Template:Bounds on growth potential mtbf rga: Difference between revisions

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(Created page with '===Bounds on Growth Potential MTBF=== ====Fisher Matrix Bounds==== ::<math>\begin{align} & MTB{{F}_{G{{P}_{L}}}}= & \frac{1}{{{r}_{U}}} \\ & MTB{{F}_{G{{P}_{U}}}}= & \frac{1}…')
 
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where  <math>{{r}_{U}}</math>  and  <math>{{r}_{L}}</math>  can be obtained from Eqn. (GPR).
where  <math>{{r}_{U}}</math>  and  <math>{{r}_{L}}</math>  can be obtained from Eqn. (GPR).
<br>
<br>
====Crow Bounds====
#REDIRECT [[Crow Extended Confidence Bounds]]
The Crow bounds for the growth potential MTBF are the same as the Fisher Matrix bounds.
<br>
<br>
'''Example 3'''
<br>
Calculate the 2-sided 90% confidence bounds on the demonstrated, projected and growth potential failure intensity for the data in Table 9.1.
<br>
<br>
'''Solution'''
<br>
The estimated demonstrated failure intensity is  <math>{{\widehat{\lambda }}_{D}}(T)=\tfrac{{{N}_{A}}+{{N}_{B}}}{T}=0.1050</math> . Based on this value, the Fisher Matrix confidence bounds for the demonstrated failure intensity at the 90% confidence level are:
 
::<math>\begin{align}
  & {{[{{\lambda }_{D}}(T)]}_{L}}= & {{{\hat{\lambda }}}_{D}}(T)+\frac{{{C}^{2}}}{2}-\sqrt{{{{\hat{\lambda }}}_{D}}(T){{C}^{2}}+\frac{{{C}^{4}}}{4}} \\
& = & 0.08152 
\end{align}</math>
 
::<math>\begin{align}
  & {{[{{\lambda }_{D}}(T)]}_{U}}= & {{{\hat{\lambda }}}_{D}}(T)+\frac{{{C}^{2}}}{2}+\sqrt{{{{\hat{\lambda }}}_{D}}(T){{C}^{2}}+\frac{{{C}^{4}}}{4}} \\
& = & 0.13525 
\end{align}</math>
 
The Crow confidence bounds for the demonstrated failure intensity at the 90% confidence level are:
 
::<math>\begin{align}
  & {{[{{\lambda }_{D}}(T)]}_{L}}= & {{\widehat{\lambda }}_{D}}(T)\frac{\chi _{(2N,1-\alpha /2)}^{2}}{2N} \\
& = & 0.07985 \\
& {{[{{\lambda }_{D}}(T)]}_{U}}= & {{\widehat{\lambda }}_{D}}(T)\frac{\chi _{(2N,\alpha /2)}^{2}}{2N} \\
& = & 0.13299 
\end{align}</math>
 
The projected failure intensity is  . Based on this value, the Fisher Matrix confidence bounds at the 90% confidence level for the projected failure intensity are:
 
::<math>\begin{align}
  & {{[{{{\hat{\lambda }}}_{P}}(T)]}_{L}}= & {{{\hat{\lambda }}}_{P}}(T){{e}^{{{z}_{\alpha }}\sqrt{Var({{{\hat{\lambda }}}_{P}}(T))}/{{{\hat{\lambda }}}_{P}}(T)}} \\
& = & 0.04902 
\end{align}</math>
 
::<math>\begin{align}
  & {{[{{{\hat{\lambda }}}_{P}}(T)]}_{U}}= & {{{\hat{\lambda }}}_{P}}(T){{e}^{-{{z}_{\alpha }}\sqrt{Var({{{\hat{\lambda }}}_{P}}(T))}/{{{\hat{\lambda }}}_{P}}(T)}} \\
& = & 0.08915 
\end{align}</math>
 
The Crow confidence bounds for the projected failure intensity are:
 
::<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}} \\
& = & 0.04807 \\
& {{[{{\lambda }_{P}}(T)]}_{U}}= & {{{\hat{\lambda }}}_{P}}(T)+\frac{{{C}^{2}}}{2}+\sqrt{{{{\hat{\lambda }}}_{P}}(T)\cdot \ \,{{C}^{2}}+\frac{{{C}^{4}}}{4}} \\
& = & 0.09090 
\end{align}</math>
 
The growth potential failure intensity is <math>\widehat{r}_{GP} (T) = \left (\frac{N_A}{T} + \sum_{i=1}^M (1-d_i) \tfrac{N_i}{T} \right ) = 0.04455 </math>.
 
