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{{template:RGA BOOK SUB|Appendix E|Confidence Bounds for Repairable Systems Analysis}}
{{template:RGA BOOK|Appendix E|Confidence Bounds for Repairable Systems Analysis}}
In this appendix, we will present the two methods used in the RGA software to estimate the confidence bounds for [[Repairable Systems Analysis|Repairable Systems Analysis]]. The Fisher Matrix approach is based on the Fisher Information Matrix and is commonly employed in the reliability field. The Crow bounds were developed by Dr. Larry Crow. 
In this appendix, we will present the two methods used in the RGA software to estimate the confidence bounds for [[Repairable Systems Analysis|Repairable Systems Analysis]]. The Fisher Matrix approach is based on the Fisher Information Matrix and is commonly employed in the reliability field. The Crow bounds were developed by Dr. Larry Crow. 


====Bounds on Beta====<!-- THIS SECTION HEADER IS LINKED FROM ANOTHER SECTION IN THIS  
===Beta===<!-- THIS SECTION HEADER IS LINKED FROM ANOTHER SECTION IN THIS  
PAGE. IF YOU RENAME THE SECTION, YOU MUST UPDATE THE LINK(S). -->
PAGE. IF YOU RENAME THE SECTION, YOU MUST UPDATE THE LINK(S). -->
=====Fisher Matrix Bounds=====
====Fisher Matrix Bounds====
The parameter <math>\beta \,\!</math> must be positive, thus <math>\ln \beta \,\!</math> is approximately treated as being normally distributed.
The parameter <math>\beta \,\!</math> must be positive, thus <math>\ln \beta \,\!</math> is approximately treated as being normally distributed.


::<math>\frac{\ln (\hat{\beta })-\ln (\beta )}{\sqrt{Var\left[ \ln (\hat{\beta }) \right]}}\ \tilde{\ }\ N(0,1)\,\!</math>
:<math>\frac{\ln (\hat{\beta })-\ln (\beta )}{\sqrt{Var\left[ \ln (\hat{\beta }) \right]}}\ \tilde{\ }\ N(0,1)\,\!</math>


::<math>C{{B}_{\beta }}=\hat{\beta }{{e}^{\pm {{z}_{\alpha }}\sqrt{Var(\hat{\beta })}/\hat{\beta }}}\,\!</math>
:<math>C{{B}_{\beta }}=\hat{\beta }{{e}^{\pm {{z}_{\alpha }}\sqrt{Var(\hat{\beta })}/\hat{\beta }}}\,\!</math>


::<math>\hat{\beta }=\frac{\underset{q=1}{\overset{K}{\mathop{\sum }}}\,{{N}_{q}}}{\hat{\lambda }\underset{q=1}{\overset{K}{\mathop{\sum }}}\,\left[ (T_{q}^{\hat{\beta }}\ln ({{T}_{q}})-S_{q}^{\hat{\beta }}\ln ({{S}_{q}}) \right]-\underset{q=1}{\overset{K}{\mathop{\sum }}}\,\underset{i=1}{\overset{{{N}_{q}}}{\mathop{\sum }}}\,\ln ({{X}_{i}}{{}_{q}})}\,\!</math>
:<math>\hat{\beta }=\frac{\underset{q=1}{\overset{K}{\mathop{\sum }}}\,{{N}_{q}}}{\hat{\lambda }\underset{q=1}{\overset{K}{\mathop{\sum }}}\,\left[ (T_{q}^{\hat{\beta }}\ln ({{T}_{q}})-S_{q}^{\hat{\beta }}\ln ({{S}_{q}}) \right]-\underset{q=1}{\overset{K}{\mathop{\sum }}}\,\underset{i=1}{\overset{{{N}_{q}}}{\mathop{\sum }}}\,\ln ({{X}_{i}}{{}_{q}})}\,\!</math>


All variance can be calculated using the Fisher Information Matrix.
All variance can be calculated using the Fisher Information Matrix. <math>\Lambda \,\!</math> is the natural log-likelihood function.


<math>\Lambda \,\!</math> is the natural log-likelihood function.
:<math>\Lambda =\underset{q=1}{\overset{K}{\mathop \sum }}\,\left[ {{N}_{q}}(\ln (\lambda )+\ln (\beta ))-\lambda (T_{q}^{\beta }-S_{q}^{\beta })+(\beta -1)\underset{i=1}{\overset{{{N}_{q}}}{\mathop \sum }}\,\ln ({{x}_{iq}}) \right]\,\!</math>


::<math>\Lambda =\underset{q=1}{\overset{K}{\mathop \sum }}\,\left[ {{N}_{q}}(\ln (\lambda )+\ln (\beta ))-\lambda (T_{q}^{\beta }-S_{q}^{\beta })+(\beta -1)\underset{i=1}{\overset{{{N}_{q}}}{\mathop \sum }}\,\ln ({{x}_{iq}}) \right]\,\!</math>
:<math>\frac{{{\partial }^{2}}\Lambda }{\partial {{\lambda }^{2}}}=-\frac{\underset{q=1}{\overset{K}{\mathop{\sum }}}\,{{N}_{q}}}{{{\lambda }^{2}}}\,\!</math>


::<math>\frac{{{\partial }^{2}}\Lambda }{\partial {{\lambda }^{2}}}=-\frac{\underset{q=1}{\overset{K}{\mathop{\sum }}}\,{{N}_{q}}}{{{\lambda }^{2}}}\,\!</math>
:<math>\frac{{{\partial }^{2}}\Lambda }{\partial \lambda \partial \beta }=-\underset{q=1}{\overset{K}{\mathop \sum }}\,\left[ T_{q}^{\beta }\ln ({{T}_{q}})-S_{q}^{\beta }\ln ({{S}_{q}}) \right]\,\!</math>


::<math>\frac{{{\partial }^{2}}\Lambda }{\partial \lambda \partial \beta }=-\underset{q=1}{\overset{K}{\mathop \sum }}\,\left[ T_{q}^{\beta }\ln ({{T}_{q}})-S_{q}^{\beta }\ln ({{S}_{q}}) \right]\,\!</math>
:<math>\frac{{{\partial }^{2}}\Lambda }{\partial {{\beta }^{2}}}=-\frac{\underset{q=1}{\overset{K}{\mathop{\sum }}}\,{{N}_{q}}}{{{\beta }^{2}}}-\lambda \underset{q=1}{\overset{K}{\mathop \sum }}\,\left[ T_{q}^{\beta }{{(\ln ({{T}_{q}}))}^{2}}-S_{q}^{\beta }{{(\ln ({{S}_{q}}))}^{2}} \right]\,\!</math>


::<math>\frac{{{\partial }^{2}}\Lambda }{\partial {{\beta }^{2}}}=-\frac{\underset{q=1}{\overset{K}{\mathop{\sum }}}\,{{N}_{q}}}{{{\beta }^{2}}}-\lambda \underset{q=1}{\overset{K}{\mathop \sum }}\,\left[ T_{q}^{\beta }{{(\ln ({{T}_{q}}))}^{2}}-S_{q}^{\beta }{{(\ln ({{S}_{q}}))}^{2}} \right]\,\!</math>
====Crow Bounds====
 
=====Crow Bounds=====
Calculate the conditional maximum likelihood estimate of <math>\tilde{\beta \,\!}\,\!</math> :
Calculate the conditional maximum likelihood estimate of <math>\tilde{\beta \,\!}\,\!</math> :


::<math>\tilde{\beta }=\frac{\underset{q=1}{\overset{K}{\mathop{\sum }}}\,{{M}_{q}}}{\underset{q=1}{\overset{K}{\mathop{\sum }}}\,\underset{i=1}{\overset{M}{\mathop{\sum }}}\,\ln \left( \tfrac{{{T}_{q}}}{{{X}_{iq}}} \right)}\,\!</math>
:<math>\tilde{\beta }=\frac{\underset{q=1}{\overset{K}{\mathop{\sum }}}\,{{M}_{q}}}{\underset{q=1}{\overset{K}{\mathop{\sum }}}\,\underset{i=1}{\overset{M}{\mathop{\sum }}}\,\ln \left( \tfrac{{{T}_{q}}}{{{X}_{iq}}} \right)}\,\!</math>


The Crow 2-sided <math>(1-a)\,\!</math> 100% confidence bounds on <math>\beta \,\!</math> are:  
The Crow 2-sided <math>(1-a)\,\!</math> 100% confidence bounds on <math>\beta \,\!</math> are:  


::<math>\begin{align}
:<math>\begin{align}
   {{\beta }_{L}}= & \tilde{\beta }\frac{\chi _{\tfrac{\alpha }{2},2M}^{2}}{2M} \\  
   {{\beta }_{L}}= & \tilde{\beta }\frac{\chi _{\tfrac{\alpha }{2},2M}^{2}}{2M} \\  
   {{\beta }_{U}}= & \tilde{\beta }\frac{\chi _{1-\tfrac{\alpha }{2},2M}^{2}}{2M}   
   {{\beta }_{U}}= & \tilde{\beta }\frac{\chi _{1-\tfrac{\alpha }{2},2M}^{2}}{2M}   
\end{align}\,\!</math>
\end{align}\,\!</math>


====Bounds on Lambda====
===Lambda===
=====Fisher Matrix Bounds=====
====Fisher Matrix Bounds====
The parameter <math>\lambda \,\!</math> must be positive, thus <math>\ln \lambda \,\!</math> is approximately treated as being normally distributed. These bounds are based on:
The parameter <math>\lambda \,\!</math> must be positive, thus <math>\ln \lambda \,\!</math> is approximately treated as being normally distributed. These bounds are based on:


::<math>\frac{\ln (\hat{\lambda })-\ln (\lambda )}{\sqrt{Var\left[ \ln (\hat{\lambda }) \right]}}\ \tilde{\ }\ N(0,1)\,\!</math>
:<math>\frac{\ln (\hat{\lambda })-\ln (\lambda )}{\sqrt{Var\left[ \ln (\hat{\lambda }) \right]}}\ \tilde{\ }\ N(0,1)\,\!</math>


The approximate confidence bounds on <math>\lambda \,\!</math> are given as:
The approximate confidence bounds on <math>\lambda \,\!</math> are given as:


::<math>C{{B}_{\lambda }}=\hat{\lambda }{{e}^{\pm {{z}_{\alpha }}\sqrt{Var(\hat{\lambda })}/\hat{\lambda }}}\,\!</math>
:<math>C{{B}_{\lambda }}=\hat{\lambda }{{e}^{\pm {{z}_{\alpha }}\sqrt{Var(\hat{\lambda })}/\hat{\lambda }}}\,\!</math>


where <math>\hat{\lambda }=\tfrac{n}{T_{K}^{{\hat{\beta }}}}\,\!</math>.
where <math>\hat{\lambda }=\tfrac{n}{T_{K}^{{\hat{\beta }}}}\,\!</math>.


The variance calculation is the same the equations given in the [[Confidence_Bounds_for_Repairable_Systems_Analysis#Bounds_on_Beta|Bounds on Beta]] section.
The variance calculation is the same the equations given in the confidence bounds on [[Confidence_Bounds_for_Repairable_Systems_Analysis#Beta|Beta]].
 
