Template:Bounds on demonstrated failure intensity rga - Revision history

Lisa Hacker: Redirected page to Crow Extended Confidence Bounds

2012-08-25T05:54:07Z

Redirected page to Crow Extended Confidence Bounds

← Older revision		Revision as of 05:54, 25 August 2012
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	~~===Bounds on Demonstrated Failure Intensity===~~		#REDIRECT [[Crow Extended Confidence Bounds]]
	~~====Fisher Matrix Bounds====~~
	~~If there are no BC failure modes, the demonstrated failure intensity is <math>{{\widehat{\lambda }}_{D}}(T)=\tfrac{{{N}_{A}}+{{N}_{BD}}}{T}</math> . Thus:~~


	~~::<math>Var({{\hat{\lambda }}_{D}}(t))=\frac{{{N}_{A}}}{{{T}^{2}}}+\frac{{{N}_{BD}}}{{{T}^{2}}}=\frac{{{\lambda }_{D}}(t)}{T}</math>~~

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

	~~::<math>\sqrt{T}\left( \frac{{{{\hat{\lambda }}}_{D}}(T)-{{\lambda }_{D}}(T)}{\sqrt{{{\lambda }_{D}}(T)}} \right)\sim N(0,1)</math>~~


	~~::<math>{{\lambda }_{D}}(T)={{\hat{\lambda }}_{D}}(T)+\frac{{{C}^{2}}}{2}\pm \sqrt{{{{\hat{\lambda }}}_{D}}(T){{C}^{2}}+\frac{{{C}^{4}}}{4}}</math>~~

	~~where <math>C=\tfrac{{{z}_{1-\alpha /2}}}{\sqrt{T}}</math> .~~
	~~<br>~~
	If there are BC failure modes, the demonstrated failure intensity, <math>{{\widehat{\lambda }}_{D}}(T)={{\widehat{\lambda }}_{CA}}</math> , is actually the instantaneous failure intensity based on all of the data. <math>{{\lambda }_{CA}}(T)</math> must be positive, thus <math>\ln {{\lambda }_{CA}}(T)</math> is approximately treated as being normally distributed.

	~~::<math>\frac{\ln {{{\hat{\lambda }}}_{CA}}(T)-\ln {{\lambda }_{CA}}(T)}{\sqrt{Var(\ln {{{\hat{\lambda }}}_{CA}}(T)})}\sim N(0,1)</math>~~

	~~The approximate confidence bounds on the instantaneous failure intensity are then estimated from:~~

	~~::<math>CB={{\hat{\lambda }}_{CA}}(T){{e}^{\pm {{z}_{\alpha }}\sqrt{Var({{{\hat{\lambda }}}_{CA}}(T))}/{{{\hat{\lambda }}}_{i}}(T)}}</math>~~

	~~where <math>{{\lambda }_{CA}}(t)=\lambda \beta {{T}^{\beta -1}}</math> .~~

	~~::<math>\begin{align}~~
	& Var({{{\hat{\lambda }}}_{CA}}(T))= & {{\left( \frac{\partial {{\lambda }_{CA}}(T)}{\partial \beta } \right)}^{2}}Var(\hat{\beta })+{{\left( \frac{\partial {{\lambda }_{CA}}(T)}{\partial \lambda } \right)}^{2}}Var(\hat{\lambda }) \\
	~~& & +2\left( \frac{\partial {{\lambda }_{CA}}(T)}{\partial \beta } \right)\left( \frac{\partial {{\lambda }_{CA}}(T)}{\partial \lambda } \right)cov(\hat{\beta },\hat{\lambda })~~
	~~\end{align}</math>~~

	~~The variance calculation is the same as described in Chapter 5.~~
	~~<br>~~
	~~====~~Crow Bounds~~====~~
	~~<br>~~
	~~If there are no BC failure modes then:~~


	~~::<math>\begin{align}~~
	~~& {{[{{\lambda }_{D}}(T)~~]~~}_{l}}= & {{\widehat{\lambda }}_{D}}(T)\frac{\chi _{(2N,1-\alpha /2)}^{2}}{2N} \\~~
	~~& {{[{{\lambda }_{D}}(T)~~]~~}_{u}}= & {{\widehat{\lambda }}_{D}}(T)\frac{\chi _{(2N,\alpha /2)}^{2}}{2N}~~
	~~\end{align}</math>~~


	~~where <math>{{\widehat{\lambda }}_{D}}(T)={{\widehat{\lambda }}_{CA}}</math> .~~
	~~<br>~~
	~~If there are BC modes then the confidence bounds on the demonstrated failure intensity are calculated as presented in Chapter 5.~~
	~~<br>~~

Nicolette Young: Created page with '===Bounds on Demonstrated Failure Intensity=== ====Fisher Matrix Bounds==== If there are no BC failure modes, the demonstrated failure intensity is ${{\widehat{\lambda }}_{…'$

