }}} \\		\ln (R)&= -{{\left( \frac{\widehat{T}}{\widehat{C}\cdot {{e}^{\tfrac{\widehat{B}}{V}}}} \right)}^{\widehat{\beta }}} \\
	& \ln (-\ln (R))= & \widehat{\beta }\left( \ln \widehat{T}-\ln \widehat{C}-\frac{\widehat{B}}{V} \right)		\ln (-\ln (R))&= \widehat{\beta }\left( \ln \widehat{T}-\ln \widehat{C}-\frac{\widehat{B}}{V} \right)
	\end{align}</math>		\end{align}</math>

Athanasios Gerokostopoulos: /* Confidence Bounds on Time */

2012-02-14T00:02:50Z

Confidence Bounds on Time

← Older revision		Revision as of 00:02, 14 February 2012
Line 1:		Line 1:
	===Confidence Bounds on Time===		====Confidence Bounds on Time====
	<br>		<br>
	The bounds on time for a given reliability are estimated by first solving the reliability function with respect to time:		The bounds on time for a given reliability are estimated by first solving the reliability function with respect to time:

Nicolette Young: Created page with '===Confidence Bounds on Time===
The bounds on time for a given reliability are estimated by first solving the reliability function with respect to time:
:: $\begin…'$

2012-01-12T22:26:56Z

Created page with '===Confidence Bounds on Time=== The bounds on time for a given reliability are estimated by first solving the reliability function with respect to time: ::<math>\begin…'

New page

===Confidence Bounds on Time===
 
The bounds on time for a given reliability are estimated by first solving the reliability function with respect to time:

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

 
or:

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

where <math>\widehat{u}=\ln \widehat{T}</math> .

 
The upper and lower bounds on <math>u</math> are 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 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})= & \frac{1}{{{\widehat{\beta }}^{4}}}{{\left[ \ln (-\ln (R)) \right]}^{2}}Var(\widehat{\beta })+\frac{1}{{{V}^{2}}}Var(\widehat{B})+\frac{1}{{{\widehat{C}}^{2}}}Var(\widehat{C}) \\
& & -\frac{2\ln (-\ln (R))}{{{\widehat{\beta }}^{2}}V}Cov(\widehat{\beta },\widehat{B})-\frac{2\ln (-\ln (R))}{{{\widehat{\beta }}^{2}}\widehat{C}}Cov(\widehat{\beta },\widehat{C}) \\
& & +\frac{2}{V\widehat{C}}Cov(\widehat{B},\widehat{C})
\end{align}</math>

 
The upper and lower bounds on time can 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. (ArreibTupper) and (ArreibTlower).

Template:Acb4weib on time - Revision history

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

Athanasios Gerokostopoulos at 01:12, 14 February 2012

Athanasios Gerokostopoulos at 01:11, 14 February 2012

Athanasios Gerokostopoulos: /* Confidence Bounds on Time */

Nicolette Young: Created page with '===Confidence Bounds on Time=== The bounds on time for a given reliability are estimated by first solving the reliability function with respect to time: ::\begin…'

Nicolette Young: Created page with '===Confidence Bounds on Time===
The bounds on time for a given reliability are estimated by first solving the reliability function with respect to time:
:: $\begin…'$