====Crow Bounds====
 
'''Failure Terminated'''
 
The confidence bounds on <math>\lambda \,\!</math> for failure terminated data are calculated using:
 
:<math>\begin{align}
  {{\lambda }_{L}}= & \frac{\chi _{\tfrac{\alpha }{2},2N}^{2}}{2\cdot \underset{q=1}{\overset{K}{\mathop{\sum }}}\,T_{q}^{^{\beta }}} \\
  {{\lambda }_{U}}= & \frac{\chi _{1-\tfrac{\alpha }{2},2N}^{2}}{2\cdot \underset{q=1}{\overset{K}{\mathop{\sum }}}\,T_{q}^{^{\beta }}} 
\end{align}\,\!</math>
 
where:
*<math>N\,\!</math> = total number of failures.
*<math>K\,\!</math> = number of systems.
*<math>{{T}_{q}}\,\!</math> = end time for the <math>{{q}^{th}}</math> system.


=====Crow Bounds=====
'''Time Terminated'''
'''Time Terminated'''


The confidence bounds on <math>\lambda \,\!</math> for time terminated data are calculated using:
The confidence bounds on <math>\lambda \,\!</math> for time terminated data are calculated using:


::<math>\begin{align}
:<math>\begin{align}
   {{\lambda }_{L}}= & \frac{\chi _{\tfrac{\alpha }{2},2N}^{2}}{2\cdot \underset{q=1}{\overset{K}{\mathop{\sum }}}\,T_{q}^{^{\beta }}} \\  
   {{\lambda }_{L}}= & \frac{\chi _{\tfrac{\alpha }{2},2N}^{2}}{2\cdot \underset{q=1}{\overset{K}{\mathop{\sum }}}\,T_{q}^{^{\beta }}} \\  
   {{\lambda }_{u}}= & \frac{\chi _{1-\tfrac{\alpha }{2},2N+2}^{2}}{2\cdot \underset{q=1}{\overset{K}{\mathop{\sum }}}\,T_{q}^{^{\beta }}}   
   {{\lambda }_{U}}= & \frac{\chi _{1-\tfrac{\alpha }{2},2N+2}^{2}}{2\cdot \underset{q=1}{\overset{K}{\mathop{\sum }}}\,T_{q}^{^{\beta }}}   
\end{align}\,\!</math>
\end{align}\,\!</math>


'''Failure Terminated'''
where:
*<math>N\,\!</math> = total number of failures.
*<math>K\,\!</math> = number of systems.
*<math>{{T}_{q}}\,\!</math> = end time for the <math>{{q}^{th}}</math> system.
 
===Cumulative Number of Failures===
====Fisher Matrix Bounds====
The cumulative number of failures, <math>N(t)\,\!</math>. must be positive, thus <math>\ln \left( N(t) \right)\,\!</math> is approximately treated as being normally distributed.
 
:<math>\frac{\ln (\hat{N}(t))-\ln (N(t))}{\sqrt{Var\left[ \ln \hat{N}(t) \right]}}\sim N(0,1)\,\!</math>
 
:<math>N(t)=\hat{N}(t){{e}^{\pm {{z}_{\alpha }}\sqrt{Var(\hat{N}(t))}/\hat{N}(t)}}\,\!</math>
 
where:


The confidence bounds on <math>\lambda \,\!</math> for failure terminated data are calculated using:
:<math>\hat{N}(t)=\hat{\lambda }{{t}^{\hat{\beta }}}\,\!</math>


::<math>\begin{align}
:<math>\begin{align}
   {{\lambda }_{L}}= & \frac{\chi _{\tfrac{\alpha }{2},2N}^{2}}{2\cdot \underset{q=1}{\overset{K}{\mathop{\sum }}}\,T_{q}^{^{\beta }}} \\
   Var(\hat{N}(t))= & {{\left( \frac{\partial N(t)}{\partial \beta } \right)}^{2}}Var(\hat{\beta })+{{\left( \frac{\partial N(t)}{\partial \lambda } \right)}^{2}}Var(\hat{\lambda }) \\
  {{\lambda }_{u}}= & \frac{\chi _{1-\tfrac{\alpha }{2},2N}^{2}}{2\cdot \underset{q=1}{\overset{K}{\mathop{\sum }}}\,T_{q}^{^{\beta }}}   
  & +2\left( \frac{\partial N(t)}{\partial \beta } \right)\left( \frac{\partial N(t)}{\partial \lambda } \right)cov(\hat{\beta },\hat{\lambda })  
\end{align}\,\!</math>
\end{align}\,\!</math>


====Bounds on Growth Rate====
The variance calculation is the same as the calculations in the confidence bounds on [[Confidence_Bounds_for_Repairable_Systems_Analysis#Beta|Beta]].
Since the growth rate is equal to <math>1-\beta \,\!</math>. the confidence bounds are:


::<math>\begin{align}
:<math>\begin{align}
   Gr.\text{ }Rat{{e}_{L}}= & 1-{{\beta }_{U}} \\  
   \frac{\partial N(t)}{\partial \beta }= & \hat{\lambda }{{t}^{\hat{\beta }}}\ln (t) \\  
   Gr.\text{ }Rat{{e}_{U}}= & 1-{{\beta }_{L}}   
   \frac{\partial N(t)}{\partial \lambda }= & t\hat{\beta }   
\end{align}\,\!</math>
\end{align}\,\!</math>


<math>{{\beta }_{L}}\,\!</math> and <math>{{\beta }_{U}}\,\!</math> are obtained using the methods described above in the [[Confidence_Bounds_for_Repairable_Systems_Analysis#Bounds_on_Beta|Bounds on Beta]] section.
====Crow Bounds====
The 2-sided confidence bounds on the cumulative number of failures are given by:


====Bounds on Cumulative MTBF====
:<math>N{{\left( t \right)}_{L}}=\frac{\chi _{\tfrac{\alpha }{2},2N}^{2}}{2\cdot S}\,\!</math>
=====Fisher Matrix Bounds=====
The cumulative MTBF, <math>{{m}_{c}}(t)\,\!</math>. must be positive, thus <math>\ln {{m}_{c}}(t)\,\!</math> is approximately treated as being normally distributed.


::<math>\frac{\ln ({{\hat{m}}_{c}}(t))-\ln ({{m}_{c}}(t))}{\sqrt{Var\left[ \ln ({{\hat{m}}_{c}}(t)) \right]}}\ \tilde{\ }\ N(0,1)\,\!</math>
:<math>N{{\left( t \right)}_{U}}=\frac{\chi _{1-\tfrac{\alpha }{2},2N+2}^{2}}{2\cdot S}\,\!</math>


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


::<math>CB={{\hat{m}}_{c}}(t){{e}^{\pm {{z}_{\alpha }}\sqrt{Var({{\hat{m}}_{c}}(t))}/{{\hat{m}}_{c}}(t)}}\,\!</math>
*<math>N\,\!</math> = total number of failures across all systems. This is not the number of failures up to time <math>t\,\!</math>.
*<math>S=\frac{\left( \frac{N}{{\hat{\lambda }}} \right)}{{{t}^{{\hat{\beta }}}}}\,\!</math>
*<math>t\,\!</math> = time at which calculations are being conducted.


:where:
===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 approximately treated as being normally distributed.


::<math>{{\hat{m}}_{c}}(t)=\frac{1}{\hat{\lambda }}{{t}^{1-\hat{\beta }}}\,\!</math>
:<math>\frac{\ln ({{\hat{\lambda }}_{c}}(t))-\ln ({{\lambda }_{c}}(t))}{\sqrt{Var\left[ \ln ({{\hat{\lambda }}_{c}}(t)) \right]}}\ \tilde{\ }\ N(0,1)\,\!</math>


::<math>\begin{align}
The approximate confidence bounds on the cumulative failure intensity are then estimated using:  
  Var({{\hat{m}}_{c}}(t))= & {{\left( \frac{\partial {{m}_{c}}(t)}{\partial \beta } \right)}^{2}}Var(\hat{\beta })+{{\left( \frac{\partial {{m}_{c}}(t)}{\partial \lambda } \right)}^{2}}Var(\hat{\lambda }) \\
  & +2\left( \frac{\partial {{m}_{c}}(t)}{\partial \beta } \right)\left( \frac{\partial {{m}_{c}}(t)}{\partial \lambda } \right)cov(\hat{\beta },\hat{\lambda })\, 
\end{align}\,\!</math>


The variance calculation is the same as the calculations given in the [[Confidence_Bounds_for_Repairable_Systems_Analysis#Bounds_on_Beta|Bounds on Beta]] section.
:<math>CB={{\hat{\lambda }}_{c}}(t){{e}^{\pm {{z}_{\alpha }}\sqrt{Var({{\hat{\lambda }}_{c}}(t))}/{{\hat{\lambda }}_{c}}(t)}}\,\!</math>


::<math>\begin{align}
where:  
  \frac{\partial {{m}_{c}}(t)}{\partial \beta }= & -\frac{1}{\hat{\lambda }}{{t}^{1-\hat{\beta }}}\ln (t) \\
  \frac{\partial {{m}_{c}}(t)}{\partial \lambda }= & -\frac{1}{{{\hat{\lambda }}^{2}}}{{t}^{1-\hat{\beta }}} 
\end{align}\,\!</math>


=====Crow Bounds=====
:<math>{{\hat{\lambda }}_{c}}(t)=\hat{\lambda }{{t}^{\hat{\beta }-1}}\,\!</math>
To calculate the Crow confidence bounds on cumulative MTBF, first calculate the Crow cumulative failure intensity confidence bounds:


::<math>C{{(t)}_{L}}=\frac{\chi _{\tfrac{\alpha }{2},2N}^{2}}{2\cdot t}\,\!</math>
and:  


::<math>C{{(t)}_{u}}=\frac{\chi _{1-\tfrac{\alpha }{2},2N+2}^{2}}{2\cdot t}\,\!</math>
:<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>


:Then
The variance calculation is the same as the calculations in the confidence bounds on [[Confidence_Bounds_for_Repairable_Systems_Analysis#Beta|Beta]].


::<math>\begin{align}
:<math>\begin{align}
   {{[MTB{{F}_{c}}]}_{L}}= & \frac{1}{C{{(t)}_{U}}} \\  
   \frac{\partial {{\lambda }_{c}}(t)}{\partial \beta }= & \hat{\lambda }{{t}^{\hat{\beta }-1}}\ln (t) \\  
   {{[MTB{{F}_{c}}]}_{U}}= & \frac{1}{C{{(t)}_{L}}}
   \frac{\partial {{\lambda }_{c}}(t)}{\partial \lambda }= & {{t}^{\hat{\beta }-1}}
\end{align}\,\!</math>
\end{align}\,\!</math>


====Bounds on Instantaneous MTBF====<!-- THIS SECTION HEADER IS LINKED FROM ANOTHER SECTION IN THIS
====Crow Bounds====
PAGE. IF YOU RENAME THE SECTION, YOU MUST UPDATE THE LINK(S). -->
The 2-sided confidence bounds on the cumulative failure intensity are given by:
=====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.