2012-01-10T20:49:05Z

Created page with '===Bounds on Demonstrated Failure Intensity=== ====Fisher Matrix Bounds==== If there are no BC failure modes, the demonstrated failure intensity is <math>{{\widehat{\lambda }}_{…'

New page

===Bounds on Demonstrated Failure Intensity===
====Fisher Matrix Bounds====
If there are no BC failure modes, the demonstrated failure intensity is <math>{{\widehat{\lambda }}_{D}}(T)=\tfrac{{{N}_{A}}+{{N}_{BD}}}{T}</math> . Thus:

::<math>Var({{\hat{\lambda }}_{D}}(t))=\frac{{{N}_{A}}}{{{T}^{2}}}+\frac{{{N}_{BD}}}{{{T}^{2}}}=\frac{{{\lambda }_{D}}(t)}{T}</math>

 
:and:

::<math>\sqrt{T}\left( \frac{{{{\hat{\lambda }}}_{D}}(T)-{{\lambda }_{D}}(T)}{\sqrt{{{\lambda }_{D}}(T)}} \right)\sim N(0,1)</math>

::<math>{{\lambda }_{D}}(T)={{\hat{\lambda }}_{D}}(T)+\frac{{{C}^{2}}}{2}\pm \sqrt{{{{\hat{\lambda }}}_{D}}(T){{C}^{2}}+\frac{{{C}^{4}}}{4}}</math>

where <math>C=\tfrac{{{z}_{1-\alpha /2}}}{\sqrt{T}}</math> .
 
If there are BC failure modes, the demonstrated failure intensity, <math>{{\widehat{\lambda }}_{D}}(T)={{\widehat{\lambda }}_{CA}}</math> , is actually the instantaneous failure intensity based on all of the data. <math>{{\lambda }_{CA}}(T)</math> must be positive, thus <math>\ln {{\lambda }_{CA}}(T)</math> is approximately treated as being normally distributed.

::<math>\frac{\ln {{{\hat{\lambda }}}_{CA}}(T)-\ln {{\lambda }_{CA}}(T)}{\sqrt{Var(\ln {{{\hat{\lambda }}}_{CA}}(T)})}\sim N(0,1)</math>

The approximate confidence bounds on the instantaneous failure intensity are then estimated from:

::<math>CB={{\hat{\lambda }}_{CA}}(T){{e}^{\pm {{z}_{\alpha }}\sqrt{Var({{{\hat{\lambda }}}_{CA}}(T))}/{{{\hat{\lambda }}}_{i}}(T)}}</math>

where <math>{{\lambda }_{CA}}(t)=\lambda \beta {{T}^{\beta -1}}</math> .

::<math>\begin{align}
& Var({{{\hat{\lambda }}}_{CA}}(T))= & {{\left( \frac{\partial {{\lambda }_{CA}}(T)}{\partial \beta } \right)}^{2}}Var(\hat{\beta })+{{\left( \frac{\partial {{\lambda }_{CA}}(T)}{\partial \lambda } \right)}^{2}}Var(\hat{\lambda }) \\
& & +2\left( \frac{\partial {{\lambda }_{CA}}(T)}{\partial \beta } \right)\left( \frac{\partial {{\lambda }_{CA}}(T)}{\partial \lambda } \right)cov(\hat{\beta },\hat{\lambda })
\end{align}</math>

The variance calculation is the same as described in Chapter 5.
 
====Crow Bounds====
 
If there are no BC failure modes then:

::<math>\begin{align}
& {{[{{\lambda }_{D}}(T)]}_{l}}= & {{\widehat{\lambda }}_{D}}(T)\frac{\chi _{(2N,1-\alpha /2)}^{2}}{2N} \\
& {{[{{\lambda }_{D}}(T)]}_{u}}= & {{\widehat{\lambda }}_{D}}(T)\frac{\chi _{(2N,\alpha /2)}^{2}}{2N}
\end{align}</math>

where <math>{{\widehat{\lambda }}_{D}}(T)={{\widehat{\lambda }}_{CA}}</math> .
 
If there are BC modes then the confidence bounds on the demonstrated failure intensity are calculated as presented in Chapter 5.

Template:Bounds on demonstrated failure intensity rga - Revision history

Lisa Hacker: Redirected page to Crow Extended Confidence Bounds

Nicolette Young: Created page with '===Bounds on Demonstrated Failure Intensity=== ====Fisher Matrix Bounds==== If there are no BC failure modes, the demonstrated failure intensity is {{\widehat{\lambda }}_{…'

Nicolette Young: Created page with '===Bounds on Demonstrated Failure Intensity=== ====Fisher Matrix Bounds==== If there are no BC failure modes, the demonstrated failure intensity is ${{\widehat{\lambda }}_{…'$