<br>
Based on this value, the Fisher Matrix and Crow confidence bounds at the 90% confidence level for the growth potential failure intensity are:
 
::<math>\begin{align}
  & {{r}_{L}}= & {{{\hat{r}}}_{GP}}+\frac{{{C}^{2}}}{2}-\sqrt{{{{\hat{r}}}_{GP}}{{C}^{2}}+\frac{{{C}^{4}}}{4}} \\
& = & 0.03020 \\
& {{r}_{U}}= & {{{\hat{r}}}_{GP}}+\frac{{{C}^{2}}}{2}+\sqrt{{{{\hat{r}}}_{GP}}{{C}^{2}}+\frac{{{C}^{4}}}{4}} \\
& = & 0.0656 
\end{align}</math>
 
Figure extendedpic7 shows the Fisher Matrix confidence bounds at the 90% confidence level for the demonstrated, projected and growth potential failure intensity. Figure extendedpic8 shows these bounds based on the Crow method.
<br>
<br>
[[Image:rga9.8.png|thumb|center|400px|Fisher Matrix confidence bounds for the failure intensity.]]
<br>
<br>
[[Image:rga9.9.png|thumb|center|400px|Crow confidence bounds for the failure intensity.]]
 
 
'''Example 4'''
<br>
Calculate the 2-sided confidence bounds at the 90% confidence level on the demonstrated, projected and growth potential MTBF for the data in Table 9.3.
<br>
<br>
'''Solution'''
<br>
For this example, there are A, BC and BD failure modes, so the estimated demonstrated failure intensity,  <math>{{\hat{\lambda }}_{D}}(T)</math> , is simply the Crow-AMSAA model applied to all A, BC, and BD data.
 
::<math>{{\hat{\lambda }}_{D}}(T)={{\widehat{\lambda }}_{CA}}=\widehat{\lambda }\widehat{\beta }{{T}^{\widehat{\beta }-1}}=0.12744</math>
 
Therefore, the demonstrated MTBF is:
 
::<math>MTB{{F}_{D}}={{[{{\hat{\lambda }}_{D}}(T)]}^{-1}}=7.84708</math>
 
Based on this value, the Fisher Matrix confidence bounds for the demonstrated failure intensity at the 90% confidence level are:
 
::<math>\begin{align}
  & {{[{{\lambda }_{D}}(T)]}_{L}}= & {{{\hat{\lambda }}}_{CA}}(T){{e}^{{{z}_{\alpha }}\sqrt{Var({{{\hat{\lambda }}}_{CA}}(T))}/{{{\hat{\lambda }}}_{i}}(T)}} \\
& = & 0.09339 
\end{align}</math>
 
::<math>\begin{align}
  & {{[{{\lambda }_{D}}(T)]}_{U}}= & {{{\hat{\lambda }}}_{CA}}(T){{e}^{-{{z}_{\alpha }}\sqrt{Var({{{\hat{\lambda }}}_{CA}}(T))}/{{{\hat{\lambda }}}_{i}}(T)}} \\
& = & 0.17390 
\end{align}</math>
 
 
The Fisher Matrix confidence bounds for the demonstrated MTBF at the 90% confidence level are:
 
::<math>\begin{align}
  & MTB{{F}_{{{D}_{L}}}}= & \frac{1}{{{[{{\lambda }_{D}}(T)]}_{U}}} \\
& = & 5.75054 \\
& MTB{{F}_{{{D}_{U}}}}= & \frac{1}{{{[{{\lambda }_{D}}(T)]}_{L}}} \\
& = & 10.70799 
\end{align}</math>
 
 
The Crow confidence bounds for the demonstrated MTBF at the 90% confidence level are:
 
::<math>\begin{align}
  & MTB{{F}_{{{D}_{L}}}}= & \frac{1}{{{[{{\lambda }_{D}}(T)]}_{U}}} \\
& = & \frac{1}{{{\widehat{\lambda }}_{D}}(T)\tfrac{{{\chi }^{2}}(2N,\alpha /2)}{2N}} \\
& = & 5.6325 \\
& MTB{{F}_{{{D}_{U}}}}= & \frac{1}{{{[{{\lambda }_{D}}(T)]}_{L}}} \\
& = & \frac{1}{{{\widehat{\lambda }}_{D}}(T)\tfrac{{{\chi }^{2}}(2N,1-\alpha /2)}{2N}} \\
& = & 10.8779 
\end{align}</math>
 
<br>
The projected failure intensity is <math>\hat{\lambda}_P (T) = \widehat{\lambda}_{CA} - \widehat{\lambda}_{BD} + \sum_{i=1}^M (1-d_i) \tfrac{N_i}{T} + \bar{d}\widehat{h}(T|BD) = 0.0885 </math>. Based on this value, the Fisher Matrix confidence bounds at the 90% confidence level for the projected failure intensity are:
 
::<math>\begin{align}
  & {{[{{\lambda }_{P}}(T)]}_{L}}= & {{{\hat{\lambda }}}_{P}}(T){{e}^{{{z}_{\alpha }}\sqrt{Var({{{\hat{\lambda }}}_{P}}(T))}/{{{\hat{\lambda }}}_{P}}(T)}} \\
& = & 0.0681 
\end{align}</math>
 
