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===Bounds on Cumulative Number of Failures===
#REDIRECT [[Crow-AMSAA_-_NHPP#Bounds_on_Cumulative_Number_of_Failures]]
====Fisher Matrix Bounds====
The cumulative number of failures,  <math>N(t)</math> , must be positive, thus  <math>\ln N(t)</math>  is treated as being normally distributed. 
 
::<math>\frac{\ln \hat{N}(t)-\ln N(t)}{\sqrt{Var(\ln \hat{N}(t)})}\ \tilde{\ }\ N(0,1)</math>
 
::<math>N(t)=\hat{N}(t){{e}^{\pm {{z}_{\alpha }}\sqrt{Var(\hat{N}(t))}/\hat{N}(t)}}</math>
 
<br>
:where:
 
::<math>\hat{N}(t)=\hat{\lambda }{{t}^{{\hat{\beta }}}}</math>
 
::<math>\begin{align}
  & Var(\hat{N}(t))= & {{\left( \frac{\partial \hat{N}(t)}{\partial \beta } \right)}^{2}}Var(\hat{\beta })+{{\left( \frac{\partial \hat{N}(t)}{\partial \lambda } \right)}^{2}}Var(\hat{\lambda }) \\
&  & +2\left( \frac{\partial \hat{N}(t)}{\partial \beta } \right)\left( \frac{\partial \hat{N}(t)}{\partial \lambda } \right)cov(\hat{\beta },\hat{\lambda }) 
\end{align}</math>
 
The variance calculation is the same as Eqn. (variance1) and:
 
::<math>\begin{align}
  & \frac{\partial \hat{N}(t)}{\partial \beta }= & \hat{\lambda }{{t}^{{\hat{\beta }}}}\ln t \\
& \frac{\partial \hat{N}(t)}{\partial \lambda }= & {{t}^{{\hat{\beta }}}} 
\end{align}</math>
 
====Crow Bounds====
<br>
The Crow cumulative number of failure confidence bounds are:
 
::<math>\begin{align}
  & {{N}_{L}}(T)= & \frac{T}{{\hat{\beta }}}{{\lambda }_{i}}{{(T)}_{L}} \\
& {{N}_{U}}(T)= & \frac{T}{{\hat{\beta }}}{{\lambda }_{i}}{{(T)}_{U}} 
\end{align}</math>
 
<br>
where  <math>{{\lambda }_{i}}{{(T)}_{L}}</math>  and  <math>{{\lambda }_{i}}{{(T)}_{U}}</math>  can be obtained from Eqn. (amsaac14).
<br>
<br>
'''Example 2'''
<br>
Calculate the 90% 2-sided confidence bounds on the cumulative and instantaneous failure intensity for the data from Example 1 given in Table 5.1. 
 
'''Solution'''
'''Fisher Matrix Bounds'''
<br>
Using  <math>\widehat{\beta }</math>  and  <math>\widehat{\lambda }</math>  estimated in Example 1, Eqns. (lambda2partial), (beta2partial) and (lambdabeta2partial) are:
 
::<math>\begin{align}
  & \frac{{{\partial }^{2}}\Lambda }{\partial {{\lambda }^{2}}}= & -\frac{22}{{{0.4239}^{2}}}=-122.43 \\
& \frac{{{\partial }^{2}}\Lambda }{\partial {{\beta }^{2}}}= & -\frac{22}{{{0.6142}^{2}}}-0.4239\cdot {{620}^{0.6142}}{{(\ln 620)}^{2}}=-967.68 \\
& \frac{{{\partial }^{2}}\Lambda }{\partial \lambda \partial \beta }= & -{{620}^{0.6142}}\ln 620=-333.64 
\end{align}</math>
 
<br>
The Fisher Matrix then becomes:
<br>
For  <math>T=620</math>  hr, the partial derivatives of the cumulative and instantaneous failure intensities are:
 
<br>
::<math>\begin{align}
  & \frac{\partial {{\lambda }_{c}}(T)}{\partial \beta }= & \widehat{\lambda }{{T}^{\widehat{\beta }-1}}\ln (T) \\
& = & 0.4239\cdot {{620}^{-0.3858}}\ln 620 \\
& = & 0.22811336 \\
& \frac{\partial {{\lambda }_{c}}(T)}{\partial \lambda }= & {{T}^{\widehat{\beta }-1}} \\
& = & {{620}^{-0.3858}} \\
& = & 0.083694185 
\end{align}</math>
 
