Template:Bounds on cumulative failure intensity camsaa-gd - Revision history

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

2012-08-24T03:48:49Z

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

← Older revision		Revision as of 03:48, 24 August 2012
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	~~===Bounds on Cumulative Failure Intensity===~~		#REDIRECT [[Crow-AMSAA_-_NHPP#Bounds_on_Cumulative_Failure_Intensity_2]]
	~~====Fisher Matrix Bounds====~~
	~~The cumulative failure intensity, <math>{{\lambda }_{c}}(t)</math> , must be positive, thus <math>\ln {{\lambda }_{c}}(t)</math> is treated as being normally distributed.~~

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

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

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

	~~:where:~~

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

	~~====Crow Bounds====~~
	~~The Crow cumulative failure intensity confidence bounds are given as:~~

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

Nicolette Young: Created page with '===Bounds on Cumulative Failure Intensity=== ====Fisher Matrix Bounds==== The cumulative failure intensity, ${{\lambda }_{c}}(t)$ , must be positive, thus $\ln …'$

2012-01-06T16:49:09Z

Created page with '===Bounds on Cumulative Failure Intensity=== ====Fisher Matrix Bounds==== The cumulative failure intensity, <math>{{\lambda }_{c}}(t)</math> , must be positive, thus <math>\ln …'

New page

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

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

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

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

:where:

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

====Crow Bounds====
The Crow cumulative failure intensity confidence bounds are given as:

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

Template:Bounds on cumulative failure intensity camsaa-gd - Revision history

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

Nicolette Young: Created page with '===Bounds on Cumulative Failure Intensity=== ====Fisher Matrix Bounds==== The cumulative failure intensity, {{\lambda }_{c}}(t) , must be positive, thus \ln …'

Nicolette Young: Created page with '===Bounds on Cumulative Failure Intensity=== ====Fisher Matrix Bounds==== The cumulative failure intensity, ${{\lambda }_{c}}(t)$ , must be positive, thus $\ln …'$