Template:Eyring-log cb on time - Revision history

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

2012-08-17T01:16:41Z

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

← Older revision		Revision as of 01:16, 17 August 2012
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	~~===Confidence Bounds on Time===~~		#REDIRECT [[Eyring_Relationship#Approximate_Confidence_Bounds_for_the_Eyring-Lognormal]]

	~~<br>~~
	~~The bounds around time for a given lognormal percentile (unreliability) are estimated by first solving the reliability equation with respect to time as follows:~~

	~~<br>~~
	~~::<math>{T}'(V;\widehat{A},\widehat{B},{{\widehat{\sigma }}_{{{T}'}}})=~~-~~\ln (V)-\widehat{A}+\frac{\widehat{B}}{V}+z\cdot {{\widehat{\sigma }}_{{{T}'}}}</math>~~

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

	~~<br>~~
	~~::<math>\begin{align}~~
	~~{T}'(V;\widehat{A},\widehat{B},{{\widehat{\sigma }}_{{{T}'}}}) &=\ \ln (T) \\~~
	~~z &=\ {{\Phi }^{-1}}\left[ F({T}') \right~~]
	~~\end{align}</math>~~

	~~<br>~~
	~~and:~~

	~~<br>~~
	~~::<math>\Phi (z)=\frac{1}{\sqrt{2\pi }}\int_{-\infty }^{z({T}')}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz</math>~~

	~~<br>~~
	~~The next step is to calculate the variance of <math>{T}'(V;\widehat{A},\widehat{B},{{\widehat{\sigma }}_{{{T}'}}}):</math>~~

	~~<br>~~
	~~::<math>\begin{align}~~
	Var({T}')= & {{\left( \frac{\partial {T}'}{\partial A} \right)}^{2}}Var(\widehat{A})+{{\left( \frac{\partial {T}'}{\partial B} \right)}^{2}}Var(\widehat{B})+{{\left( \frac{\partial {T}'}{\partial {{\sigma }_{{{T}'}}}} \right)}^{2}}Var({{\widehat{\sigma }}_{{{T}'}}}) +2\left( \frac{\partial {T}'}{\partial A} \right)\left( \frac{\partial {T}'}{\partial B} \right)Cov\left( \widehat{A},\widehat{B} \right) \\
	& +2\left( \frac{\partial {T}'}{\partial A} \right)\left( \frac{\partial {T}'}{\partial {{\sigma }_{{{T}'}}}} \right)Cov\left( \widehat{A},{{\widehat{\sigma }}_{{{T}'}}} \right) +2\left( \frac{\partial {T}'}{\partial B} \right)\left( \frac{\partial {T}'}{\partial {{\sigma }_{{{T}'}}}} \right)Cov\left( \widehat{B},{{\widehat{\sigma }}_{{{T}'}}} \right)
	~~\end{align}</math>~~

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

	~~<br>~~
	~~::<math>\begin{align}~~
	& Var({T}')= Var(\widehat{A})+\frac{1}{V}Var(\widehat{B})+{{\widehat{z}}^{2}}Var({{\widehat{\sigma }}_{{{T}'}}}) -\frac{2}{V}Cov\left( \widehat{A},\widehat{B} \right) -2\widehat{z}Cov\left( \widehat{A},{{\widehat{\sigma }}_{{{T}'}}} \right) +\frac{2\widehat{z}}{V}Cov\left( \widehat{B},{{\widehat{\sigma }}_{{{T}'}}} \right)
	~~\end{align}</math>~~

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

	~~<br>~~
	~~::<math>\begin{align}~~
	~~& T_{U}^{\prime }= & \ln {{T}_{U}}={T}'+{{K}_{\alpha }}\sqrt{Var({T}')} \\~~
	~~& T_{L}^{\prime }= & \ln {{T}_{L}}={T}'-{{K}_{\alpha }}\sqrt{Var({T}')}~~
	~~\end{align}</math>~~

	~~<br>~~
	~~Solving for <math>{{T}_{U}}</math> and <math>{{T}_{L}}</math> yields:~~

	~~<br>~~
	~~::<math>\begin{align}~~
	~~& {{T}_{U}}= & {{e}^{T_{U}^{\prime }}}\text{ (Upper bound)} \\~~
	~~& {{T}_{L}}= & {{e}^{T_{L}^{\prime }}}\text{ (Lower bound)}~~
	~~\end{align}</math>~~

Athanasios Gerokostopoulos: /* Confidence Bounds on Time */

2012-02-15T00:42:17Z

Confidence Bounds on Time

← Older revision		Revision as of 00:42, 15 February 2012
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	<br>		<br>
	:where:		where:

	<br>		<br>
	::<math>\begin{align}		::<math>\begin{align}
	& {T}'(V;\widehat{A},\widehat{B},{{\widehat{\sigma }}_{{{T}'}}})= & \ln (T) \\		{T}'(V;\widehat{A},\widehat{B},{{\widehat{\sigma }}_{{{T}'}}}) &=\ \ln (T) \\
	& z= & {{\Phi }^{-1}}\left[ F({T}') \right]		z &=\ {{\Phi }^{-1}}\left[ F({T}') \right]
	\end{align}</math>		\end{align}</math>

	<br>		<br>
	:and:		and:

	<br>		<br>
	::<math>\Phi (z)=\frac{1}{\sqrt{2\pi }}\~~mathop{}_~~{-\infty }^{z({T}')}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz</math>		::<math>\Phi (z)=\frac{1}{\sqrt{2\pi }}\int_{-\infty }^{z({T}')}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz</math>

	<br>		<br>
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	<br>		<br>
	::<math>\begin{align}		::<math>\begin{align}
	& Var({T}')= & {{\left( \frac{\partial {T}'}{\partial A} \right)}^{2}}Var(\widehat{A})+{{\left( \frac{\partial {T}'}{\partial B} \right)}^{2}}Var(\widehat{B})+{{\left( \frac{\partial {T}'}{\partial {{\sigma }_{{{T}'}}}} \right)}^{2}}Var({{\widehat{\sigma }}_{{{T}'}}}) \\		Var({T}')= & {{\left( \frac{\partial {T}'}{\partial A} \right)}^{2}}Var(\widehat{A})+{{\left( \frac{\partial {T}'}{\partial B} \right)}^{2}}Var(\widehat{B})+{{\left( \frac{\partial {T}'}{\partial {{\sigma }_{{{T}'}}}} \right)}^{2}}Var({{\widehat{\sigma }}_{{{T}'}}}) +2\left( \frac{\partial {T}'}{\partial A} \right)\left( \frac{\partial {T}'}{\partial B} \right)Cov\left( \widehat{A},\widehat{B} \right) \\
	~~& &~~ +2\left( \frac{\partial {T}'}{\partial A} \right)\left( \frac{\partial {T}'}{\partial B} \right)Cov\left( \widehat{A},\widehat{B} \right) \\		& +2\left( \frac{\partial {T}'}{\partial A} \right)\left( \frac{\partial {T}'}{\partial {{\sigma }_{{{T}'}}}} \right)Cov\left( \widehat{A},{{\widehat{\sigma }}_{{{T}'}}} \right) +2\left( \frac{\partial {T}'}{\partial B} \right)\left( \frac{\partial {T}'}{\partial {{\sigma }_{{{T}'}}}} \right)Cov\left( \widehat{B},{{\widehat{\sigma }}_{{{T}'}}} \right)
	& & +2\left( \frac{\partial {T}'}{\partial A} \right)\left( \frac{\partial {T}'}{\partial {{\sigma }_{{{T}'}}}} \right)Cov\left( \widehat{A},{{\widehat{\sigma }}_{{{T}'}}} \right) \\
	~~& &~~ +2\left( \frac{\partial {T}'}{\partial B} \right)\left( \frac{\partial {T}'}{\partial {{\sigma }_{{{T}'}}}} \right)Cov\left( \widehat{B},{{\widehat{\sigma }}_{{{T}'}}} \right)
	\end{align}</math>		\end{align}</math>

	<br>		<br>
	:or:		or:

	<br>		<br>
	::<math>\begin{align}		::<math>\begin{align}
	& Var({T}')= & Var(\widehat{A})+\frac{1}{V}Var(\widehat{B})+{{\widehat{z}}^{2}}Var({{\widehat{\sigma }}_{{{T}'}}}) \\		& Var({T}')= Var(\widehat{A})+\frac{1}{V}Var(\widehat{B})+{{\widehat{z}}^{2}}Var({{\widehat{\sigma }}_{{{T}'}}}) -\frac{2}{V}Cov\left( \widehat{A},\widehat{B} \right) -2\widehat{z}Cov\left( \widehat{A},{{\widehat{\sigma }}_{{{T}'}}} \right) +\frac{2\widehat{z}}{V}Cov\left( \widehat{B},{{\widehat{\sigma }}_{{{T}'}}} \right)
	~~& &~~ -\frac{2}{V}Cov\left( \widehat{A},\widehat{B} \right) \\
	~~& &~~ -2\widehat{z}Cov\left( \widehat{A},{{\widehat{\sigma }}_{{{T}'}}} \right) \\
	~~& &~~ +\frac{2\widehat{z}}{V}Cov\left( \widehat{B},{{\widehat{\sigma }}_{{{T}'}}} \right)
	\end{align}</math>		\end{align}</math>