<br>
::<math>\begin{align}
  & \frac{\partial {{\lambda }_{i}}(T)}{\partial \beta }= & \widehat{\lambda }{{T}^{\widehat{\beta }-1}}+\widehat{\lambda }\widehat{\beta }{{T}^{\widehat{\beta }-1}}\ln T \\
& = & 0.4239\cdot {{620}^{-0.3858}}+0.4239\cdot 0.6142\cdot {{620}^{-0.3858}}\ln 620 \\
& = & 0.17558519 
\end{align}</math>
<br>
::<math>\begin{align}
  & \frac{\partial {{\lambda }_{i}}(T)}{\partial \lambda }= & \widehat{\beta }{{T}^{\widehat{\beta }-1}} \\
& = & 0.6142\cdot {{620}^{-0.3858}} \\
& = & 0.051404969 
\end{align}</math>
 
Therefore, the variances become:
 
<br>
The cumulative and instantaneous failure intensities at  <math>T=620</math>  hr are:
<br>
::<math>\begin{align}
  & {{\lambda }_{c}}(T)= & 0.03548 \\
& {{\lambda }_{i}}(T)= & 0.02179 
\end{align}</math>
 
So, at the 90% confidence level and for  <math>T=620</math>  hr, the Fisher Matrix confidence bounds for the cumulative failure intensity are:
 
::<math>\begin{align}
  & {{[{{\lambda }_{c}}(T)]}_{L}}= & 0.02499 \\
& {{[{{\lambda }_{c}}(T)]}_{U}}= & 0.05039 
\end{align}</math>
 
The confidence bounds for the instantaneous failure intensity are:
 
::<math>\begin{align}
  & {{[{{\lambda }_{i}}(T)]}_{L}}= & 0.01327 \\
& {{[{{\lambda }_{i}}(T)]}_{U}}= & 0.03579 
\end{align}</math>
 
Figures 4fig82 and 4fig83 display plots of the Fisher Matrix confidence bounds for the cumulative and instantaneous failure intensity, respectively.
<br>
[[Image:rga5.2.png|thumb|center|400px|Cumulative failure intensity with 2-sided 90% Fisher Matrix confidence bounds.]]
<br>
[[Image:rga5.3.png|thumb|center|400px|Instantaneous failure intensity with 2-sided 90% Fisher Matrix confidence bounds.]]
<br>
'''Crow Bounds'''
<br>
The Crow confidence bounds for the cumulative failure intensity at the 90% confidence level and for  <math>T=620</math>  hr are:
 
::<math>\begin{align}
  & {{[{{\lambda }_{c}}(T)]}_{L}}= & \frac{\chi _{\tfrac{\alpha }{2},2N}^{2}}{2\cdot t} \\
& = & \frac{29.787476}{2*620} \\
& = & 0.02402 \\
& {{[{{\lambda }_{c}}(T)]}_{U}}= & \frac{\chi _{1-\tfrac{\alpha }{2},2N+2}^{2}}{2\cdot t} \\
& = & \frac{62.8296}{2*620} \\
& = & 0.05067 
\end{align}</math>
 
The Crow confidence bounds for the instantaneous failure intensity at the 90% confidence level and for  <math>T=620</math>  hr are:
 
::<math>\begin{align}
  & {{[{{\lambda }_{i}}(t)]}_{L}}= & \frac{1}{{{[MTB{{F}_{i}}]}_{U}}} \\
& = & \frac{1}{MTB{{F}_{i}}\cdot U} \\
& = & 0.01179 
\end{align}</math>
 
::<math>\begin{align}
  & {{[{{\lambda }_{i}}(t)]}_{U}}= & \frac{1}{{{[MTB{{F}_{i}}]}_{L}}} \\
& = & \frac{1}{MTB{{F}_{i}}\cdot L} \\
& = & 0.03253 
\end{align}</math>
 