::<math>\begin{align}
  & {{[{{\lambda }_{P}}(T)]}_{U}}= & {{{\hat{\lambda }}}_{P}}(T){{e}^{-{{z}_{\alpha }}\sqrt{Var({{{\hat{\lambda }}}_{P}}(T))}/{{{\hat{\lambda }}}_{P}}(T)}} \\
& = & 0.1152 
\end{align}</math>
 
 
The Fisher Matrix confidence bounds for the projected MTBF at the 90% confidence level are:
 
::<math>\begin{align}
  & MTB{{F}_{{{P}_{L}}}}= & \frac{1}{{{[{{\lambda }_{P}}(T)]}_{U}}} \\
& = & 8.6818 \\
& MTB{{F}_{{{P}_{U}}}}= & \frac{1}{{{[{{\lambda }_{P}}(T)]}_{L}}} \\
& = & 14.6926 
\end{align}</math>
 
The Crow confidence bounds for the projected failure intensity are:
 
::<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}} \\
& = & 0.0672 \\
& {{[{{\lambda }_{P}}(T)]}_{U}}= & {{{\hat{\lambda }}}_{P}}(T)+\frac{{{C}^{2}}}{2}+\sqrt{{{{\hat{\lambda }}}_{P}}(T)\cdot {{C}^{2}}+\frac{{{C}^{4}}}{4}} \\
& = & 0.1166 
\end{align}</math>
 
 
The Crow confidence bounds for the projected MTBF at the 90% confidence level are:
 
::<math>\begin{align}
  & MTB{{F}_{{{P}_{L}}}}= & \frac{1}{{{[{{\widehat{\lambda }}_{P}}(T)]}_{U}}} \\
& = & 8.5743 \\
& MTB{{F}_{{{P}_{U}}}}= & \frac{1}{{{[{{\widehat{\lambda }}_{P}}(T)]}_{L}}} \\
& = & 14.8769 
\end{align}</math>
 
 
The growth potential failure intensity is  <math>\widehat{\lambda}_{GP} = \widehat{\lambda}_{CA} - \widehat{\lambda}_{BD} + \sum_{i=1}^M (1-d_i) \tfrac{N_i}{T} = 0.0670 </math>.
<math>\hat{\lambda}_P (T) = \widehat{\lambda}_{CA} - \widehat{\lambda}_{BD} + \sum_{i=1}^M (1-d_i) \tfrac{N_i}{T} + \bar{d}\widehat{h}(T|BD) = 0.0885 </math>.Based on this value, the Fisher Matrix and Crow confidence bounds at the 90% confidence level for the growth potential failure intensity are:
 
 
::<math>\begin{align}
  & {{r}_{L}}= & {{{\hat{r}}}_{GP}}+\frac{{{C}^{2}}}{2}-\sqrt{{{{\hat{r}}}_{GP}}{{C}^{2}}+\frac{{{C}^{4}}}{4}} \\
& = & 0.0488 \\
& {{r}_{U}}= & {{{\hat{r}}}_{GP}}+\frac{{{C}^{2}}}{2}+\sqrt{{{{\hat{r}}}_{GP}}{{C}^{2}}+\frac{{{C}^{4}}}{4}} \\
& = & 0.0919 
\end{align}</math>
 
 
The Fisher Matrix and Crow confidence bounds for the growth potential MTBF at the 90% confidence level are:
 
::<math>\begin{align}
  & MTB{{F}_{G{{P}_{L}}}}= & \frac{1}{{{r}_{U}}} \\
& = & 10.8790 \\
& MTB{{F}_{G{{P}_{U}}}}= & \frac{1}{{{r}_{L}}} \\
& = & 20.4855 
\end{align}</math>
 
 
Figure extendedpic9 shows the Fisher Matrix confidence bounds at the 90% confidence level for the demonstrated, projected and growth potential MTBF. Figure extendedpic10 shows these bounds based on the Crow method.
[[Image:rga9.10.png|thumb|center|400px|Fisher Matrix confidence bounds on MTBF.]]
<br>
<br>
[[Image:rga9.11.png|thumb|center|400px|Crow confidence bounds on MTBF.]]

Revision as of 05:36, 25 August 2012

Bounds on Growth Potential MTBF

Fisher Matrix Bounds

[math]\displaystyle{ \begin{align} & MTB{{F}_{G{{P}_{L}}}}= & \frac{1}{{{r}_{U}}} \\ & MTB{{F}_{G{{P}_{U}}}}= & \frac{1}{{{r}_{L}}} \end{align} }[/math]

where [math]\displaystyle{ {{r}_{U}} }[/math] and [math]\displaystyle{ {{r}_{L}} }[/math] can be obtained from Eqn. (GPR).

  1. REDIRECT Crow Extended Confidence Bounds
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