::<math>\frac{\ln ({{\hat{m}}_{i}}(t))-\ln ({{m}_{i}}(t))}{\sqrt{Var\left[ \ln ({{\hat{m}}_{i}}(t)) \right]}}\ \tilde{\ }\ N(0,1)\,\!</math>
:<math>CFI_L=\frac{\chi _{\tfrac{\alpha }{2},2N}^{2}}{2\cdot t \cdot S}\,\!</math>


The approximate confidence bounds on the instantaneous MTBF are then estimated from:  
:<math>CFI_U=\frac{\chi _{1-\tfrac{\alpha }{2},2N+2}^{2}}{2\cdot t \cdot S}\,\!</math>


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


:where:
*<math>N\,\!</math> = total number of failures across all systems. This is not the number of failures up to time <math>t\,\!</math>.
*<math>S=\frac{\left( \frac{N}{{\hat{\lambda }}} \right)}{{{t}^{{\hat{\beta }}}}}\,\!</math>
*<math>t\,\!</math> = time at which calculations are being conducted.


::<math>{{\hat{m}}_{i}}(t)=\frac{1}{\lambda \beta {{t}^{\beta -1}}}\,\!</math>
===Cumulative MTBF===
====Fisher Matrix Bounds====
The cumulative MTBF, <math>{{m}_{c}}(t)\,\!</math>. must be positive, thus <math>\ln {{m}_{c}}(t)\,\!</math> is approximately treated as being normally distributed.


::<math>\begin{align}
:<math>\frac{\ln ({{\hat{m}}_{c}}(t))-\ln ({{m}_{c}}(t))}{\sqrt{Var\left[ \ln ({{\hat{m}}_{c}}(t)) \right]}}\ \tilde{\ }\ N(0,1)\,\!</math>
  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 the calculations given in the [[Confidence_Bounds_for_Repairable_Systems_Analysis#Bounds_on_Beta|Bounds on Beta]] section.
The approximate confidence bounds on the cumulative MTBF are then estimated from:


::<math>\begin{align}
:<math>CB={{\hat{m}}_{c}}(t){{e}^{\pm {{z}_{\alpha }}\sqrt{Var({{\hat{m}}_{c}}(t))}/{{\hat{m}}_{c}}(t)}}\,\!</math>
  \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=====
where:
'''Failure Terminated Data'''


To calculate the bounds for failure terminated data, consider the following equation:
:<math>{{\hat{m}}_{c}}(t)=\frac{1}{\hat{\lambda }}{{t}^{1-\hat{\beta }}}\,\!</math>


::<math>G(\mu |n)=\mathop{}_{0}^{\infty }\frac{{{e}^{-x}}{{x}^{n-2}}}{(n-2)!}\underset{i=0}{\overset{n-1}{\mathop \sum }}\,\frac{1}{i!}{{\left( \frac{\mu }{x} \right)}^{i}}\exp (-\frac{\mu }{x})\,dx\,\!</math>
:<math>\begin{align}
  Var({{\hat{m}}_{c}}(t))= & {{\left( \frac{\partial {{m}_{c}}(t)}{\partial \beta } \right)}^{2}}Var(\hat{\beta })+{{\left( \frac{\partial {{m}_{c}}(t)}{\partial \lambda } \right)}^{2}}Var(\hat{\lambda }) \\  
  & +2\left( \frac{\partial {{m}_{c}}(t)}{\partial \beta } \right)\left( \frac{\partial {{m}_{c}}(t)}{\partial \lambda } \right)cov(\hat{\beta },\hat{\lambda })\,
\end{align}\,\!</math>


Find the values <math>{{p}_{1}}\,\!</math> and <math>{{p}_{2}}\,\!</math> by finding the solution <math>c\,\!</math> to <math>G({{n}^{2}}/c|n)=\xi \,\!</math> for <math>\xi =\tfrac{\alpha }{2}\,\!</math> and <math>\xi =1-\tfrac{\alpha }{2}\,\!</math>. respectively. If using the biased parameters, <math>\hat{\beta }\,\!</math> and <math>\hat{\lambda }\,\!</math>. then the upper and lower confidence bounds are:
The variance calculation is the same as the calculations given in the confidence bounds on [[Confidence_Bounds_for_Repairable_Systems_Analysis#Beta|Beta]].


::<math>\begin{align}
:<math>\begin{align}
   {{[MTB{{F}_{i}}]}_{L}}= & MTB{{F}_{i}}\cdot {{p}_{1}} \\  
   \frac{\partial {{m}_{c}}(t)}{\partial \beta }= & -\frac{1}{\hat{\lambda }}{{t}^{1-\hat{\beta }}}\ln (t) \\  
   {{[MTB{{F}_{i}}]}_{U}}= & MTB{{F}_{i}}\cdot {{p}_{2}}   
   \frac{\partial {{m}_{c}}(t)}{\partial \lambda }= & -\frac{1}{{{\hat{\lambda }}^{2}}}{{t}^{1-\hat{\beta }}}   
\end{align}\,\!</math>
\end{align}\,\!</math>


where <math>MTB{{F}_{i}}=\tfrac{1}{\hat{\lambda }\hat{\beta }{{t}^{\hat{\beta }-1}}}\,\!</math>. If using the unbiased parameters, <math>\bar{\beta }\,\!</math> and <math>\bar{\lambda }\,\!</math>. then the upper and lower confidence bounds are:
====Crow Bounds====
The 2-sided confidence bounds on the cumulative MTBF <math>(CMTBF)\,\!</math> are given by:


::<math>\begin{align}
:<math>\begin{align}
  {{[MTB{{F}_{i}}]}_{L}}= & MTB{{F}_{i}}\cdot \left( \frac{N-2}{N} \right)\cdot {{p}_{1}} \\  
& CMTBF_{L}=\frac{1}{CFI_{U}} \\  
  {{[MTB{{F}_{i}}]}_{U}}= & MTB{{F}_{i}}\cdot \left( \frac{N-2}{N} \right)\cdot {{p}_{2}}   
& CMTBF_{U}=\frac{1}{CFI_{L}}   
\end{align}\,\!</math>
\end{align}\,\!</math>


where <math>MTB{{F}_{i}}=\tfrac{1}{\hat{\lambda }\hat{\beta }{{t}^{\hat{\beta }-1}}}\,\!</math>.
where <math>CFI_L\,\!</math> and <math>CFI_U\,\!</math> are calculated using the process for the confidence bounds on [[Confidence Bounds for Repairable Systems Analysis#Crow_Bounds_4|cumulative failure intensity]].


'''Time Terminated Data'''
===Instantaneous MTBF===<!-- THIS SECTION HEADER IS LINKED FROM ANOTHER SECTION IN THIS
PAGE. IF YOU RENAME THE SECTION, YOU MUST UPDATE THE LINK(S). -->
====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.
 
:<math>\frac{\ln ({{\hat{m}}_{i}}(t))-\ln ({{m}_{i}}(t))}{\sqrt{Var\left[ \ln ({{\hat{m}}_{i}}(t)) \right]}}\ \tilde{\ }\ N(0,1)\,\!</math>


To calculate the bounds for time terminated data, consider the following equation where <math>{{I}_{1}}(.)\,\!</math> is the modified Bessel function of order one:  
The approximate confidence bounds on the instantaneous MTBF are then estimated from:  


::<math>H(x|k)=\underset{j=1}{\overset{k}{\mathop \sum }}\,\frac{{{x}^{2j-1}}}{{{2}^{2j-1}}(j-1)!j!{{I}_{1}}(x)}\,\!</math>
:<math>CB={{\hat{m}}_{i}}(t){{e}^{\pm {{z}_{\alpha }}\sqrt{Var({{\hat{m}}_{i}}(t))}/{{\hat{m}}_{i}}(t)}}\,\!</math>


Find the values <math>{{\Pi }_{1}}\,\!</math> and <math>{{\Pi }_{2}}\,\!</math> by finding the solution <math>x\,\!</math> to <math>H(x|k)=\tfrac{\alpha }{2}\,\!</math> and <math>H(x|k)=1-\tfrac{\alpha }{2}\,\!</math> in the cases corresponding to the lower and upper bounds, respectively.
where:


Calculate <math>\Pi =\tfrac{{{n}^{2}}}{4{{x}^{2}}}\,\!</math> for each case. If using the biased parameters, <math>\hat{\beta }\,\!</math> and <math>\hat{\lambda }\,\!</math>. then the upper and lower confidence bounds are:
:<math>{{\hat{m}}_{i}}(t)=\frac{1}{\lambda \beta {{t}^{\beta -1}}}\,\!</math>


::<math>\begin{align}
:<math>\begin{align}
   {{[MTB{{F}_{i}}]}_{L}}= & MTB{{F}_{i}}\cdot {{\Pi }_{1}} \\  
   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 }) \\
  {{[MTB{{F}_{i}}]}_{U}}= & MTB{{F}_{i}}\cdot {{\Pi }_{2}}   
  & +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>
\end{align}\,\!</math>


where <math>MTB{{F}_{i}}=\tfrac{1}{\hat{\lambda }\hat{\beta }{{t}^{\hat{\beta }-1}}}\,\!</math>. If using the unbiased parameters, <math>\bar{\beta }\,\!</math> and <math>\bar{\lambda }\,\!</math>. then the upper and lower confidence bounds are:
The variance calculation is the same as the calculations given in the confidence bounds on [[Confidence_Bounds_for_Repairable_Systems_Analysis#Beta|Beta]].


::<math>\begin{align}
:<math>\begin{align}
   {{[MTB{{F}_{i}}]}_{L}}= & MTB{{F}_{i}}\cdot \left( \frac{N-1}{N} \right)\cdot {{\Pi }_{1}} \\  
   \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) \\  
   {{[MTB{{F}_{i}}]}_{U}}= & MTB{{F}_{i}}\cdot \left( \frac{N-1}{N} \right)\cdot {{\Pi }_{2}}   
   \frac{\partial {{m}_{i}}(t)}{\partial \lambda }= & -\frac{1}{{{\hat{\lambda }}^{2}}\hat{\beta }}{{t}^{1-\hat{\beta }}}   
\end{align}\,\!</math>
\end{align}\,\!</math>


where <math>MTB{{F}_{i}}=\tfrac{1}{\hat{\lambda }\hat{\beta }{{t}^{\hat{\beta }-1}}}\,\!</math>.
====Crow Bounds====
'''Failure Terminated'''


====Bounds on Cumulative Failure Intensity====
For failure terminated data and the 2-sided confidence bounds on instantaneous MTBF <math>(IMTBF)\,\!</math>, consider the following equation:
=====Fisher Matrix Bounds=====
The cumulative failure intensity, <math>{{\lambda }_{c}}(t)\,\!</math> must be positive, thus <math>\ln {{\lambda }_{c}}(t)\,\!</math> is approximately treated as being normally distributed.


