Template:Acb-l conf bounds for weib2 - Revision history

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

2012-08-16T05:58:19Z

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

← Older revision		Revision as of 05:58, 16 August 2012
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	~~===Approximate Confidence Bounds for the Arrhenius~~-Weibull~~===~~		#REDIRECT [[Arrhenius_Relationship#Approximate_Confidence_Bounds_for_the_Arrhenius-Weibull]]
	~~<br>~~
	~~{{acb4weib on parameters}}~~

	~~{{acb4weib on reliability}}~~

	~~{{acb4weib on time}}~~

	~~{{appr conf bounds for arr-weibull}}~~

Athanasios Gerokostopoulos at 00:01, 14 February 2012

2012-02-14T00:01:55Z

← Older revision		Revision as of 00:01, 14 February 2012
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	===Approximate Confidence Bounds for the Arrhenius-Weibull:===		===Approximate Confidence Bounds for the Arrhenius-Weibull===
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	{{acb4weib on parameters}}		{{acb4weib on parameters}}

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

2012-02-14T00:01:45Z

Approximate Confidence Bounds for the Arrhenius-Weibull:

← Older revision		Revision as of 00:01, 14 February 2012
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	==Approximate Confidence Bounds for the Arrhenius-Weibull:==		===Approximate Confidence Bounds for the Arrhenius-Weibull:===
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Nicolette Young: /* Confidence Bounds on Time */

2012-01-12T22:26:51Z

Confidence Bounds on Time

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Nicolette Young: /* Confidence Bounds on Reliability */

2012-01-12T22:25:59Z

Confidence Bounds on Reliability

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Nicolette Young: /* Bounds on the Parameters */

2012-01-12T22:25:34Z

Bounds on the Parameters

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 ==Approximate Confidence Bounds for the Arrhenius-Weibull:==
 <br>
-===Bounds on the Parameters===
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 ===Confidence Bounds on Reliability===

Nicolette Young: Created page with '==Approximate Confidence Bounds for the Arrhenius-Weibull:==
===Bounds on the Parameters===
From the asymptotically normal property of the maximum likelihood estimators…'

2012-01-12T22:24:46Z

Created page with '==Approximate Confidence Bounds for the Arrhenius-Weibull:== ===Bounds on the Parameters=== From the asymptotically normal property of the maximum likelihood estimators…'

New page

==Approximate Confidence Bounds for the Arrhenius-Weibull:==
 
===Bounds on the Parameters===
 
From the asymptotically normal property of the maximum likelihood estimators, and since <math>\widehat{\beta },</math> and <math>\widehat{C}</math> are positive parameters, <math>\ln (\widehat{\beta }),</math> and <math>\ln (\widehat{C})</math> can then be treated as normally distributed. After performing this transformation, the bounds on the parameters can be estimated from:

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

 
also:

 
::<math>\begin{align}
& {{B}_{U}}= & \widehat{B}+{{K}_{\alpha }}\sqrt{Var(\widehat{B})} \\
& {{B}_{L}}= & \widehat{B}-{{K}_{\alpha }}\sqrt{Var(\widehat{B})}
\end{align}</math>

 
and:

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

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

 
::<math>\left[ \begin{matrix}
Var(\widehat{\beta }) & Cov(\widehat{\beta },\widehat{B}) & Cov(\widehat{\beta },\widehat{C}) \\
Cov(\widehat{B},\widehat{\beta }) & Var(\widehat{B}) & Cov(\widehat{B},\widehat{C}) \\
Cov(\widehat{C},\widehat{\beta }) & Cov(\widehat{C},\widehat{B}) & Var(\widehat{C}) \\
\end{matrix} \right]={{\left[ \begin{matrix}
-\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\beta }^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial \beta \partial B} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial \beta \partial C} \\
-\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial \beta } & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{B}^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial C} \\
-\tfrac{{{\partial }^{2}}\Lambda }{\partial C\partial \beta } & -\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 reliability function for the Arrhenius-Weibull model (ML estimate) is given by:

 
::<math>\widehat{R}(T,V)={{e}^{-{{\left( \tfrac{T}{\widehat{C}\cdot {{e}^{\tfrac{\widehat{B}}{V}}}} \right)}^{\widehat{\beta }}}}}</math>

 
or:

 
::<math>\widehat{R}(T)={{e}^{-{{e}^{\ln \left[ {{\left( \tfrac{T}{\widehat{C}\cdot {{e}^{\tfrac{\widehat{B}}{V}}}} \right)}^{\widehat{\beta }}} \right]}}}}</math>

 
Setting:

 
::<math>\widehat{u}=\ln \left[ {{\left( \frac{T}{\widehat{C}\cdot {{e}^{\tfrac{\widehat{B}}{V}}}} \right)}^{\widehat{\beta }}} \right]</math>

 
or:

 
::<math>\widehat{u}=\widehat{\beta }\left[ \ln (T)-\ln (\widehat{C})-\frac{\widehat{B}}{V} \right]</math>

 
The reliability function now becomes:

 
::<math>\widehat{R}(T,V)={{e}^{-{{e}^{\widehat{u}}}}}</math>

 
The next step is to find the upper and lower bounds on <math>\widehat{u}\ \ :</math>

 
::<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})= & {{\left( \frac{\widehat{u}}{\widehat{\beta }} \right)}^{2}}Var(\widehat{\beta })+{{\left( \frac{\widehat{\beta }}{V} \right)}^{2}}Var(\widehat{B})+{{\left( \frac{\widehat{\beta }}{\widehat{C}} \right)}^{2}}Var(\widehat{C}) \\
& & -\frac{2\widehat{u}}{V}Cov(\widehat{\beta },\widehat{B})-\frac{2\widehat{u}}{\widehat{C}}Cov(\widehat{\beta },\widehat{C})+\frac{2{{\widehat{\beta }}^{2}}}{V\widehat{C}}Cov(\widehat{B},\widehat{C})
\end{align}</math>

 
The upper and lower bounds on reliability are:

 
::<math>\begin{align}
& {{R}_{U}}(T,V)= & {{e}^{-{{e}^{\left( {{u}_{L}} \right)}}}} \\
& {{R}_{L}}(T,V)= & {{e}^{-{{e}^{\left( {{u}_{U}} \right)}}}}
\end{align}</math>

where <math>{{u}_{U}}</math> and <math>{{u}_{L}}</math> are estimated from Eqns. (ArreibRupper) and (ArreibRlower).
 

===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).

{{appr conf bounds for arr-weibull}}