Template:Acb on the mean life - Revision history

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

2012-08-16T05:48:50Z

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

← Older revision		Revision as of 05:48, 16 August 2012
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	~~====Confidence Bounds on the Mean Life====~~		#REDIRECT [[Arrhenius_Relationship#Approximate_Confidence_Bounds_for_the_Arrhenius-Exponential]]
	~~<br>~~
	The Arrhenius-exponential distribution is given by setting <math>m=L(V)</math> in the exponential <math>pdf</math> equation. The upper <math>({{m}_{U}})</math> and lower <math>({{m}_{L}})</math> bounds on the mean life are then estimated by:

	~~<br>~~
	~~::<math>\begin{align}~~
	~~& {{m}_{U}}= \widehat{m}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{m})}}{\widehat{m}}}} \\~~
	~~& {{m}_{L}}= \widehat{m}\cdot {{e}^{-\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{m})}}{\widehat{m}}}}~~
	~~\end{align}</math>~~

	~~<br>~~
	~~where <math>{{K}_{\alpha }}</math> is defined by:~~

	~~<br>~~
	~~::<math>\alpha =\frac{1}{\sqrt{2\pi }}\int_{{{K}_{\alpha }}}^{\infty }{{e}^{-\tfrac{{{t}^{2}}}{2}}}dt=1-\Phi ({{K}_{\alpha }})</math>~~

	~~<br>~~
	If <math>\delta </math> is the confidence level (i.e., 95%=0.95), then <math>\alpha =\tfrac{1-\delta }{2}</math> for the two-sided bounds, and <math>\alpha =1-\delta </math> for the one-sided bounds. The variance of <math>\widehat{m}</math> is given by:

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


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

	~~<br>~~
	~~::<math>Var(\widehat{m})={{e}^{\tfrac{2\widehat{B}}{V}}}\left~~[ ~~Var(\widehat{C})+\frac{{{\widehat{C}}^{2}}}{{{V}^{2}}}Var(\widehat{B})+\frac{2\widehat{C}}{V}Cov(\widehat{B},\widehat{C}) \right]</math>~~

	~~<br>~~
	The variances and covariance of <math>B</math> and <math>C</math> are estimated from the local Fisher matrix (evaluated at <math>\widehat{B}</math> , <math>\widehat{C})</math> as follows:

	~~<br>~~
	~~::<math>\left~~[ ~~\begin{matrix}~~
	~~Var(\widehat{B}) & Cov(\widehat{B},\widehat{C}) \\~~
	~~Cov(\widehat{C},\widehat{B}) & Var(\widehat{C}) \\~~
	~~\end{matrix} \right~~]~~={{\left[ \begin{matrix}~~
	~~-\tfrac{{{\partial }^{2}}\Lambda }{\partial {{B}^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial C} \\~~
	~~-\tfrac{{{\partial }^{2}}\Lambda }{\partial C\partial B} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{C}^{2}}} \\~~
	~~\end{matrix} \right~~]~~}^{-1}}</math>~~

Athanasios Gerokostopoulos at 00:57, 14 February 2012

2012-02-14T00:57:24Z

← Older revision		Revision as of 00:57, 14 February 2012
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	<br>		<br>
	::<math>\begin{align}		::<math>\begin{align}
	& Var(\widehat{m})= & {{\left( \frac{\partial m}{\partial C} \right)}^{2}}Var(\widehat{C})+{{\left( \frac{\partial m}{\partial B} \right)}^{2}}Var(\widehat{B}) \\		& Var(\widehat{m})= & {{\left( \frac{\partial m}{\partial C} \right)}^{2}}Var(\widehat{C})+{{\left( \frac{\partial m}{\partial B} \right)}^{2}}Var(\widehat{B}) +2\left( \frac{\partial m}{\partial C} \right)\left( \frac{\partial m}{\partial B} \right)Cov(\widehat{B},\widehat{C})
	~~& &~~ +2\left( \frac{\partial m}{\partial C} \right)\left( \frac{\partial m}{\partial B} \right)Cov(\widehat{B},\widehat{C})
	\end{align}</math>		\end{align}</math>

Athanasios Gerokostopoulos: /* Confidence Bounds on the Mean Life */

2012-02-14T00:56:27Z

Confidence Bounds on the Mean Life

← Older revision		Revision as of 00:56, 14 February 2012
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	====Confidence Bounds on the Mean Life====		====Confidence Bounds on the Mean Life====
	<br>		<br>
	The Arrhenius-exponential distribution is given ~~by Eqn. (arrhenius)~~ by setting <math>m=L(V)</math> ~~as shown~~ in ~~Eqn. (Arrean)~~. The upper <math>({{m}_{U}})</math> and lower <math>({{m}_{L}})</math> bounds on the mean life are then estimated by:		The Arrhenius-exponential distribution is given by setting <math>m=L(V)</math> in the exponential <math>pdf</math> equation. The upper <math>({{m}_{U}})</math> and lower <math>({{m}_{L}})</math> bounds on the mean life are then estimated by:

	<br>		<br>
	::<math>\begin{align}		::<math>\begin{align}
	& {{m}_{U}}= & \widehat{m}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{m})}}{\widehat{m}}}} \\		& {{m}_{U}}= \widehat{m}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{m})}}{\widehat{m}}}} \\
	& {{m}_{L}}= & \widehat{m}\cdot {{e}^{-\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{m})}}{\widehat{m}}}}		& {{m}_{L}}= \widehat{m}\cdot {{e}^{-\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{m})}}{\widehat{m}}}}
	\end{align}</math>		\end{align}</math>

