Template:Bounds on projected failure intensity rga - Revision history

Lisa Hacker: Redirected page to Crow Extended Confidence Bounds

2012-08-25T05:42:39Z

Redirected page to Crow Extended Confidence Bounds

← Older revision		Revision as of 05:42, 25 August 2012
Line 1:		Line 1:
	~~===Bounds on Projected Failure Intensity===~~		#REDIRECT [[Crow Extended Confidence Bounds]]
	~~====Fisher Matrix Bounds====~~
	~~The projected failure intensity <math>{{\lambda }_{P}}(T)</math> must be positive, thus <math>\ln {{\lambda }_{P}}(T)</math> is approximately treated as being normally distributed as well:~~

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


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

	~~<br>~~
	~~where:~~
	~~<br>~~
	:• <math>{{\hat{\lambda }}_{P}}(T)=\tfrac{{{N}_{A}}}{T}+\underset{i=1}{\overset{M}{\mathop{\sum }}}\,(1-{{d}_{i}})\tfrac{{{N}_{i}}}{T}+\overline{d}\tfrac{M}{T}\bar{\beta }</math> when there are no BC modes.
	:• <math>{{\hat{\lambda }}_{P}}(T)={{\widehat{\lambda }}_{EM}}={{\widehat{\lambda }}_{CA}}-{{\widehat{\lambda }}_{BD}}+\underset{i=1}{\overset{M}{\mathop{\sum }}}\,(1-{{d}_{i}})\tfrac{{{N}_{i}}}{T}+\overline{d}\widehat{h}(T\|BD)</math> when there are BC modes.
	~~:• <math>{{N}_{i}}</math> is the total failure number of the <math>{{i}^{th}}</math> distinct BD mode.~~
	~~<br>~~
	~~You can then get:~~

	~~::<math>Var({{\lambda }_{P}}(T))\approx Var({{\hat{\gamma }}_{GP}})+\mu _{d}^{2}Var(h(T))\approx \frac{{{{\hat{r}}}_{GP}}}{T}+\mu _{d}^{2}Var(h(T))</math>~~

	~~:where:~~

	~~::<math>\begin{align}~~
	~~& \hat{h}(T)= & \frac{M}{T}\bar{\beta } \\~~
	~~& Var(\hat{h}(T))= & {{(\frac{M}{T})}^{2}}Var(\bar{\beta })={{(\frac{M}{T})}^{2}}{{(\frac{M}{M-1})}^{2}}Var(\hat{\beta })=\frac{{{M}^{4}}}{{{T}^{2}}{{(M-1)}^{2}}}Var(\hat{\beta })~~
	~~\end{align}</math>~~

	~~The <math>Var(\hat{\beta })</math> can be obtained from Fisher Matrix based on <math>M</math> distinct BD modes.~~
	~~<br>~~

	~~====~~Crow Bounds~~====~~
	~~::<math>\begin{align}~~
	~~& {{[{{\lambda }_{P}}(T)~~]~~}_{L}}= & {{{\hat{\lambda }}}_{P}}(T)+\frac{{{C}^{2}}}{2}-\sqrt{{{{\hat{\lambda }}}_{P}}(T)\cdot {{C}^{2}}+\frac{{{C}^{4}}}{4}} \\~~
	~~& {{[{{\lambda }_{P}}(T)~~]~~}_{U}}= & {{{\hat{\lambda }}}_{P}}(T)+\frac{{{C}^{2}}}{2}+\sqrt{{{{\hat{\lambda }}}_{P}}(T)\cdot \ \,{{C}^{2}}+\frac{{{C}^{4}}}{4}}~~
	~~\end{align}</math>~~


	~~where <math>C=\tfrac{{{z}_{1-\alpha /2}}}{\sqrt{T}}</math> .~~
	~~<br>~~

Lisa Hacker: Reverted edits by Lisa Hacker (Talk) to last revision by Nicolette Young

2012-08-25T05:40:38Z

Reverted edits by Lisa Hacker (Talk) to last revision by Nicolette Young

← Older revision		Revision as of 05:40, 25 August 2012
Line 1:		Line 1:
	~~#REDIRECT [[~~Crow ~~Extended Confidence~~ Bounds]]		===Bounds on Projected Failure Intensity===
			====Fisher Matrix Bounds====
			The projected failure intensity <math>{{\lambda }_{P}}(T)</math> must be positive, thus <math>\ln {{\lambda }_{P}}(T)</math> is approximately treated as being normally distributed as well:

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


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

			<br>
			where:
			<br>
			:• <math>{{\hat{\lambda }}_{P}}(T)=\tfrac{{{N}_{A}}}{T}+\underset{i=1}{\overset{M}{\mathop{\sum }}}\,(1-{{d}_{i}})\tfrac{{{N}_{i}}}{T}+\overline{d}\tfrac{M}{T}\bar{\beta }</math> when there are no BC modes.
			:• <math>{{\hat{\lambda }}_{P}}(T)={{\widehat{\lambda }}_{EM}}={{\widehat{\lambda }}_{CA}}-{{\widehat{\lambda }}_{BD}}+\underset{i=1}{\overset{M}{\mathop{\sum }}}\,(1-{{d}_{i}})\tfrac{{{N}_{i}}}{T}+\overline{d}\widehat{h}(T\|BD)</math> when there are BC modes.
			:• <math>{{N}_{i}}</math> is the total failure number of the <math>{{i}^{th}}</math> distinct BD mode.
			<br>
			You can then get:

			::<math>Var({{\lambda }_{P}}(T))\approx Var({{\hat{\gamma }}_{GP}})+\mu _{d}^{2}Var(h(T))\approx \frac{{{{\hat{r}}}_{GP}}}{T}+\mu _{d}^{2}Var(h(T))</math>

			:where:

			::<math>\begin{align}
			& \hat{h}(T)= & \frac{M}{T}\bar{\beta } \\
			& Var(\hat{h}(T))= & {{(\frac{M}{T})}^{2}}Var(\bar{\beta })={{(\frac{M}{T})}^{2}}{{(\frac{M}{M-1})}^{2}}Var(\hat{\beta })=\frac{{{M}^{4}}}{{{T}^{2}}{{(M-1)}^{2}}}Var(\hat{\beta })
			\end{align}</math>

			The <math>Var(\hat{\beta })</math> can be obtained from Fisher Matrix based on <math>M</math> distinct BD modes.
			<br>

			====Crow Bounds====
			::<math>\begin{align}
			& {{[{{\lambda }_{P}}(T)]}_{L}}= & {{{\hat{\lambda }}}_{P}}(T)+\frac{{{C}^{2}}}{2}-\sqrt{{{{\hat{\lambda }}}_{P}}(T)\cdot {{C}^{2}}+\frac{{{C}^{4}}}{4}} \\
			& {{[{{\lambda }_{P}}(T)]}_{U}}= & {{{\hat{\lambda }}}_{P}}(T)+\frac{{{C}^{2}}}{2}+\sqrt{{{{\hat{\lambda }}}_{P}}(T)\cdot \ \,{{C}^{2}}+\frac{{{C}^{4}}}{4}}
			\end{align}</math>


			where <math>C=\tfrac{{{z}_{1-\alpha /2}}}{\sqrt{T}}</math> .
			<br>

Lisa Hacker: Redirected page to Crow Extended Confidence Bounds

2012-08-25T05:37:18Z

Redirected page to Crow Extended Confidence Bounds

← Older revision		Revision as of 05:37, 25 August 2012
Line 1:		Line 1:
	~~===Bounds on Projected Failure Intensity===~~		#REDIRECT [[Crow Extended Confidence Bounds]]
	~~====Fisher Matrix Bounds====~~
	~~The projected failure intensity <math>{{\lambda }_{P}}(T)</math> must be positive, thus <math>\ln {{\lambda }_{P}}(T)</math> is approximately treated as being normally distributed as well:~~

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


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

	~~<br>~~
	~~where:~~
	~~<br>~~
	:• <math>{{\hat{\lambda }}_{P}}(T)=\tfrac{{{N}_{A}}}{T}+\underset{i=1}{\overset{M}{\mathop{\sum }}}\,(1-{{d}_{i}})\tfrac{{{N}_{i}}}{T}+\overline{d}\tfrac{M}{T}\bar{\beta }</math> when there are no BC modes.
	:• <math>{{\hat{\lambda }}_{P}}(T)={{\widehat{\lambda }}_{EM}}={{\widehat{\lambda }}_{CA}}-{{\widehat{\lambda }}_{BD}}+\underset{i=1}{\overset{M}{\mathop{\sum }}}\,(1-{{d}_{i}})\tfrac{{{N}_{i}}}{T}+\overline{d}\widehat{h}(T\|BD)</math> when there are BC modes.
	~~:• <math>{{N}_{i}}</math> is the total failure number of the <math>{{i}^{th}}</math> distinct BD mode.~~
	~~<br>~~
	~~You can then get:~~

	~~::<math>Var({{\lambda }_{P}}(T))\approx Var({{\hat{\gamma }}_{GP}})+\mu _{d}^{2}Var(h(T))\approx \frac{{{{\hat{r}}}_{GP}}}{T}+\mu _{d}^{2}Var(h(T))</math>~~

	~~:where:~~

	~~::<math>\begin{align}~~
	~~& \hat{h}(T)= & \frac{M}{T}\bar{\beta } \\~~
	~~& Var(\hat{h}(T))= & {{(\frac{M}{T})}^{2}}Var(\bar{\beta })={{(\frac{M}{T})}^{2}}{{(\frac{M}{M-1})}^{2}}Var(\hat{\beta })=\frac{{{M}^{4}}}{{{T}^{2}}{{(M-1)}^{2}}}Var(\hat{\beta })~~
	~~\end{align}</math>~~

	~~The <math>Var(\hat{\beta })</math> can be obtained from Fisher Matrix based on <math>M</math> distinct BD modes.~~
	~~<br>~~