::<math>\frac{\ln ({{\hat{\lambda }}_{c}}(t))-\ln ({{\lambda }_{c}}(t))}{\sqrt{Var\left[ \ln ({{\hat{\lambda }}_{c}}(t)) \right]}}\ \tilde{\ }\ N(0,1)\,\!</math>
:<math>G(\mu |n)=\mathop{}_{0}^{\infty }\frac{{{e}^{-x}}{{x}^{n-2}}}{(n-2)!}\underset{i=0}{\overset{n-1}{\mathop \sum }}\,\frac{1}{i!}{{\left( \frac{\mu }{x} \right)}^{i}}\exp (-\frac{\mu }{x})\,dx\,\!</math>


The approximate confidence bounds on the cumulative failure intensity are then estimated using:
Find the values <math>{{p}_{1}}\,\!</math> and <math>{{p}_{2}}\,\!</math> by finding the solution
<math>G\left( \left. \frac{{{n}^{2}}}{c} \right|n \right)=\frac{\alpha }{2}</math> and <math>G\left( \left. \frac{{{n}^{2}}}{c} \right|n \right)=1-\frac{\alpha }{2}</math> for the lower and upper bounds, respectively.


::<math>CB={{\hat{\lambda }}_{c}}(t){{e}^{\pm {{z}_{\alpha }}\sqrt{Var({{\hat{\lambda }}_{c}}(t))}/{{\hat{\lambda }}_{c}}(t)}}\,\!</math>
If using the biased parameters, <math>\hat{\beta }\,\!</math> and <math>\hat{\lambda }\,\!</math>, then the upper and lower confidence bounds are:


:where:
:<math>\begin{align}
  {{IMTBF}_{L}}= & IMTBF\cdot {{p}_{1}} \\
  {{IMTBF}_{U}}= & IMTBF\cdot {{p}_{2}} 
\end{align}\,\!</math>


::<math>{{\hat{\lambda }}_{c}}(t)=\hat{\lambda }{{t}^{\hat{\beta }-1}}\,\!</math>
where <math>IMTBF=\tfrac{1}{\hat{\lambda }\hat{\beta }{{t}^{\hat{\beta }-1}}}\,\!</math>.


:and:  
If using the unbiased parameters, <math>\bar{\beta }\,\!</math> and <math>\bar{\lambda }\,\!</math>, then the upper and lower confidence bounds are:


::<math>\begin{align}
:<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 }) \\  
   {{IMTBF}_{L}}= & IMTBF\cdot \left( \frac{N-2}{N} \right)\cdot {{p}_{1}} \\  
  & +2\left( \frac{\partial {{\lambda }_{c}}(t)}{\partial \beta } \right)\left( \frac{\partial {{\lambda }_{c}}(t)}{\partial \lambda } \right)cov(\hat{\beta },\hat{\lambda })  
  {{IMTBF}_{U}}= & IMTBF\cdot \left( \frac{N-2}{N} \right)\cdot {{p}_{2}}   
\end{align}\,\!</math>
\end{align}\,\!</math>


The variance calculation is the same as the calculations in the [[Confidence_Bounds_for_Repairable_Systems_Analysis#Bounds_on_Beta|Bounds on Beta]] section.
where <math>IMTBF=\tfrac{1}{\bar{\lambda }\bar{\beta }{{t}^{\bar{\beta }-1}}}\,\!</math>.


::<math>\begin{align}
'''Time Terminated'''
  \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}}   
Consider the following equation where <math>{{I}_{1}}(.)\,\!</math> is the modified Bessel function of order one:  
 
:<math>H(x|k)=\underset{j=1}{\overset{k}{\mathop \sum }}\,\frac{{{x}^{2j-1}}}{{{2}^{2j-1}}(j-1)!j!{{I}_{1}}(x)}\,\!</math>
 
Find the values <math>{{\Pi }_{1}}\,\!</math> and <math>{{\Pi }_{2}}\,\!</math> by finding the solution <math>x\,\!</math> to <math>H(x|k)=\tfrac{\alpha }{2}\,\!</math> and <math>H(x|k)=1-\tfrac{\alpha }{2}\,\!</math> in the cases corresponding to the lower and upper bounds, respectively. Calculate <math>\Pi =\tfrac{4{{n}^{2}}}{{{x}^{2}}}\,\!</math> for each case.
 
If using the biased parameters, <math>\hat{\beta }\,\!</math> and <math>\hat{\lambda }\,\!</math>, then the upper and lower confidence bounds are:
 
:<math>\begin{align}
  {{IMTBF}_{L}}= & IMTBF\cdot {{\Pi }_{1}} \\  
  {{IMTBF}_{U}}= & IMTBF\cdot {{\Pi }_{2}}   
\end{align}\,\!</math>
\end{align}\,\!</math>


=====Crow Bounds=====
where <math>IMTBF=\tfrac{1}{\hat{\lambda }\hat{\beta }{{t}^{\hat{\beta }-1}}}\,\!</math>.
The Crow cumulative failure intensity confidence bounds are given by:


::<math>C{{(t)}_{L}}=\frac{\chi _{\tfrac{\alpha }{2},2N}^{2}}{2\cdot t}\,\!</math>
If using the unbiased parameters, <math>\bar{\beta }\,\!</math> and <math>\bar{\lambda }\,\!</math>, then the upper and lower confidence bounds are:


:<math>\begin{align}
  {{IMTBF}_{L}}= & IMTBF\cdot \left( \frac{N-1}{N} \right)\cdot {{\Pi }_{1}} \\
  {{IMTBF}_{U}}= & IMTBF\cdot \left( \frac{N-1}{N} \right)\cdot {{\Pi }_{2}} 
\end{align}\,\!</math>


::<math>C{{(t)}_{u}}=\frac{\chi _{1-\tfrac{\alpha }{2},2N+2}^{2}}{2\cdot t}\,\!</math>
where <math>IMTBF=\tfrac{1}{\bar{\lambda }\bar{\beta }{{t}^{\bar{\beta }-1}}}\,\!</math>.


====Bounds on Instantaneous Failure Intensity====<!-- THIS SECTION HEADER IS LINKED FROM ANOTHER SECTION IN THIS PAGE. IF YOU RENAME THE SECTION, YOU MUST UPDATE THE LINK(S). -->
===Instantaneous Failure Intensity===<!-- THIS SECTION HEADER IS LINKED FROM ANOTHER SECTION IN THIS PAGE. IF YOU RENAME THE SECTION, YOU MUST UPDATE THE LINK(S). -->
=====Fisher Matrix Bounds=====
====Fisher Matrix Bounds====
The instantaneous failure intensity, <math>{{\lambda }_{i}}(t)\,\!</math>. must be positive, thus <math>\ln {{\lambda }_{i}}(t)\,\!</math> is approximately treated as being normally distributed.  
The instantaneous failure intensity, <math>{{\lambda }_{i}}(t)\,\!</math>. must be positive, thus <math>\ln {{\lambda }_{i}}(t)\,\!</math> is approximately treated as being normally distributed.  


::<math>\frac{\ln ({{\hat{\lambda }}_{i}}(t))-\ln ({{\lambda }_{i}}(t))}{\sqrt{Var\left[ \ln ({{\hat{\lambda }}_{i}}(t)) \right]}}\sim N(0,1)\,\!</math>
:<math>\frac{\ln ({{\hat{\lambda }}_{i}}(t))-\ln ({{\lambda }_{i}}(t))}{\sqrt{Var\left[ \ln ({{\hat{\lambda }}_{i}}(t)) \right]}}\sim N(0,1)\,\!</math>


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


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


where <math>{{\lambda }_{i}}(t)=\lambda \beta {{t}^{\beta -1}}\,\!</math> and:  
where <math>{{\lambda }_{i}}(t)=\lambda \beta {{t}^{\beta -1}}\,\!</math> and:  


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


The variance calculation is the same as the calculations in the [[Confidence_Bounds_for_Repairable_Systems_Analysis#Bounds_on_Beta|Bounds on Beta]] section.
The variance calculation is the same as the calculations in the confidence bounds on [[Confidence_Bounds_for_Repairable_Systems_Analysis#Beta|Beta]].


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


=====Crow Bounds=====
====Crow Bounds====
The Crow instantaneous failure intensity confidence bounds are given as:  
The 2-sided confidence bounds on the instantaneous failure intensity <math>(IFI)\,\!</math> are given by:


::<math>\begin{align}
:<math>\begin{align}
   {{[{{\lambda }_{i}}(t)]}_{L}}= & \frac{1}{{{[MTB{{F}_{i}}]}_{U}}} \\  
   {IFI_{L}}= & \frac{1}{{IMTBF}_{U}} \\  
   {{[{{\lambda }_{i}}(t)]}_{U}}= & \frac{1}{{{[MTB{{F}_{i}}]}_{L}}}   
   {IFI_{U}}= & \frac{1}{{IMTBF}_{L}}   
\end{align}\,\!</math>
\end{align}\,\!</math>


====Bounds on Time Given Cumulative MTBF====
where <math>IMTB{{F}_{L}}\,\!</math> and <math>IMTB{{F}_{U}}\,\!</math> are calculated using the process presented for the confidence bounds on the [[Confidence Bounds for Repairable Systems Analysis#Instantaneous_MTBF|instantaneous MTBF]].
=====Fisher Matrix Bounds=====
 
===Time Given Cumulative Failure Intensity===
====Fisher Matrix Bounds====
The time, <math>T\,\!</math>. must be positive, thus <math>\ln T\,\!</math> is approximately treated as being normally distributed.  
The time, <math>T\,\!</math>. must be positive, thus <math>\ln T\,\!</math> is approximately treated as being normally distributed.  


::<math>\frac{\ln (\hat{T})-\ln (T)}{\sqrt{Var\left[ \ln (\hat{T}) \right]}}\ \tilde{\ }\ N(0,1)\,\!</math>
:<math>\frac{\ln (\hat{T})-\ln (T)}{\sqrt{Var\left[ \ln \hat{T} \right]}}\ \tilde{\ }\ N(0,1)\,\!</math>


The confidence bounds on the time are given by:  
The confidence bounds on the time are given by:  


::<math>CB=\hat{T}{{e}^{\pm {{z}_{\alpha }}\sqrt{Var(\hat{T})}/\hat{T}}}\,\!</math>
:<math>CB=\hat{T}{{e}^{\pm {{z}_{\alpha }}\sqrt{Var(\hat{T})}/\hat{T}}}\,\!</math>


:where:  
where:  


::<math>Var(\hat{T})={{\left( \frac{\partial T}{\partial \beta } \right)}^{2}}Var(\hat{\beta })+{{\left( \frac{\partial T}{\partial \lambda } \right)}^{2}}Var(\hat{\lambda })+2\left( \frac{\partial T}{\partial \beta } \right)\left( \frac{\partial T}{\partial \lambda } \right)cov(\hat{\beta },\hat{\lambda })\,\!</math>
:<math>Var(\hat{T})={{\left( \frac{\partial T}{\partial \beta } \right)}^{2}}Var(\hat{\beta })+{{\left( \frac{\partial T}{\partial \lambda } \right)}^{2}}Var(\hat{\lambda })+2\left( \frac{\partial T}{\partial \beta } \right)\left( \frac{\partial T}{\partial \lambda } \right)cov(\hat{\beta },\hat{\lambda })\,\!</math>


The variance calculation is the same as the calculations in the [[Confidence_Bounds_for_Repairable_Systems_Analysis#Bounds_on_Beta|Bounds on Beta]] section.
The variance calculation is the same as the calculations given in the confidence bounds on [[Confidence_Bounds_for_Repairable_Systems_Analysis#Beta|Beta]].


