Template:Bounds on cumulative mtbf camsaa-gd - Revision history

Richard House: Redirected page to Crow-AMSAA - NHPP#Bounds on Cumulative MTBF 2

2012-08-24T03:47:16Z

Redirected page to Crow-AMSAA - NHPP#Bounds on Cumulative MTBF 2

← Older revision		Revision as of 03:47, 24 August 2012
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	~~===Bounds on Cumulative MTBF===~~		#REDIRECT [[Crow-AMSAA_-_NHPP#Bounds_on_Cumulative_MTBF_2]]
	~~====Fisher Matrix Bounds====~~
	~~The cumulative MTBF, <math>{{m}_{c}}(t)</math> , must be positive, thus <math>\ln {{m}_{c}}(t)</math> is treated as being normally distributed as well.~~

	~~::<math>\frac{\ln {{{\hat{m}}}_{c}}(t)-\ln {{m}_{c}}(t)}{\sqrt{Var(\ln {{{\hat{m}}}_{c}}(t)})}\ \tilde{\ }\ N(0,1)</math>~~

	~~The approximate confidence bounds on the cumulative MTBF are then estimated from:~~

	~~::<math>CB={{\hat{m}}_{c}}(t){{e}^{\pm {{z}_{\alpha }}\sqrt{Var({{{\hat{m}}}_{c}}(t))}/{{{\hat{m}}}_{c}}(t)}}</math>~~

	~~:where:~~

	~~<br>~~
	~~::<math>{{\hat{m}}_{c}}(t)=\frac{1}{{\hat{\lambda }}}{{t}^{1-\hat{\beta }}}</math>~~

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

	~~The variance calculation is the same as Eqn. (variances) and:~~

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

	~~====Crow Bounds====~~
	~~Calculate the Crow cumulative failure intensity confidence bounds:~~

	~~::<math>C{{(t)}_{L}}=\frac{\chi _{\tfrac{\alpha }{2},2N}^{2}}{2\cdot t}</math>~~

	~~::<math>C{{(t)}_{U}}=\frac{\chi _{1~~-~~\tfrac{\alpha }{2},2N+2}^{2}}{2\cdot t}</math>~~

	~~:Then:~~

	~~::<math>\begin{align}~~
	~~& {{[MTB{{F}_{c}}~~]~~}_{L}}= & \frac{1}{C{{(t)}_{U}}} \\~~
	~~& {{[MTB{{F}_{c}}~~]~~}_{U}}= & \frac{1}{C{{(t)}_{L}}}~~
	~~\end{align}</math>~~

Nicolette Young: Created page with '===Bounds on Cumulative MTBF=== ====Fisher Matrix Bounds==== The cumulative MTBF, ${{m}_{c}}(t)$ , must be positive, thus $\ln {{m}_{c}}(t)$ is treated a…'

2012-01-05T23:12:52Z

Created page with '===Bounds on Cumulative MTBF=== ====Fisher Matrix Bounds==== The cumulative MTBF, <math>{{m}_{c}}(t)</math> , must be positive, thus <math>\ln {{m}_{c}}(t)</math> is treated a…'

New page

===Bounds on Cumulative MTBF===
====Fisher Matrix Bounds====
The cumulative MTBF, <math>{{m}_{c}}(t)</math> , must be positive, thus <math>\ln {{m}_{c}}(t)</math> is treated as being normally distributed as well.

::<math>\frac{\ln {{{\hat{m}}}_{c}}(t)-\ln {{m}_{c}}(t)}{\sqrt{Var(\ln {{{\hat{m}}}_{c}}(t)})}\ \tilde{\ }\ N(0,1)</math>

The approximate confidence bounds on the cumulative MTBF are then estimated from:

::<math>CB={{\hat{m}}_{c}}(t){{e}^{\pm {{z}_{\alpha }}\sqrt{Var({{{\hat{m}}}_{c}}(t))}/{{{\hat{m}}}_{c}}(t)}}</math>

:where:

<br>
::<math>{{\hat{m}}_{c}}(t)=\frac{1}{{\hat{\lambda }}}{{t}^{1-\hat{\beta }}}</math>

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

The variance calculation is the same as Eqn. (variances) and:

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

====Crow Bounds====
Calculate the Crow cumulative failure intensity confidence bounds:

::<math>C{{(t)}_{L}}=\frac{\chi _{\tfrac{\alpha }{2},2N}^{2}}{2\cdot t}</math>

::<math>C{{(t)}_{U}}=\frac{\chi _{1-\tfrac{\alpha }{2},2N+2}^{2}}{2\cdot t}</math>

:Then:

::<math>\begin{align}
& {{[MTB{{F}_{c}}]}_{L}}= & \frac{1}{C{{(t)}_{U}}} \\
& {{[MTB{{F}_{c}}]}_{U}}= & \frac{1}{C{{(t)}_{L}}}
\end{align}</math>

Template:Bounds on cumulative mtbf camsaa-gd - Revision history

Richard House: Redirected page to Crow-AMSAA - NHPP#Bounds on Cumulative MTBF 2

Nicolette Young: Created page with '===Bounds on Cumulative MTBF=== ====Fisher Matrix Bounds==== The cumulative MTBF, {{m}_{c}}(t) , must be positive, thus \ln {{m}_{c}}(t) is treated a…'

Nicolette Young: Created page with '===Bounds on Cumulative MTBF=== ====Fisher Matrix Bounds==== The cumulative MTBF, ${{m}_{c}}(t)$ , must be positive, thus $\ln {{m}_{c}}(t)$ is treated a…'