Template:Appr conf bounds for arr-exp - Revision history

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

2012-08-16T05:47:40Z

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

← Older revision		Revision as of 05:47, 16 August 2012
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	~~===Approximate Confidence Bounds for the Arrhenius~~-Exponential~~===~~		#REDIRECT [[Arrhenius_Relationship#Approximate_Confidence_Bounds_for_the_Arrhenius-Exponential]]
	~~<br>~~
	There are different methods for computing confidence bounds. ALTA utilizes confidence bounds that are based on the asymptotic theory for maximum likelihood estimates, most commonly referred to as the Fisher matrix bounds.
	~~<br>~~
	~~{{acb on the mean life}}~~

	~~{{acb on reliability}}~~

	~~{{acb on time}}~~

Athanasios Gerokostopoulos: /* Approximate Confidence Bounds for the Arrhenius-Exponential */

2012-02-13T23:59:55Z

Approximate Confidence Bounds for the Arrhenius-Exponential

← Older revision		Revision as of 23:59, 13 February 2012
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	==Approximate Confidence Bounds for the Arrhenius-Exponential==		===Approximate Confidence Bounds for the Arrhenius-Exponential===
	<br>		<br>
	There are different methods for computing confidence bounds. ALTA utilizes confidence bounds that are based on the asymptotic theory for maximum likelihood estimates, most commonly referred to as the Fisher matrix bounds.		There are different methods for computing confidence bounds. ALTA utilizes confidence bounds that are based on the asymptotic theory for maximum likelihood estimates, most commonly referred to as the Fisher matrix bounds.

Nicolette Young: /* Confidence Bounds on Time */

2012-01-12T22:18:09Z

Confidence Bounds on Time

← Older revision		Revision as of 22:18, 12 January 2012
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	{{acb on reliability}}		{{acb on reliability}}

	~~===Confidence Bounds on Time===~~		{{acb on time}}
	~~<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>\widehat{T}=-\widehat{m}\cdot \ln (R)</math>~~

	~~<br>~~
	~~The corresponding confidence bounds are then estimated from:~~

	~~<br>~~
	~~::<math>\begin{align}~~
	~~& {{T}_{U}}= & -{{m}_{U}}\cdot \ln (R) \\~~
	~~& {{T}_{L}}= & -{{m}_{L}}\cdot \ln (R)~~
	~~\end{align}</math>~~


	~~where <math>{{m}_{U}}</math> and <math>{{m}_{L~~}}~~</math> are estimated using Eqns. (ArrhuUpper) and (ArrhuLower).~~

Nicolette Young: /* Confidence Bounds on Reliability */

2012-01-12T22:17:36Z

Confidence Bounds on Reliability

← Older revision		Revision as of 22:17, 12 January 2012
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	{{acb on the mean life}}		{{acb on the mean life}}

	~~===Confidence Bounds on Reliability===~~		{{acb on reliability}}
	~~<br>~~
	~~The bounds~~ on reliability ~~for any given time, <math>T</math> , are estimated by:~~
	~~<br>~~
	~~::<math>\begin{align}~~
	~~& {{R}_{U}}(T)= & {{e}^{-\tfrac{T}{{{m}_{U}}}}} \\~~
	~~& {{R}_{L}}(T)= & {{e}^{-\tfrac{T}{{{m}_{L}}}}}~~
	~~\end{align}</math>~~

	~~<br>~~
	~~where <math>{{m}_{U}}</math> and <math>{{m}_{L~~}}~~</math> are estimated using Eqns. (ArrhuUpper) and (ArrhuLower).~~
	~~<br>~~

	===Confidence Bounds on Time===		===Confidence Bounds on Time===

Nicolette Young: /* Confidence Bounds on the Mean Life */

2012-01-12T22:17:01Z

Confidence Bounds on the Mean Life

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 There are different methods for computing confidence bounds. ALTA utilizes confidence bounds that are based on the asymptotic theory for maximum likelihood estimates, most commonly referred to as the Fisher matrix bounds.
 <br>
-===Confidence Bounds on the Mean Life===
+{{acb on the mean life}}
 ===Confidence Bounds on Reliability===

Nicolette Young: Created page with '==Approximate Confidence Bounds for the Arrhenius-Exponential==
There are different methods for computing confidence bounds. ALTA utilizes confidence bounds that are based o…'

2012-01-12T22:16:20Z

Created page with '==Approximate Confidence Bounds for the Arrhenius-Exponential== There are different methods for computing confidence bounds. ALTA utilizes confidence bounds that are based o…'

New page

==Approximate Confidence Bounds for the Arrhenius-Exponential==
 
There are different methods for computing confidence bounds. ALTA utilizes confidence bounds that are based on the asymptotic theory for maximum likelihood estimates, most commonly referred to as the Fisher matrix bounds.
 
===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>

===Confidence Bounds on Reliability===
 
The bounds on reliability for any given time, <math>T</math> , are estimated by:
 
::<math>\begin{align}
& {{R}_{U}}(T)= & {{e}^{-\tfrac{T}{{{m}_{U}}}}} \\
& {{R}_{L}}(T)= & {{e}^{-\tfrac{T}{{{m}_{L}}}}}
\end{align}</math>

 
where <math>{{m}_{U}}</math> and <math>{{m}_{L}}</math> are estimated using Eqns. (ArrhuUpper) and (ArrhuLower).
 

===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>\widehat{T}=-\widehat{m}\cdot \ln (R)</math>

 
The corresponding confidence bounds are then estimated from:

 
::<math>\begin{align}
& {{T}_{U}}= & -{{m}_{U}}\cdot \ln (R) \\
& {{T}_{L}}= & -{{m}_{L}}\cdot \ln (R)
\end{align}</math>

where <math>{{m}_{U}}</math> and <math>{{m}_{L}}</math> are estimated using Eqns. (ArrhuUpper) and (ArrhuLower).