::<math>\hat{T}={{(\lambda \cdot {{m}_{c}})}^{1/(1-\beta )}}\,\!</math>
:<math>\hat{T}={{\left( \frac{{{\lambda }_{c}}(T)}{\lambda } \right)}^{1/(\beta -1)}}\,\!</math>


::<math>\begin{align}
:<math>\begin{align}
   \frac{\partial T}{\partial \beta }= & \frac{{{(\lambda \cdot {{m}_{c}})}^{1/(1-\beta )}}\ln (\lambda \cdot {{m}_{c}})}{{{(1-\beta )}^{2}}} \\  
   \frac{\partial T}{\partial \beta }= & \frac{-{{\left( \tfrac{{{\lambda }_{c}}(T)}{\lambda } \right)}^{1/(\beta -1)}}\ln \left( \tfrac{{{\lambda }_{c}}(T)}{\lambda } \right)}{{{(1-\beta )}^{2}}} \\  
   \frac{\partial T}{\partial \lambda }= & \frac{{{(\lambda \cdot {{m}_{c}})}^{1/(1-\beta )}}}{\lambda (1-\beta )}   
   \frac{\partial T}{\partial \lambda }= & {{\left( \frac{{{\lambda }_{c}}(T)}{\lambda } \right)}^{1/(\beta -1)}}\frac{1}{\lambda (1-\beta )}   
\end{align}\,\!</math>
\end{align}\,\!</math>


====Crow Bounds====
The 2-sided confidence bounds on time given cumulative failure intensity <math>(CFI)\,\!</math> are given by:


=====Crow Bounds=====
:<math>\hat{t}={{\left( \frac{CFI}{{\hat{\lambda }}} \right)}^{\tfrac{1}{\hat{\beta }-1}}}\,\!</math>
Step 1: Calculate:


::<math>\hat{T}={{\left( \frac{{{\lambda }_{c}}(T)}{{\hat{\lambda }}} \right)}^{\tfrac{1}{\beta -1}}}\,\!</math>
Then estimate, the number of failures, <math>N\,\!</math>, such that:


Step 2: Estimate the number of failures:
:<math>N=\hat{\lambda }{{\hat{t}}^{{\hat{\beta }}}}\,\!</math>


::<math>N(\hat{T})=\hat{\lambda }{{\hat{T}}^{{\hat{\beta }}}}\,\!</math>
The lower and upper confidence bounds on time are then estimated using:


Step 3: Obtain the confidence bounds on time given the cumulative failure intensity by solving for <math>{{t}_{l}}\,\!</math> and <math>{{t}_{u}}\,\!</math> in the following equations:
:<math>\begin{align}
 
  {{t}_{L}}= & \frac{\chi _{\tfrac{\alpha }{2},2N}^{2}}{2\cdot CFI} \\  
::<math>\begin{align}
  {{t}_{U}}= & \frac{\chi _{1-\tfrac{\alpha }{2},2N+2}^{2}}{2\cdot CFI}   
  & {{t}_{l}}= & \frac{\chi _{\tfrac{\alpha }{2},2N}^{2}}{2\cdot {{\lambda }_{c}}(T)} \\  
& {{t}_{u}}= & \frac{\chi _{1-\tfrac{\alpha }{2},2N+2}^{2}}{2\cdot {{\lambda }_{c}}(T)}   
\end{align}\,\!</math>
\end{align}\,\!</math>


<br>
===Time Given Cumulative MTBF===
 
====Fisher Matrix Bounds====
====Bounds on Time Given Instantaneous MTBF====
=====Fisher Matrix Bounds=====
The time, <math>T\,\!</math>. must be positive, thus <math>\ln T\,\!</math> is approximately treated as being normally distributed.  
The time, <math>T\,\!</math>. must be positive, thus <math>\ln T\,\!</math> is approximately treated as being normally distributed.  


::<math>\frac{\ln (\hat{T})-\ln (T)}{\sqrt{Var\left[ \ln (\hat{T}) \right]}}\ \tilde{\ }\ N(0,1)\,\!</math>
:<math>\frac{\ln (\hat{T})-\ln (T)}{\sqrt{Var\left[ \ln (\hat{T}) \right]}}\ \tilde{\ }\ N(0,1)\,\!</math>


The confidence bounds on the time are given by:  
The confidence bounds on the time are given by:  


::<math>CB=\hat{T}{{e}^{\pm {{z}_{\alpha }}\sqrt{Var(\hat{T})}/\hat{T}}}\,\!</math>
:<math>CB=\hat{T}{{e}^{\pm {{z}_{\alpha }}\sqrt{Var(\hat{T})}/\hat{T}}}\,\!</math>


:where:  
where:  


::<math>Var(\hat{T})={{\left( \frac{\partial T}{\partial \beta } \right)}^{2}}Var(\hat{\beta })+{{\left( \frac{\partial T}{\partial \lambda } \right)}^{2}}Var(\hat{\lambda })+2\left( \frac{\partial T}{\partial \beta } \right)\left( \frac{\partial T}{\partial \lambda } \right)cov(\hat{\beta },\hat{\lambda })\,\!</math>
:<math>Var(\hat{T})={{\left( \frac{\partial T}{\partial \beta } \right)}^{2}}Var(\hat{\beta })+{{\left( \frac{\partial T}{\partial \lambda } \right)}^{2}}Var(\hat{\lambda })+2\left( \frac{\partial T}{\partial \beta } \right)\left( \frac{\partial T}{\partial \lambda } \right)cov(\hat{\beta },\hat{\lambda })\,\!</math>


The variance calculation is the same as the calculations in the [[Confidence_Bounds_for_Repairable_Systems_Analysis#Bounds_on_Beta|Bounds on Beta]] section.
The variance calculation is the same as the calculations in the confidence bounds on [[Confidence_Bounds_for_Repairable_Systems_Analysis#Beta|Beta]].


::<math>\hat{T}={{(\lambda \beta \cdot MTB{{F}_{i}})}^{1/(1-\beta )}}\,\!</math>
:<math>\hat{T}={{(\lambda \cdot {{m}_{c}})}^{1/(1-\beta )}}\,\!</math>


::<math>\begin{align}
:<math>\begin{align}
   \frac{\partial T}{\partial \beta }= & {{\left( \lambda \beta \cdot MTB{{F}_{i}} \right)}^{1/(1-\beta )}}[\frac{1}{{{(1-\beta )}^{2}}}\ln (\lambda \beta \cdot MTB{{F}_{i}})+\frac{1}{\beta (1-\beta )}] \\  
   \frac{\partial T}{\partial \beta }= & \frac{{{(\lambda \cdot {{m}_{c}})}^{1/(1-\beta )}}\ln (\lambda \cdot {{m}_{c}})}{{{(1-\beta )}^{2}}} \\  
   \frac{\partial T}{\partial \lambda }= & \frac{{{(\lambda \beta \cdot MTB{{F}_{i}})}^{1/(1-\beta )}}}{\lambda (1-\beta )}   
   \frac{\partial T}{\partial \lambda }= & \frac{{{(\lambda \cdot {{m}_{c}})}^{1/(1-\beta )}}}{\lambda (1-\beta )}   
\end{align}\,\!</math>
\end{align}\,\!</math>


=====Crow Bounds=====
====Crow Bounds====
Step 1: Calculate the confidence bounds on the instantaneous MTBF as presented in Section 5.5.2.[Per Kate: THE SECTION 5.5.2 IS ABOUT THE CHI-SQUARED TEST FOR GROUPED DATA. NEED TO ASK SME TO VERIFY THE CORRECT SECTION TO LINK TO.]
The 2-sided confidence bounds on time given cumulative MTBF <math>(CMTBF)\,\!</math> are estimated using the process for the confidence bounds on [[Confidence Bounds for Repairable Systems Analysis#Time_Given_Cumulative_Failure_Intensity|time given cumulative failure intensity]] <math>(CFI)\,\!</math> where <math>CFI=\frac{1}{CMTBF}\,\!</math>.


Step 2: Calculate the bounds on time as follows.
===Time Given Instantaneous MTBF===
====Fisher Matrix Bounds====
The time, <math>T\,\!</math>. must be positive, thus <math>\ln T\,\!</math> is approximately treated as being normally distributed.  


'''Failure Terminated Data'''
:<math>\frac{\ln (\hat{T})-\ln (T)}{\sqrt{Var\left[ \ln (\hat{T}) \right]}}\ \tilde{\ }\ N(0,1)\,\!</math>


::<math>\hat{T}={{(\frac{\lambda \beta \cdot MTB{{F}_{i}}}{c})}^{1/(1-\beta )}}\,\!</math>
The confidence bounds on the time are given by:  


So the lower an upper bounds on time are:
:<math>CB=\hat{T}{{e}^{\pm {{z}_{\alpha }}\sqrt{Var(\hat{T})}/\hat{T}}}\,\!</math>


::<math>{{\hat{T}}_{L}}={{(\frac{\lambda \beta \cdot MTB{{F}_{i}}}{{{c}_{1}}})}^{1/(1-\beta )}}\,\!</math>
where:  


:<math>Var(\hat{T})={{\left( \frac{\partial T}{\partial \beta } \right)}^{2}}Var(\hat{\beta })+{{\left( \frac{\partial T}{\partial \lambda } \right)}^{2}}Var(\hat{\lambda })+2\left( \frac{\partial T}{\partial \beta } \right)\left( \frac{\partial T}{\partial \lambda } \right)cov(\hat{\beta },\hat{\lambda })\,\!</math>


::<math>{{\hat{T}}_{U}}={{(\frac{\lambda \beta \cdot MTB{{F}_{i}}}{{{c}_{2}}})}^{1/(1-\beta )}}\,\!</math>
The variance calculation is the same as the calculations in the confidence bounds on [[Confidence_Bounds_for_Repairable_Systems_Analysis#Beta|Beta]].