Figures 4fig84 and 4fig85 display plots of the Crow confidence bounds for the cumulative and instantaneous failure intensity, respectively.
<br>
[[Image:rga5.4.png|thumb|center|400px|Cumulative failure intensity with 2-sided 90% Crow confidence bounds.]]
<br>
[[Image:rga5.5.png|thumb|center|400px|Instantaneous failure intensity with 2-sided 90% Crow confidence bounds.]]
<br>
::<math>\begin{align}
  & Var(\widehat{\lambda })= & 0.13519969 \\
& Var(\widehat{\beta })= & 0.017105343 \\
& Cov(\widehat{\beta },\widehat{\lambda })= & -0.046614609 
\end{align}</math>
'''Example 3'''
<br>
Calculate the confidence bounds on the cumulative and instantaneous MTBF for the data in Table 5.1.
<br>
'''Solution'''
<br>
'''Fisher Matrix Bounds'''
<br>
From the previous example:
 
And for  <math>T=620</math>  hr, the partial derivatives of the cumulative and instantaneous MTBF are:
 
::<math>\begin{align}
  & \frac{\partial {{m}_{c}}(T)}{\partial \beta }= & -\frac{1}{\widehat{\lambda }}{{T}^{1-\widehat{\beta }}}\ln T \\
& = & -\frac{1}{0.4239}{{620}^{0.3858}}\ln 620 \\
& = & -181.23135 \\
& \frac{\partial {{m}_{c}}(T)}{\partial \lambda }= & -\frac{1}{{{\widehat{\lambda }}^{2}}}{{T}^{1-\widehat{\beta }}} \\
& = & -\frac{1}{{{0.4239}^{2}}}{{620}^{0.3858}} \\
& = & -66.493299 \\
& \frac{\partial {{m}_{i}}(T)}{\partial \beta }= & -\frac{1}{\widehat{\lambda }{{\widehat{\beta }}^{2}}}{{T}^{1-\beta }}-\frac{1}{\widehat{\lambda }\widehat{\beta }}{{T}^{1-\widehat{\beta }}}\ln T \\
& = & -\frac{1}{0.4239\cdot {{0.6142}^{2}}}{{620}^{0.3858}}-\frac{1}{0.4239\cdot 0.6142}{{620}^{0.3858}}\ln 620 \\
& = & -369.78634 \\
& \frac{\partial {{m}_{i}}(T)}{\partial \lambda }= & -\frac{1}{{{\widehat{\lambda }}^{2}}\widehat{\beta }}{{T}^{1-\widehat{\beta }}} \\
& = & -\frac{1}{{{0.4239}^{2}}\cdot 0.6142}\cdot {{620}^{0.3858}} \\
& = & -108.26001 
\end{align}</math>
 
Therefore, the variances become:
 
::<math>\begin{align}
  & Var({{\widehat{m}}_{c}}(T))= & {{\left( -181.23135 \right)}^{2}}\cdot 0.017105343+{{\left( -66.493299 \right)}^{2}}\cdot 0.13519969 \\
&  & -2\cdot \left( -181.23135 \right)\cdot \left( -66.493299 \right)\cdot 0.046614609 \\
& = & 36.113376 
\end{align}</math>
 
::<math>\begin{align}
  & Var({{\widehat{m}}_{i}}(T))= & {{\left( -369.78634 \right)}^{2}}\cdot 0.017105343+{{\left( -108.26001 \right)}^{2}}\cdot 0.13519969 \\
&  & -2\cdot \left( -369.78634 \right)\cdot \left( -108.26001 \right)\cdot 0.046614609 \\
& = & 191.33709 
\end{align}</math>
 
So, at 90% confidence level and  <math>T=620</math>  hr, the Fisher Matrix confidence bounds are:
 
