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

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

2012-08-24T03:48:11Z

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

← Older revision		Revision as of 03:48, 24 August 2012
Line 1:		Line 1:
	~~===Bounds on Instantaneous MTBF===~~		#REDIRECT [[Crow-AMSAA_-_NHPP#Bounds_on_Instantaneous_MTBF_2]]
	~~====Fisher Matrix Bounds====~~
	~~The instantaneous MTBF, <math>{{m}_{i}}(t)</math> , must be positive, thus <math>\ln {{m}_{i}}(t)</math> is approximately treated as being normally distributed as well.~~

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

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

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

	~~:where:~~

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

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

	~~====~~Crow ~~Bounds====~~
	~~:Step 1: Calculate <math>P(i)=\tfrac{{{T}_{i}}}{{{T}_{K}}},\,\,i=1,2,\ldots ,K</math> .~~
	~~:Step 2: Calculate:~~

	~~::<math>A=\underset{i=1}{\overset{K}{\mathop \sum }}\,\frac{{{\left[ P{{(i)}^{{\hat{\beta }}}}\ln P{{(i)}^{{\hat{\beta }}}}~~-~~P{{(i~~-~~1)}^{\widehat{\beta }}}\ln P{{(i-1)}^{{\hat{\beta }}}} \right~~]~~}^{2}}}{\left[ P{{(i)}^{{\hat{\beta }}}}-P{{(i-1)}^{{\hat{\beta }}}} \right~~]~~}</math>~~

	:Step 3: Calculate <math>D=\sqrt{\tfrac{1}{A}+1}</math> and <math>W=\tfrac{({{z}_{1-\alpha /2}})\cdot D}{\sqrt{N}}</math> . Thus an approximate 2-sided <math>(1-\alpha )</math> 100-percent confidence interval on <math>{{\hat{m}}_{i}}(t)</math> is:

	~~::<math>MTB{{F}_{i}}={{\widehat{m}}_{i}}(1\pm W)</math>~~

Nicolette Young: Created page with '===Bounds on Instantaneous MTBF=== ====Fisher Matrix Bounds==== The instantaneous MTBF, ${{m}_{i}}(t)$ , must be positive, thus $\ln {{m}_{i}}(t)$ is app…'

2012-01-06T16:48:23Z

Created page with '===Bounds on Instantaneous MTBF=== ====Fisher Matrix Bounds==== The instantaneous MTBF, <math>{{m}_{i}}(t)</math> , must be positive, thus <math>\ln {{m}_{i}}(t)</math> is app…'

New page

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

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

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

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

:where:

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

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

====Crow Bounds====
:Step 1: Calculate <math>P(i)=\tfrac{{{T}_{i}}}{{{T}_{K}}},\,\,i=1,2,\ldots ,K</math> .
:Step 2: Calculate:

::<math>A=\underset{i=1}{\overset{K}{\mathop \sum }}\,\frac{{{\left[ P{{(i)}^{{\hat{\beta }}}}\ln P{{(i)}^{{\hat{\beta }}}}-P{{(i-1)}^{\widehat{\beta }}}\ln P{{(i-1)}^{{\hat{\beta }}}} \right]}^{2}}}{\left[ P{{(i)}^{{\hat{\beta }}}}-P{{(i-1)}^{{\hat{\beta }}}} \right]}</math>

:Step 3: Calculate <math>D=\sqrt{\tfrac{1}{A}+1}</math> and <math>W=\tfrac{({{z}_{1-\alpha /2}})\cdot D}{\sqrt{N}}</math> . Thus an approximate 2-sided <math>(1-\alpha )</math> 100-percent confidence interval on <math>{{\hat{m}}_{i}}(t)</math> is:

::<math>MTB{{F}_{i}}={{\widehat{m}}_{i}}(1\pm W)</math>

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

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

Nicolette Young: Created page with '===Bounds on Instantaneous MTBF=== ====Fisher Matrix Bounds==== The instantaneous MTBF, {{m}_{i}}(t) , must be positive, thus \ln {{m}_{i}}(t) is app…'

Nicolette Young: Created page with '===Bounds on Instantaneous MTBF=== ====Fisher Matrix Bounds==== The instantaneous MTBF, ${{m}_{i}}(t)$ , must be positive, thus $\ln {{m}_{i}}(t)$ is app…'