'''Time Terminated Data'''
:<math>\hat{T}={{(\lambda \beta \cdot MTB{{F}_{i}})}^{1/(1-\beta )}}\,\!</math>


::<math>\hat{T}={{(\frac{\lambda \beta \cdot MTB{{F}_{i}}}{\Pi })}^{1/(1-\beta )}}\,\!</math>
:<math>\begin{align}
  \frac{\partial T}{\partial \beta }= & {{\left( \lambda \beta \cdot MTB{{F}_{i}} \right)}^{1/(1-\beta )}}[\frac{1}{{{(1-\beta )}^{2}}}\ln (\lambda \beta \cdot MTB{{F}_{i}})+\frac{1}{\beta (1-\beta )}] \\
  \frac{\partial T}{\partial \lambda }= & \frac{{{(\lambda \beta \cdot MTB{{F}_{i}})}^{1/(1-\beta )}}}{\lambda (1-\beta )} 
\end{align}\,\!</math>


So the lower and upper bounds on time are:
====Crow Bounds====
 
::<math>{{\hat{T}}_{L}}={{(\frac{\lambda \beta \cdot MTB{{F}_{i}}}{{{\Pi }_{1}}})}^{1/(1-\beta )}}\,\!</math>


'''Failure Terminated'''


::<math>{{\hat{T}}_{U}}={{(\frac{\lambda \beta \cdot MTB{{F}_{i}}}{{{\Pi }_{2}}})}^{1/(1-\beta )}}\,\!</math>
Calculate the constants <math>p_1\,\!</math> and <math>p_2\,\!</math> using procedures described for the confidence bounds on [[Confidence Bounds for Repairable Systems Analysis#Instantaneous_MTBF|instantaneous MTBF]]. The lower and upper confidence bounds on time are then given by:


====Bounds on Time Given Cumulative Failure Intensity====
:<math>{{\hat{t}}_{L}}={{\left( \frac{\lambda \beta \cdot IMTBF}{{{p}_{1}}} \right)}^{\tfrac{1}{1-\beta }}}</math>
=====Fisher Matrix Bounds=====
The time, <math>T\,\!</math>. must be positive, thus <math>\ln T\,\!</math> is approximately treated as being normally distributed.


::<math>\frac{\ln (\hat{T})-\ln (T)}{\sqrt{Var\left[ \ln \hat{T} \right]}}\ \tilde{\ }\ N(0,1)\,\!</math>
:<math>{{\hat{t}}_{U}}={{\left( \frac{\lambda \beta \cdot IMTBF}{{{p}_{2}}} \right)}^{\tfrac{1}{1-\beta }}}</math>
 
The confidence bounds on the time are given by:
 
::<math>CB=\hat{T}{{e}^{\pm {{z}_{\alpha }}\sqrt{Var(\hat{T})}/\hat{T}}}\,\!</math>
 
:where:
 
::<math>Var(\hat{T})={{\left( \frac{\partial T}{\partial \beta } \right)}^{2}}Var(\hat{\beta })+{{\left( \frac{\partial T}{\partial \lambda } \right)}^{2}}Var(\hat{\lambda })+2\left( \frac{\partial T}{\partial \beta } \right)\left( \frac{\partial T}{\partial \lambda } \right)cov(\hat{\beta },\hat{\lambda })\,\!</math>
 
The variance calculation is the same as the calculations given in the [[Confidence_Bounds_for_Repairable_Systems_Analysis#Bounds_on_Beta|Bounds on Beta]] section.
 
::<math>\hat{T}={{\left( \frac{{{\lambda }_{c}}(T)}{\lambda } \right)}^{1/(\beta -1)}}\,\!</math>


'''Time Terminated'''


::<math>\begin{align}
Calculate the constants <math>{{\Pi }_{1}}\,\!</math> and <math>{{\Pi }_{2}}\,\!</math> using procedures described for the confidence bounds on [[Confidence Bounds for Repairable Systems Analysis#Instantaneous_MTBF|instantaneous MTBF]]. The lower and upper confidence bounds on time are then given by:
  \frac{\partial T}{\partial \beta }= & \frac{-{{\left( \tfrac{{{\lambda }_{c}}(T)}{\lambda } \right)}^{1/(\beta -1)}}\ln \left( \tfrac{{{\lambda }_{c}}(T)}{\lambda } \right)}{{{(1-\beta )}^{2}}} \\
  \frac{\partial T}{\partial \lambda }= & {{\left( \frac{{{\lambda }_{c}}(T)}{\lambda } \right)}^{1/(\beta -1)}}\frac{1}{\lambda (1-\beta )} 
\end{align}\,\!</math>


=====Crow Bounds=====
:<math>{{\hat{t}}_{L}}={{\left( \frac{\lambda \beta \cdot IMTBF}{{{\Pi }_{1}}} \right)}^{\tfrac{1}{1-\beta }}}\,\!</math>
Step 1: Calculate:


::<math>\hat{T}={{\left( \frac{{{\lambda }_{c}}(T)}{{\hat{\lambda }}} \right)}^{\tfrac{1}{\beta -1}}}\,\!</math>
:<math>{{\hat{t}}_{U}}={{\left( \frac{\lambda \beta \cdot IMTBF}{{{\Pi }_{2}}} \right)}^{\tfrac{1}{1-\beta }}}\,\!</math>


Step 2: Estimate the number of failures:
===Time Given Instantaneous Failure Intensity===
 
====Fisher Matrix Bounds====
::<math>N(\hat{T})=\hat{\lambda }{{\hat{T}}^{{\hat{\beta }}}}\,\!</math>
 
Step 3: Obtain the confidence bounds on time given the cumulative failure intensity by solving for <math>{{t}_{l}}\,\!</math> and <math>{{t}_{u}}\,\!</math> in the following equations:
 
::<math>\begin{align}
  {{t}_{l}}= & \frac{\chi _{\tfrac{\alpha }{2},2N}^{2}}{2\cdot {{\lambda }_{c}}(T)} \\
  {{t}_{u}}= & \frac{\chi _{1-\tfrac{\alpha }{2},2N+2}^{2}}{2\cdot {{\lambda }_{c}}(T)} 
\end{align}\,\!</math>
 
====Bounds on Time Given Instantaneous Failure Intensity====
=====Fisher Matrix Bounds=====
These bounds are based on:  
These bounds are based on:  


::<math>\frac{\ln (\hat{T})-\ln (T)}{\sqrt{Var\left[ \ln (\hat{T}) \right]}}\sim N(0,1)\,\!</math>
:<math>\frac{\ln (\hat{T})-\ln (T)}{\sqrt{Var\left[ \ln (\hat{T}) \right]}}\sim N(0,1)\,\!</math>


The confidence bounds on the time are given by:
The confidence bounds on the time are given by:


::<math>CB=\hat{T}{{e}^{\pm {{z}_{\alpha }}\sqrt{Var(\hat{T})}/\hat{T}}}\,\!</math>
:<math>CB=\hat{T}{{e}^{\pm {{z}_{\alpha }}\sqrt{Var(\hat{T})}/\hat{T}}}\,\!</math>


:where:  
where:  


::<math>\begin{align}
:<math>\begin{align}
   Var(\hat{T})= & {{\left( \frac{\partial T}{\partial \beta } \right)}^{2}}Var(\hat{\beta })+{{\left( \frac{\partial T}{\partial \lambda } \right)}^{2}}Var(\hat{\lambda }) \\  
   Var(\hat{T})= & {{\left( \frac{\partial T}{\partial \beta } \right)}^{2}}Var(\hat{\beta })+{{\left( \frac{\partial T}{\partial \lambda } \right)}^{2}}Var(\hat{\lambda }) \\  
   & +2\left( \frac{\partial T}{\partial \beta } \right)\left( \frac{\partial T}{\partial \lambda } \right)cov(\hat{\beta },\hat{\lambda })   
   & +2\left( \frac{\partial T}{\partial \beta } \right)\left( \frac{\partial T}{\partial \lambda } \right)cov(\hat{\beta },\hat{\lambda })   
\end{align}\,\!</math>
\end{align}\,\!</math>


The variance calculation is the same as the calculations given in the [[Confidence_Bounds_for_Repairable_Systems_Analysis#Bounds_on_Beta|Bounds on Beta]] section.
The variance calculation is the same as the calculations given in the confidence bounds on [[Confidence_Bounds_for_Repairable_Systems_Analysis#Beta|Beta]].


::<math>\hat{T}={{\left( \frac{{{\lambda }_{i}}(T)}{\lambda \cdot \beta } \right)}^{1/(\beta -1)}}\,\!</math>
:<math>\hat{T}={{\left( \frac{{{\lambda }_{i}}(T)}{\lambda \cdot \beta } \right)}^{1/(\beta -1)}}\,\!</math>


::<math>\begin{align}
:<math>\begin{align}
   \frac{\partial T}{\partial \beta }= & {{\left( \frac{{{\lambda }_{i}}(T)}{\lambda \cdot \beta } \right)}^{1/(\beta -1)}}[-\frac{\ln (\tfrac{{{\lambda }_{i}}(T)}{\lambda \cdot \beta })}{{{(\beta -1)}^{2}}}+\frac{1}{\beta (1-\beta )}] \\  
   \frac{\partial T}{\partial \beta }= & {{\left( \frac{{{\lambda }_{i}}(T)}{\lambda \cdot \beta } \right)}^{1/(\beta -1)}}[-\frac{\ln (\tfrac{{{\lambda }_{i}}(T)}{\lambda \cdot \beta })}{{{(\beta -1)}^{2}}}+\frac{1}{\beta (1-\beta )}] \\  
   \frac{\partial T}{\partial \lambda }= & {{\left( \frac{{{\lambda }_{i}}(T)}{\lambda \cdot \beta } \right)}^{1/(\beta -1)}}\frac{1}{\lambda (1-\beta )}   
   \frac{\partial T}{\partial \lambda }= & {{\left( \frac{{{\lambda }_{i}}(T)}{\lambda \cdot \beta } \right)}^{1/(\beta -1)}}\frac{1}{\lambda (1-\beta )}   
\end{align}\,\!</math>
\end{align}\,\!</math>


=====Crow Bounds=====
====Crow Bounds====
Step 1: Calculate <math>{{\lambda }_{i}}(T)=\tfrac{1}{MTB{{F}_{i}}}\,\!</math>.
The 2-sided confidence bounds on time given instantaneous failure intensity <math>(IFI)\,\!</math> are estimated using the process for the confidence bounds on [[Confidence Bounds for Repairable Systems Analysis#Time_Given_Instantaneous_MTBF|time given instantaneous MTBF]] where <math>IMTBF=\frac{1}{IFI}\,\!</math>.