::<math>\begin{align}
  & {{[{{m}_{c}}(T)]}_{L}}= & {{{\hat{m}}}_{c}}(t){{e}^{-{{z}_{\alpha }}\sqrt{Var({{{\hat{m}}}_{c}}(t))}/{{{\hat{m}}}_{c}}(t)}} \\
& = & 19.84581 \\
& {{[{{m}_{c}}(T)]}_{U}}= & {{{\hat{m}}}_{c}}(t){{e}^{{{z}_{\alpha }}\sqrt{Var({{{\hat{m}}}_{c}}(t))}/{{{\hat{m}}}_{c}}(t)}} \\
& = & 40.01927 
\end{align}</math>
<br>
::<math>\begin{align}
  & {{[{{m}_{i}}(T)]}_{L}}= & {{{\hat{m}}}_{i}}(t){{e}^{-{{z}_{\alpha }}\sqrt{Var({{{\hat{m}}}_{i}}(t))}/{{{\hat{m}}}_{i}}(t)}} \\
& = & 27.94261 \\
& {{[{{m}_{i}}(T)]}_{U}}= & {{{\hat{m}}}_{i}}(t){{e}^{{{z}_{\alpha }}\sqrt{Var({{{\hat{m}}}_{i}}(t))}/{{{\hat{m}}}_{i}}(t)}} \\
& = & 75.34193 
\end{align}</math>
 
Figures 4fig86 and 4fig87 show plots of the Fisher Matrix confidence bounds for the cumulative and instantaneous MTBFs.
<br>
[[Image:rga5.6.png|thumb|center|400px|Cumulative MTBF with 2-sided 90% Fisher Matrix confidence bounds.]]
<br>
<br>
[[Image:rga5.7.png|thumb|center|400px|Instantaneous MTBF with 2-sided Fisher Matrix confidence bounds.]]
<br>
'''Crow Bounds'''
<br>
The Crow confidence bounds for the cumulative MTBF and the instantaneous MTBF at the 90% confidence level and for  <math>T=620</math>  hr are:
 
::<math>\begin{align}
  & {{[{{m}_{c}}(T)]}_{L}}= & \frac{1}{{{[{{\lambda }_{c}}(T)]}_{U}}} \\
& = & 20.5023 \\
& {{[{{m}_{c}}(T)]}_{U}}= & \frac{1}{{{[{{\lambda }_{c}}(T)]}_{L}}} \\
& = & 41.6282 
\end{align}</math>
 
::<math>\begin{align}
  & {{[MTB{{F}_{i}}]}_{L}}= & MTB{{F}_{i}}\cdot {{\Pi }_{1}} \\
& = & 30.7445 \\
& {{[MTB{{F}_{i}}]}_{U}}= & MTB{{F}_{i}}\cdot {{\Pi }_{2}} \\
& = & 84.7972 
\end{align}</math>
 
Figures 4fig88 and 4fig89 show plots of the Crow confidence bounds for the cumulative and instantaneous MTBF.
<br>
[[Image:rga5.8.png|thumb|center|400px|Cumulative MTBF with 2-sided 90% Crow confidence bounds.]]
<br>
[[Image:rga5.9.png|thumb|center|400px|Instantaneous MTBF with 2-sided 90% Crow confidence bounds.]]
Confidence bounds can also be obtained on the parameters  <math>\widehat{\beta }</math>  and  <math>\widehat{\lambda }</math> . For Fisher Matrix confidence bounds:
 
::<math>\begin{align}
  & {{\beta }_{L}}= & \hat{\beta }{{e}^{{{z}_{\alpha }}\sqrt{Var(\hat{\beta })}/\hat{\beta }}} \\
& = & 0.4325 \\
& {{\beta }_{U}}= & \hat{\beta }{{e}^{-{{z}_{\alpha }}\sqrt{Var(\hat{\beta })}/\hat{\beta }}} \\
& = & 0.8722 
\end{align}</math>
<br>
:and:
<br>
::<math>\begin{align}
  & {{\lambda }_{L}}= & \hat{\lambda }{{e}^{{{z}_{\alpha }}\sqrt{Var(\hat{\lambda })}/\hat{\lambda }}} \\
& = & 0.1016 \\
& {{\lambda }_{U}}= & \hat{\lambda }{{e}^{-{{z}_{\alpha }}\sqrt{Var(\hat{\lambda })}/\hat{\lambda }}} \\
& = & 1.7691 
\end{align}</math>
<br>
For Crow confidence bounds:
 
::<math>\begin{align}
  & {{\beta }_{L}}= & 0.4527 \\
& {{\beta }_{U}}= & 0.9350 
\end{align}</math>
<br>
:and:
<br>
::<math>\begin{align}
  & {{\lambda }_{L}}= & 0.2870 \\
& {{\lambda }_{U}}= & 0.5827 
\end{align}</math>

Latest revision as of 04:15, 24 August 2012

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