Template:Bounds on time given cumulative mtbf camsaa-cb - Revision history

Richard House: Redirected page to Crow-AMSAA - NHPP#Bounds on Time Given Cumulative MTBF

2012-08-24T04:12:19Z

Redirected page to Crow-AMSAA - NHPP#Bounds on Time Given Cumulative MTBF

← Older revision		Revision as of 04:12, 24 August 2012
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	~~===Bounds on Time Given Cumulative MTBF===~~		#REDIRECT [[Crow-AMSAA_-_NHPP#Bounds_on_Time_Given_Cumulative_MTBF]]
	~~====Fisher Matrix Bounds====~~
	~~The time, <math>T</math> , must be positive, thus <math>\ln T</math> is treated as being normally distributed.~~
	~~::<math>\frac{\ln \hat{T}-\ln T}{\sqrt{Var(\ln \hat{T}})}\ \tilde{\ }\ N(0,1)</math>~~
	~~<br>~~
	~~Confidence bounds on the time are given by:~~
	~~<br>~~
	~~::<math>CB=\hat{T}{{e}^{\pm {{z}_{\alpha }}\sqrt{Var(\hat{T})}/\hat{T}}}</math>~~
	~~:where:~~
	~~::<math>\begin{align}~~
	~~& Var(\hat{T})= & {{\left( \frac{\partial T}{\partial \beta } \right)}^{2}}Var(\hat{\beta })+{{\left( \frac{\partial T}{\partial \lambda } \right)}^{2}}Var(\hat{\lambda }) \\~~
	~~& & +2\left( \frac{\partial T}{\partial \beta } \right)\left( \frac{\partial T}{\partial \lambda } \right)cov(\hat{\beta },\,\,\,\hat{\lambda })~~
	~~\end{align}</math>~~
	~~<br>~~
	~~The variance calculation is the same as Eqn. (variance1) and:~~
	~~<br>~~
	~~::<math>\hat{T}={{(\lambda \cdot {{m}_{c}})}^{1/(1-\beta )}}</math>~~

	~~::<math>\begin{align}~~
	~~& \frac{\partial T}{\partial \beta }= & \frac{{{(\lambda \cdot \,{{m}_{c}})}^{1/(1-\beta )}}\ln (\lambda \cdot \text{ }{{m}_{c}})}{{{(1~~-~~\beta )}^{2}}} \\~~
	~~& \frac{\partial T}{\partial \lambda }= & \frac{{{(\lambda \text{ }\cdot \text{ }{{m}_{c}})}^{1/(1~~-~~\beta )}}}{\lambda (1-\beta )}~~
	~~\end{align}</math>~~

	~~====Crow Bounds====~~
	~~:Step 1: Calculate <math>{{\lambda }_{c}}(T)=\tfrac{1}{MTB{{F}_{c}}}</math> .~~
	~~:Step 2: Use the equations from 5.2.8.2 to calculate the bounds on time given the cumulative failure intensity.~~

Nicolette Young: Created page with '===Bounds on Time Given Cumulative MTBF=== ====Fisher Matrix Bounds==== The time, $T$ , must be positive, thus $\ln T$ is treated as being normally distr…'

2012-01-05T23:01:52Z

Created page with '===Bounds on Time Given Cumulative MTBF=== ====Fisher Matrix Bounds==== The time, <math>T</math> , must be positive, thus <math>\ln T</math> is treated as being normally distr…'

New page

===Bounds on Time Given Cumulative MTBF===
====Fisher Matrix Bounds====
The time, <math>T</math> , must be positive, thus <math>\ln T</math> is treated as being normally distributed.
::<math>\frac{\ln \hat{T}-\ln T}{\sqrt{Var(\ln \hat{T}})}\ \tilde{\ }\ N(0,1)</math>
<br>
Confidence bounds on the time are given by:
<br>
::<math>CB=\hat{T}{{e}^{\pm {{z}_{\alpha }}\sqrt{Var(\hat{T})}/\hat{T}}}</math>
:where:
::<math>\begin{align}
& Var(\hat{T})= & {{\left( \frac{\partial T}{\partial \beta } \right)}^{2}}Var(\hat{\beta })+{{\left( \frac{\partial T}{\partial \lambda } \right)}^{2}}Var(\hat{\lambda }) \\
& & +2\left( \frac{\partial T}{\partial \beta } \right)\left( \frac{\partial T}{\partial \lambda } \right)cov(\hat{\beta },\,\,\,\hat{\lambda })
\end{align}</math>
<br>
The variance calculation is the same as Eqn. (variance1) and:
<br>
::<math>\hat{T}={{(\lambda \cdot {{m}_{c}})}^{1/(1-\beta )}}</math>

::<math>\begin{align}
& \frac{\partial T}{\partial \beta }= & \frac{{{(\lambda \cdot \,{{m}_{c}})}^{1/(1-\beta )}}\ln (\lambda \cdot \text{ }{{m}_{c}})}{{{(1-\beta )}^{2}}} \\
& \frac{\partial T}{\partial \lambda }= & \frac{{{(\lambda \text{ }\cdot \text{ }{{m}_{c}})}^{1/(1-\beta )}}}{\lambda (1-\beta )}
\end{align}</math>

====Crow Bounds====
:Step 1: Calculate <math>{{\lambda }_{c}}(T)=\tfrac{1}{MTB{{F}_{c}}}</math> .
:Step 2: Use the equations from 5.2.8.2 to calculate the bounds on time given the cumulative failure intensity.

Template:Bounds on time given cumulative mtbf camsaa-cb - Revision history

Richard House: Redirected page to Crow-AMSAA - NHPP#Bounds on Time Given Cumulative MTBF

Nicolette Young: Created page with '===Bounds on Time Given Cumulative MTBF=== ====Fisher Matrix Bounds==== The time, T , must be positive, thus \ln T is treated as being normally distr…'

Nicolette Young: Created page with '===Bounds on Time Given Cumulative MTBF=== ====Fisher Matrix Bounds==== The time, $T$ , must be positive, thus $\ln T$ is treated as being normally distr…'