Step 2: Use the equations in the [[Confidence_Bounds_for_Repairable_Systems_Analysis#Bounds_on_Time_Given_Instantaneous_MTBF|Bounds on Time Given Instantaneous MTBF]] section to calculate the bounds on time given the instantaneous failure intensity.
===Reliability===
 
====Fisher Matrix Bounds====
====Bounds on Reliability====
=====Fisher Matrix Bounds=====
These bounds are based on:  
These bounds are based on:  


::<math>\log it(\hat{R}(t))\sim N(0,1)\,\!</math>
:<math>\log it(\hat{R}(t))\sim N(0,1)\,\!</math>


 
:<math>\log it(\hat{R}(t))=\ln \left\{ \frac{\hat{R}(t)}{1-\hat{R}(t)} \right\}\,\!</math>
::<math>\log it(\hat{R}(t))=\ln \left\{ \frac{\hat{R}(t)}{1-\hat{R}(t)} \right\}\,\!</math>


The confidence bounds on reliability are given by:  
The confidence bounds on reliability are given by:  


::<math>CB=\frac{\hat{R}(t)}{\hat{R}(t)+(1-\hat{R}(t)){{e}^{\pm {{z}_{\alpha }}\sqrt{Var(\hat{R}(t))}/\left[ \hat{R}(t)(1-\hat{R}(t)) \right]}}}\,\!</math>
:<math>CB=\frac{\hat{R}(t)}{\hat{R}(t)+(1-\hat{R}(t)){{e}^{\pm {{z}_{\alpha }}\sqrt{Var(\hat{R}(t))}/\left[ \hat{R}(t)(1-\hat{R}(t)) \right]}}}\,\!</math>


:<math>Var(\hat{R}(t))={{\left( \frac{\partial R}{\partial \beta } \right)}^{2}}Var(\hat{\beta })+{{\left( \frac{\partial R}{\partial \lambda } \right)}^{2}}Var(\hat{\lambda })+2\left( \frac{\partial R}{\partial \beta } \right)\left( \frac{\partial R}{\partial \lambda } \right)cov(\hat{\beta },\hat{\lambda })\,\!</math>


::<math>Var(\hat{R}(t))={{\left( \frac{\partial R}{\partial \beta } \right)}^{2}}Var(\hat{\beta })+{{\left( \frac{\partial R}{\partial \lambda } \right)}^{2}}Var(\hat{\lambda })+2\left( \frac{\partial R}{\partial \beta } \right)\left( \frac{\partial R}{\partial \lambda } \right)cov(\hat{\beta },\hat{\lambda })\,\!</math>
The variance calculation is the same as the calculations in the confidence bounds on [[Confidence_Bounds_for_Repairable_Systems_Analysis#Beta|Beta]].


The variance calculation is the same as the calculations in the [[Confidence_Bounds_for_Repairable_Systems_Analysis#Bounds_on_Beta|Bounds on Beta]] section.
:<math>\begin{align}
 
::<math>\begin{align}
   \frac{\partial R}{\partial \beta }= & {{e}^{-[\hat{\lambda }{{(t+d)}^{\hat{\beta }}}-\hat{\lambda }{{t}^{\hat{\beta }}}]}}[\lambda {{t}^{\hat{\beta }}}\ln (t)-\lambda {{(t+d)}^{\hat{\beta }}}\ln (t+d)] \\  
   \frac{\partial R}{\partial \beta }= & {{e}^{-[\hat{\lambda }{{(t+d)}^{\hat{\beta }}}-\hat{\lambda }{{t}^{\hat{\beta }}}]}}[\lambda {{t}^{\hat{\beta }}}\ln (t)-\lambda {{(t+d)}^{\hat{\beta }}}\ln (t+d)] \\  
   \frac{\partial R}{\partial \lambda }= & {{e}^{-[\hat{\lambda }{{(t+d)}^{\hat{\beta }}}-\hat{\lambda }{{t}^{\hat{\beta }}}]}}[{{t}^{\hat{\beta }}}-{{(t+d)}^{\hat{\beta }}}]   
   \frac{\partial R}{\partial \lambda }= & {{e}^{-[\hat{\lambda }{{(t+d)}^{\hat{\beta }}}-\hat{\lambda }{{t}^{\hat{\beta }}}]}}[{{t}^{\hat{\beta }}}-{{(t+d)}^{\hat{\beta }}}]   
\end{align}\,\!</math>
\end{align}\,\!</math>


=====Crow Bounds=====
====Crow Bounds====
'''Failure Terminated Data'''
'''Failure Terminated'''


With failure terminated data, the 100( <math>1-\alpha \,\!</math> )% confidence interval for the current reliability at time <math>t\,\!</math> in a specified mission time <math>d\,\!</math> is:  
For failure terminated data, the 100( <math>1-\alpha \,\!</math> )% confidence interval on the current reliability at time <math>t\,\!</math> for a specified mission duration <math>d\,\!</math> is:  


::<math>({{[\hat{R}(d)]}^{\tfrac{1}{{{p}_{1}}}}},{{[\hat{R}(d)]}^{\tfrac{1}{{{p}_{2}}}}})\,\!</math>
:<math>\left( {{\left[ \hat{R}\left( d \right) \right]}^{\tfrac{1}{{{p}_{1}}}}},{{\left[ \hat{R}\left( d \right) \right]}^{\tfrac{1}{{{p}_{2}}}}} \right)\,\!</math>


:where
where:


::<math>\hat{R}(\tau )={{e}^{-[\hat{\lambda }{{(t+\tau )}^{\hat{\beta }}}-\hat{\lambda }{{t}^{\hat{\beta }}}]}}\,\!</math>
*<math>\hat{R}\left( d \right)={{e}^{-\left[ \hat{\lambda }{{\left( t+d \right)}^{{\hat{\beta }}}}-\hat{\lambda }{{t}^{{\hat{\beta }}}} \right]}}\,\!</math>
*<math>p_1\,\!</math> and <math>p_2\,\!</math> are obtained from the confidence bounds on [[Confidence Bounds for Repairable Systems Analysis#Instantaneous_MTBF|instantaneous MTBF]] for failure terminated data.


<math>{{p}_{1}}\,\!</math> and <math>{{p}_{2}}\,\!</math> can be obtained from the equation for failure terminated data in the [[Confidence_Bounds_for_Repairable_Systems_Analysis#Bounds_on_Instantaneous_MTBF|Bounds on Instantaneous MTBF]] section.
'''Time Terminated'''


'''Time Terminated Data'''
For time terminated data, the 100( <math>1-\alpha \,\!</math> )% confidence interval on the current reliability at time <math>t\,\!</math> for a specified mission duration <math>d\,\!</math> is:


With time terminated data, the 100( <math>1-\alpha \,\!</math> )% confidence interval for the current reliability at time <math>t\,\!</math> in a specified mission time <math>\tau \,\!</math> is:
:<math>\left( {{\left[ \hat{R}\left( d \right) \right]}^{\tfrac{1}{{{\Pi }_{1}}}}},{{\left[ \hat{R}\left( d \right) \right]}^{\tfrac{1}{{{\Pi }_{2}}}}} \right)\,\!</math>  


::<math>({{[\hat{R}(d)]}^{\tfrac{1}{{{p}_{1}}}}},{{[\hat{R}(d)]}^{\tfrac{1}{{{p}_{2}}}}})\,\!</math>
where:  


:where:
*<math>\hat{R}\left( d \right)={{e}^{-\left[ \hat{\lambda }{{\left( t+d \right)}^{{\hat{\beta }}}}-\hat{\lambda }{{t}^{{\hat{\beta }}}} \right]}}\,\!</math>
* <math>{{\Pi }_{1}}\,\!</math> and <math>{{\Pi }_{2}}\,\!</math> are obtained from the confidence bounds on [[Confidence Bounds for Repairable Systems Analysis#Instantaneous_MTBF|instantaneous MTBF]] for time terminated data.


::<math>\hat{R}(d)={{e}^{-[\hat{\lambda }{{(t+d)}^{\hat{\beta }}}-\hat{\lambda }{{t}^{\hat{\beta }}}]}}\,\!</math>
===Time Given Reliability and Mission Time===
 
====Fisher Matrix Bounds====
<math>{{p}_{1}}\,\!</math> and <math>{{p}_{2}}\,\!</math> can be obtained from the equation for time terminated data in the [[Confidence_Bounds_for_Repairable_Systems_Analysis#Bounds_on_Instantaneous_MTBF|Bounds on Instantaneous MTBF]] section.
 
====Bounds on Time Given Reliability and Mission Time====
=====Fisher Matrix Bounds=====
The time, <math>t\,\!</math>. must be positive, thus <math>\ln t\,\!</math> is approximately treated as being normally distributed.  
The time, <math>t\,\!</math>. must be positive, thus <math>\ln t\,\!</math> is approximately treated as being normally distributed.  


::<math>\frac{\ln (\hat{t})-\ln (t)}{\sqrt{Var\left[ \ln (\hat{t}) \right]}}\sim N(0,1)\,\!</math>
:<math>\frac{\ln (\hat{t})-\ln (t)}{\sqrt{Var\left[ \ln (\hat{t}) \right]}}\sim N(0,1)\,\!</math>


The confidence bounds on time are calculated by using:
The confidence bounds on time are calculated by using:


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


:where:
where:


::<math>Var(\hat{t})={{\left( \frac{\partial t}{\partial \beta } \right)}^{2}}Var(\hat{\beta })+{{\left( \frac{\partial t}{\partial \lambda } \right)}^{2}}Var(\hat{\lambda })+2\left( \frac{\partial t}{\partial \beta } \right)\left( \frac{\partial t}{\partial \lambda } \right)cov(\hat{\beta },\hat{\lambda })\,\!</math>
:<math>Var(\hat{t})={{\left( \frac{\partial t}{\partial \beta } \right)}^{2}}Var(\hat{\beta })+{{\left( \frac{\partial t}{\partial \lambda } \right)}^{2}}Var(\hat{\lambda })+2\left( \frac{\partial t}{\partial \beta } \right)\left( \frac{\partial t}{\partial \lambda } \right)cov(\hat{\beta },\hat{\lambda })\,\!</math>


<math>\hat{t}\,\!</math> is calculated numerically from:
<math>\hat{t}\,\!</math> is calculated numerically from:


::<math>\hat{R}(d)={{e}^{-[\hat{\lambda }{{(\hat{t}+d)}^{\hat{\beta }}}-\hat{\lambda }{{{\hat{t}}}^{\hat{\beta }}}]}}\text{ };\text{ }d\text{ = mission time}\,\!</math>
:<math>\hat{R}(d)={{e}^{-[\hat{\lambda }{{(\hat{t}+d)}^{\hat{\beta }}}-\hat{\lambda }{{{\hat{t}}}^{\hat{\beta }}}]}}\text{ };\text{ }d\text{ = mission time}\,\!</math>


The variance calculations are done by:
The variance calculations are done by:


::<math>\begin{align}
:<math>\begin{align}
   \frac{\partial t}{\partial \beta }= & \frac{{{{\hat{t}}}^{{\hat{\beta }}}}\ln (\hat{t})-{{(\hat{t}+d)}^{{\hat{\beta }}}}\ln (\hat{t}+d)}{\hat{\beta }{{(\hat{t}+d)}^{\hat{\beta }-1}}-\hat{\beta }{{{\hat{t}}}^{\hat{\beta }-1}}} \\  
   \frac{\partial t}{\partial \beta }= & \frac{{{{\hat{t}}}^{{\hat{\beta }}}}\ln (\hat{t})-{{(\hat{t}+d)}^{{\hat{\beta }}}}\ln (\hat{t}+d)}{\hat{\beta }{{(\hat{t}+d)}^{\hat{\beta }-1}}-\hat{\beta }{{{\hat{t}}}^{\hat{\beta }-1}}} \\  
   \frac{\partial t}{\partial \lambda }= & \frac{{{{\hat{t}}}^{{\hat{\beta }}}}-{{(\hat{t}+d)}^{{\hat{\beta }}}}}{\hat{\lambda }\hat{\beta }{{(\hat{t}+d)}^{\hat{\beta }-1}}-\hat{\lambda }\hat{\beta }{{{\hat{t}}}^{\hat{\beta }-1}}}   
   \frac{\partial t}{\partial \lambda }= & \frac{{{{\hat{t}}}^{{\hat{\beta }}}}-{{(\hat{t}+d)}^{{\hat{\beta }}}}}{\hat{\lambda }\hat{\beta }{{(\hat{t}+d)}^{\hat{\beta }-1}}-\hat{\lambda }\hat{\beta }{{{\hat{t}}}^{\hat{\beta }-1}}}   
\end{align}\,\!</math>
\end{align}\,\!</math>