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	<br>		<br>
	::<math>\alpha =\frac{1}{\sqrt{2\pi }}\~~mathop{}_~~{{{K}_{\alpha }}}^{\infty }{{e}^{-\tfrac{{{t}^{2}}}{2}}}dt=1-\Phi ({{K}_{\alpha }})</math>		::<math>\alpha =\frac{1}{\sqrt{2\pi }}\int_{{{K}_{\alpha }}}^{\infty }{{e}^{-\tfrac{{{t}^{2}}}{2}}}dt=1-\Phi ({{K}_{\alpha }})</math>

	<br>		<br>

Athanasios Gerokostopoulos: /* Confidence Bounds on the Mean Life */

2012-02-14T00:00:18Z

Confidence Bounds on the Mean Life

← Older revision		Revision as of 00:00, 14 February 2012
Line 1:		Line 1:
	===Confidence Bounds on the Mean Life===		====Confidence Bounds on the Mean Life====
	<br>		<br>
	The Arrhenius-exponential distribution is given by Eqn. (arrhenius) by setting <math>m=L(V)</math> as shown in Eqn. (Arrean). The upper <math>({{m}_{U}})</math> and lower <math>({{m}_{L}})</math> bounds on the mean life are then estimated by:		The Arrhenius-exponential distribution is given by Eqn. (arrhenius) by setting <math>m=L(V)</math> as shown in Eqn. (Arrean). The upper <math>({{m}_{U}})</math> and lower <math>({{m}_{L}})</math> bounds on the mean life are then estimated by:

Nicolette Young: Created page with '===Confidence Bounds on the Mean Life===
The Arrhenius-exponential distribution is given by Eqn. (arrhenius) by setting $m=L(V)$ as shown in Eqn. (Arrean). The …'

2012-01-12T22:17:07Z

Created page with '===Confidence Bounds on the Mean Life=== The Arrhenius-exponential distribution is given by Eqn. (arrhenius) by setting <math>m=L(V)</math> as shown in Eqn. (Arrean). The …'

New page

===Confidence Bounds on the Mean Life===
 
The Arrhenius-exponential distribution is given by Eqn. (arrhenius) by setting <math>m=L(V)</math> as shown in Eqn. (Arrean). The upper <math>({{m}_{U}})</math> and lower <math>({{m}_{L}})</math> bounds on the mean life are then estimated by:

 
::<math>\begin{align}
& {{m}_{U}}= & \widehat{m}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{m})}}{\widehat{m}}}} \\
& {{m}_{L}}= & \widehat{m}\cdot {{e}^{-\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{m})}}{\widehat{m}}}}
\end{align}</math>

 
where <math>{{K}_{\alpha }}</math> is defined by:

 
::<math>\alpha =\frac{1}{\sqrt{2\pi }}\mathop{}_{{{K}_{\alpha }}}^{\infty }{{e}^{-\tfrac{{{t}^{2}}}{2}}}dt=1-\Phi ({{K}_{\alpha }})</math>

 
If <math>\delta </math> is the confidence level (i.e., 95%=0.95), then <math>\alpha =\tfrac{1-\delta }{2}</math> for the two-sided bounds, and <math>\alpha =1-\delta </math> for the one-sided bounds. The variance of <math>\widehat{m}</math> is given by:

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

 
or:

 
::<math>Var(\widehat{m})={{e}^{\tfrac{2\widehat{B}}{V}}}\left[ Var(\widehat{C})+\frac{{{\widehat{C}}^{2}}}{{{V}^{2}}}Var(\widehat{B})+\frac{2\widehat{C}}{V}Cov(\widehat{B},\widehat{C}) \right]</math>

 
The variances and covariance of <math>B</math> and <math>C</math> are estimated from the local Fisher matrix (evaluated at <math>\widehat{B}</math> , <math>\widehat{C})</math> as follows:

 
::<math>\left[ \begin{matrix}
Var(\widehat{B}) & Cov(\widehat{B},\widehat{C}) \\
Cov(\widehat{C},\widehat{B}) & Var(\widehat{C}) \\
\end{matrix} \right]={{\left[ \begin{matrix}
-\tfrac{{{\partial }^{2}}\Lambda }{\partial {{B}^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial C} \\
-\tfrac{{{\partial }^{2}}\Lambda }{\partial C\partial B} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{C}^{2}}} \\
\end{matrix} \right]}^{-1}}</math>

Template:Acb on the mean life - Revision history

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

Athanasios Gerokostopoulos at 00:57, 14 February 2012

Athanasios Gerokostopoulos: /* Confidence Bounds on the Mean Life */

Athanasios Gerokostopoulos: /* Confidence Bounds on the Mean Life */

Nicolette Young: Created page with '===Confidence Bounds on the Mean Life=== The Arrhenius-exponential distribution is given by Eqn. (arrhenius) by setting m=L(V) as shown in Eqn. (Arrean). The …'

Nicolette Young: Created page with '===Confidence Bounds on the Mean Life===
The Arrhenius-exponential distribution is given by Eqn. (arrhenius) by setting $m=L(V)$ as shown in Eqn. (Arrean). The …'