	~~====~~Crow Bounds~~====~~
	~~::<math>\begin{align}~~
	~~& {{[{{\lambda }_{P}}(T)~~]~~}_{L}}= & {{{\hat{\lambda }}}_{P}}(T)+\frac{{{C}^{2}}}{2}-\sqrt{{{{\hat{\lambda }}}_{P}}(T)\cdot {{C}^{2}}+\frac{{{C}^{4}}}{4}} \\~~
	~~& {{[{{\lambda }_{P}}(T)~~]~~}_{U}}= & {{{\hat{\lambda }}}_{P}}(T)+\frac{{{C}^{2}}}{2}+\sqrt{{{{\hat{\lambda }}}_{P}}(T)\cdot \ \,{{C}^{2}}+\frac{{{C}^{4}}}{4}}~~
	~~\end{align}</math>~~


	~~where <math>C=\tfrac{{{z}_{1-\alpha /2}}}{\sqrt{T}}</math> .~~
	~~<br>~~

Nicolette Young: Created page with '===Bounds on Projected Failure Intensity=== ====Fisher Matrix Bounds==== The projected failure intensity ${{\lambda }_{P}}(T)$ must be positive, thus $\ln {{\l…'$

2012-01-10T20:50:49Z

Created page with '===Bounds on Projected Failure Intensity=== ====Fisher Matrix Bounds==== The projected failure intensity <math>{{\lambda }_{P}}(T)</math> must be positive, thus <math>\ln {{\l…'

New page

===Bounds on Projected Failure Intensity===
====Fisher Matrix Bounds====
The projected failure intensity <math>{{\lambda }_{P}}(T)</math> must be positive, thus <math>\ln {{\lambda }_{P}}(T)</math> is approximately treated as being normally distributed as well:

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

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

 
where:
 
:• <math>{{\hat{\lambda }}_{P}}(T)=\tfrac{{{N}_{A}}}{T}+\underset{i=1}{\overset{M}{\mathop{\sum }}}\,(1-{{d}_{i}})\tfrac{{{N}_{i}}}{T}+\overline{d}\tfrac{M}{T}\bar{\beta }</math> when there are no BC modes.
:• <math>{{\hat{\lambda }}_{P}}(T)={{\widehat{\lambda }}_{EM}}={{\widehat{\lambda }}_{CA}}-{{\widehat{\lambda }}_{BD}}+\underset{i=1}{\overset{M}{\mathop{\sum }}}\,(1-{{d}_{i}})\tfrac{{{N}_{i}}}{T}+\overline{d}\widehat{h}(T|BD)</math> when there are BC modes.
:• <math>{{N}_{i}}</math> is the total failure number of the <math>{{i}^{th}}</math> distinct BD mode.
 
You can then get:

::<math>Var({{\lambda }_{P}}(T))\approx Var({{\hat{\gamma }}_{GP}})+\mu _{d}^{2}Var(h(T))\approx \frac{{{{\hat{r}}}_{GP}}}{T}+\mu _{d}^{2}Var(h(T))</math>

:where:

::<math>\begin{align}
& \hat{h}(T)= & \frac{M}{T}\bar{\beta } \\
& Var(\hat{h}(T))= & {{(\frac{M}{T})}^{2}}Var(\bar{\beta })={{(\frac{M}{T})}^{2}}{{(\frac{M}{M-1})}^{2}}Var(\hat{\beta })=\frac{{{M}^{4}}}{{{T}^{2}}{{(M-1)}^{2}}}Var(\hat{\beta })
\end{align}</math>

The <math>Var(\hat{\beta })</math> can be obtained from Fisher Matrix based on <math>M</math> distinct BD modes.
 

====Crow Bounds====
::<math>\begin{align}
& {{[{{\lambda }_{P}}(T)]}_{L}}= & {{{\hat{\lambda }}}_{P}}(T)+\frac{{{C}^{2}}}{2}-\sqrt{{{{\hat{\lambda }}}_{P}}(T)\cdot {{C}^{2}}+\frac{{{C}^{4}}}{4}} \\
& {{[{{\lambda }_{P}}(T)]}_{U}}= & {{{\hat{\lambda }}}_{P}}(T)+\frac{{{C}^{2}}}{2}+\sqrt{{{{\hat{\lambda }}}_{P}}(T)\cdot \ \,{{C}^{2}}+\frac{{{C}^{4}}}{4}}
\end{align}</math>

where <math>C=\tfrac{{{z}_{1-\alpha /2}}}{\sqrt{T}}</math> .

Template:Bounds on projected failure intensity rga - Revision history

Lisa Hacker: Redirected page to Crow Extended Confidence Bounds

Lisa Hacker: Reverted edits by Lisa Hacker (Talk) to last revision by Nicolette Young

Lisa Hacker: Redirected page to Crow Extended Confidence Bounds

Nicolette Young: Created page with '===Bounds on Projected Failure Intensity=== ====Fisher Matrix Bounds==== The projected failure intensity {{\lambda }_{P}}(T) must be positive, thus \ln {{\l…'

Nicolette Young: Created page with '===Bounds on Projected Failure Intensity=== ====Fisher Matrix Bounds==== The projected failure intensity ${{\lambda }_{P}}(T)$ must be positive, thus $\ln {{\l…'$