Template:Bounds on instantaneous mtbf camsaa-gd: Difference between revisions

From ReliaWiki
Jump to navigation Jump to search
(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…')
 
 
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>

Latest revision as of 03:48, 24 August 2012

Redirect to:

Retrieved from "https://www.reliawiki.com/index.php?title=Template:Bounds_on_instantaneous_mtbf_camsaa-gd&oldid=33774"