Nicolette Young: Created page with '===Confidence Bounds on Time===
The bounds around time for a given lognormal percentile (unreliability) are estimated by first solving the reliability equation with respect…'

2012-01-12T23:46:55Z

Created page with '===Confidence Bounds on Time=== The bounds around time for a given lognormal percentile (unreliability) are estimated by first solving the reliability equation with respect…'

New page

===Confidence Bounds on Time===

 
The bounds around time for a given lognormal percentile (unreliability) are estimated by first solving the reliability equation with respect to time as follows:

 
::<math>{T}'(V;\widehat{A},\widehat{B},{{\widehat{\sigma }}_{{{T}'}}})=-\ln (V)-\widehat{A}+\frac{\widehat{B}}{V}+z\cdot {{\widehat{\sigma }}_{{{T}'}}}</math>

 
:where:

 
::<math>\begin{align}
& {T}'(V;\widehat{A},\widehat{B},{{\widehat{\sigma }}_{{{T}'}}})= & \ln (T) \\
& z= & {{\Phi }^{-1}}\left[ F({T}') \right]
\end{align}</math>

 
:and:

 
::<math>\Phi (z)=\frac{1}{\sqrt{2\pi }}\mathop{}_{-\infty }^{z({T}')}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz</math>

 
The next step is to calculate the variance of <math>{T}'(V;\widehat{A},\widehat{B},{{\widehat{\sigma }}_{{{T}'}}}):</math>

 
::<math>\begin{align}
& Var({T}')= & {{\left( \frac{\partial {T}'}{\partial A} \right)}^{2}}Var(\widehat{A})+{{\left( \frac{\partial {T}'}{\partial B} \right)}^{2}}Var(\widehat{B})+{{\left( \frac{\partial {T}'}{\partial {{\sigma }_{{{T}'}}}} \right)}^{2}}Var({{\widehat{\sigma }}_{{{T}'}}}) \\
& & +2\left( \frac{\partial {T}'}{\partial A} \right)\left( \frac{\partial {T}'}{\partial B} \right)Cov\left( \widehat{A},\widehat{B} \right) \\
& & +2\left( \frac{\partial {T}'}{\partial A} \right)\left( \frac{\partial {T}'}{\partial {{\sigma }_{{{T}'}}}} \right)Cov\left( \widehat{A},{{\widehat{\sigma }}_{{{T}'}}} \right) \\
& & +2\left( \frac{\partial {T}'}{\partial B} \right)\left( \frac{\partial {T}'}{\partial {{\sigma }_{{{T}'}}}} \right)Cov\left( \widehat{B},{{\widehat{\sigma }}_{{{T}'}}} \right)
\end{align}</math>

 
:or:

 
::<math>\begin{align}
& Var({T}')= & Var(\widehat{A})+\frac{1}{V}Var(\widehat{B})+{{\widehat{z}}^{2}}Var({{\widehat{\sigma }}_{{{T}'}}}) \\
& & -\frac{2}{V}Cov\left( \widehat{A},\widehat{B} \right) \\
& & -2\widehat{z}Cov\left( \widehat{A},{{\widehat{\sigma }}_{{{T}'}}} \right) \\
& & +\frac{2\widehat{z}}{V}Cov\left( \widehat{B},{{\widehat{\sigma }}_{{{T}'}}} \right)
\end{align}</math>

 
The upper and lower bounds are then found by:

 
::<math>\begin{align}
& T_{U}^{\prime }= & \ln {{T}_{U}}={T}'+{{K}_{\alpha }}\sqrt{Var({T}')} \\
& T_{L}^{\prime }= & \ln {{T}_{L}}={T}'-{{K}_{\alpha }}\sqrt{Var({T}')}
\end{align}</math>

 
Solving for <math>{{T}_{U}}</math> and <math>{{T}_{L}}</math> yields:

 
::<math>\begin{align}
& {{T}_{U}}= & {{e}^{T_{U}^{\prime }}}\text{ (Upper bound)} \\
& {{T}_{L}}= & {{e}^{T_{L}^{\prime }}}\text{ (Lower bound)}
\end{align}</math>

Template:Eyring-log cb on time - Revision history

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

Athanasios Gerokostopoulos: /* Confidence Bounds on Time */

Nicolette Young: Created page with '===Confidence Bounds on Time=== The bounds around time for a given lognormal percentile (unreliability) are estimated by first solving the reliability equation with respect…'

Nicolette Young: Created page with '===Confidence Bounds on Time===
The bounds around time for a given lognormal percentile (unreliability) are estimated by first solving the reliability equation with respect…'