=====Crow Bounds=====
====Crow Bounds====
'''Failure Terminated Data'''
'''Failure Terminated'''
 
For failure terminated data, the 2-sided confidence bounds on time given reliability and mission time estimated by calculating:
 
:<math>\left( {{{\hat{R}}}_{L}},{{{\hat{R}}}_{U}} \right)=\left( {{R}^{\tfrac{1}{{{p}_{1}}}}},{{R}^{\tfrac{1}{{{p}_{2}}}}}\, \right)\,\!</math>


Step 1: Calculate <math>({{\hat{R}}_{lower}},{{\hat{R}}_{upper}})=({{R}^{\tfrac{1}{{{p}_{1}}}}},{{R}^{\tfrac{1}{{{p}_{2}}}}})\,\!</math>.
where <math>p_1\,\!</math> and <math>p_2\,\!</math> are obtained from the confidence bounds on [[Confidence Bounds for Repairable Systems Analysis#Instantaneous_MTBF|instantaneous MTBF]] for failure terminated data.


Step 2: Let <math>R={{\hat{R}}_{lower}}\,\!</math> and solve numerically for <math>{{t}_{1}}\,\!</math> using <math>R={{e}^{-[\hat{\lambda }{{({{{\hat{t}}}_{1}}+d)}^{\hat{\beta }}}-\hat{\lambda }\hat{t}_{1}^{\hat{\beta }}]}}\,\!</math>.
Let <math>R={{\hat{R}}_{L}}\,\!</math> and solve numerically for <math>{{t}_{1}}\,\!</math> using <math>R={{e}^{-[\hat{\lambda }{{({{{\hat{t}}}_{1}}+d)}^{\hat{\beta }}}-\hat{\lambda }\hat{t}_{1}^{\hat{\beta }}]}}\,\!</math>.


Step 3: Let <math>R={{\hat{R}}_{upper}}\,\!</math> and solve numerically for <math>{{t}_{2}}\,\!</math> using <math>R={{e}^{-[\hat{\lambda }{{({{{\hat{t}}}_{2}}+d)}^{\hat{\beta }}}-\hat{\lambda }\hat{t}_{2}^{\hat{\beta }}]}}\,\!</math>.
Let <math>R={{\hat{R}}_{U}}\,\!</math> and solve numerically for <math>{{t}_{2}}\,\!</math> using <math>R={{e}^{-[\hat{\lambda }{{({{{\hat{t}}}_{2}}+d)}^{\hat{\beta }}}-\hat{\lambda }\hat{t}_{2}^{\hat{\beta }}]}}\,\!</math>.


Step 4: If <math>{{t}_{1}}<{{t}_{2}}\,\!</math>. then <math>{{t}_{lower}}={{t}_{1}}\,\!</math> and <math>{{t}_{upper}}={{t}_{2}}\,\!</math>. If <math>{{t}_{1}}>{{t}_{2}}\,\!</math>. then <math>{{t}_{lower}}={{t}_{2}}\,\!</math> and <math>{{t}_{upper}}={{t}_{1}}\,\!</math>.
If <math>{{t}_{1}}<{{t}_{2}}\,\!</math> then <math>{{t}_{L}}={{t}_{1}}\,\!</math> and <math>{{t}_{U}}={{t}_{2}}\,\!</math>. If <math>{{t}_{1}}>{{t}_{2}}\,\!</math> then <math>{{t}_{L}}={{t}_{2}}\,\!</math> and <math>{{t}_{U}}={{t}_{1}}\,\!</math>.


'''Time Terminated'''


'''Time Terminated Data'''
For time terminated data, the 2-sided confidence bounds on time given reliability and mission time estimated by calculating:
<br>
 
Step 1: Calculate <math>({{\hat{R}}_{lower}},{{\hat{R}}_{upper}})=({{R}^{\tfrac{1}{{{\Pi }_{1}}}}},{{R}^{\tfrac{1}{{{\Pi }_{2}}}}})\,\!</math>.
:<math>\left( {{{\hat{R}}}_{L}},{{{\hat{R}}}_{U}} \right)=\left( {{R}^{\tfrac{1}{{{\Pi }_{1}}}}},{{R}^{\tfrac{1}{{{\Pi }_{2}}}}}\, \right)</math>.
 
where <math>\Pi_1\,\!</math> and <math>\Pi_2\,\!</math> are obtained from the confidence bounds on [[Confidence Bounds for Repairable Systems Analysis#Instantaneous_MTBF|instantaneous MTBF]] for time terminated data.


Step 2: Let <math>R={{\hat{R}}_{lower}}\,\!</math> and solve numerically for <math>{{t}_{1}}\,\!</math> using <math>R={{e}^{-[\hat{\lambda }{{({{{\hat{t}}}_{1}}+d)}^{\hat{\beta }}}-\hat{\lambda }\hat{t}_{1}^{\hat{\beta }}]}}\,\!</math>.
Let <math>R={{\hat{R}}_{L}}\,\!</math> and solve numerically for <math>{{t}_{1}}\,\!</math> using <math>R={{e}^{-[\hat{\lambda }{{({{{\hat{t}}}_{1}}+d)}^{\hat{\beta }}}-\hat{\lambda }\hat{t}_{1}^{\hat{\beta }}]}}\,\!</math>.


Step 3: Let <math>R={{\hat{R}}_{upper}}\,\!</math> and solve numerically for <math>{{t}_{2}}\,\!</math> using <math>R={{e}^{-[\hat{\lambda }{{({{{\hat{t}}}_{2}}+d)}^{\hat{\beta }}}-\hat{\lambda }\hat{t}_{2}^{\hat{\beta }}]}}\,\!</math>.
Let <math>R={{\hat{R}}_{U}}\,\!</math> and solve numerically for <math>{{t}_{2}}\,\!</math> using <math>R={{e}^{-[\hat{\lambda }{{({{{\hat{t}}}_{2}}+d)}^{\hat{\beta }}}-\hat{\lambda }\hat{t}_{2}^{\hat{\beta }}]}}\,\!</math>.


Step 4: If <math>{{t}_{1}}<{{t}_{2}}\,\!</math>. then <math>{{t}_{lower}}={{t}_{1}}\,\!</math> and <math>{{t}_{upper}}={{t}_{2}}\,\!</math>. If <math>{{t}_{1}}>{{t}_{2}}\,\!</math>. then <math>{{t}_{lower}}={{t}_{2}}\,\!</math> and <math>{{t}_{upper}}={{t}_{1}}\,\!</math>.
If <math>{{t}_{1}}<{{t}_{2}}\,\!</math>. then <math>{{t}_{L}}={{t}_{1}}\,\!</math> and <math>{{t}_{U}}={{t}_{2}}\,\!</math>. If <math>{{t}_{1}}>{{t}_{2}}\,\!</math>. then <math>{{t}_{L}}={{t}_{2}}\,\!</math> and <math>{{t}_{U}}={{t}_{1}}\,\!</math>.


====Bounds on Mission Time Given Reliability and Time====
===Mission Time Given Reliability and Time===
=====Fisher Matrix Bounds=====
====Fisher Matrix Bounds====
The mission time, <math>d\,\!</math>. must be positive, thus <math>\ln \left( d \right)\,\!</math> is approximately treated as being normally distributed.  
The mission time, <math>d\,\!</math>. must be positive, thus <math>\ln \left( d \right)\,\!</math> is approximately treated as being normally distributed.  


::<math>\frac{\ln (\hat{d})-\ln (d)}{\sqrt{Var\left[ \ln (\hat{d}) \right]}}\sim N(0,1)\,\!</math>
:<math>\frac{\ln (\hat{d})-\ln (d)}{\sqrt{Var\left[ \ln (\hat{d}) \right]}}\sim N(0,1)\,\!</math>


The confidence bounds on mission time are given by using:
The confidence bounds on mission time are given by using:


::<math>CB=\hat{d}{{e}^{\pm {{z}_{\alpha }}\sqrt{Var(\hat{d})}/\hat{d}}}\,\!</math>
:<math>CB=\hat{d}{{e}^{\pm {{z}_{\alpha }}\sqrt{Var(\hat{d})}/\hat{d}}}\,\!</math>


:where:
where:


::<math>Var(\hat{d})={{\left( \frac{\partial d}{\partial \beta } \right)}^{2}}Var(\hat{\beta })+{{\left( \frac{\partial d}{\partial \lambda } \right)}^{2}}Var(\hat{\lambda })+2\left( \frac{\partial td}{\partial \beta } \right)\left( \frac{\partial d}{\partial \lambda } \right)cov(\hat{\beta },\hat{\lambda })\,\!</math>
:<math>Var(\hat{d})={{\left( \frac{\partial d}{\partial \beta } \right)}^{2}}Var(\hat{\beta })+{{\left( \frac{\partial d}{\partial \lambda } \right)}^{2}}Var(\hat{\lambda })+2\left( \frac{\partial td}{\partial \beta } \right)\left( \frac{\partial d}{\partial \lambda } \right)cov(\hat{\beta },\hat{\lambda })\,\!</math>


Calculate <math>\hat{d}\,\!</math> from:
Calculate <math>\hat{d}\,\!</math> from:


::<math>\hat{d}={{\left[ {{t}^{{\hat{\beta }}}}-\frac{\ln (R)}{{\hat{\lambda }}} \right]}^{\tfrac{1}{{\hat{\beta }}}}}-t\,\!</math>
:<math>\hat{d}={{\left[ {{t}^{{\hat{\beta }}}}-\frac{\ln (R)}{{\hat{\lambda }}} \right]}^{\tfrac{1}{{\hat{\beta }}}}}-t\,\!</math>


The variance calculations are done by:
The variance calculations are done by:


::<math>\begin{align}
:<math>\begin{align}
   \frac{\partial d}{\partial \beta }= & \left[ \frac{{{t}^{{\hat{\beta }}}}\ln (t)}{{{(t+\hat{d})}^{{\hat{\beta }}}}}-\ln (t+\hat{d}) \right]\cdot \frac{t+\hat{d}}{{\hat{\beta }}} \\  
   \frac{\partial d}{\partial \beta }= & \left[ \frac{{{t}^{{\hat{\beta }}}}\ln (t)}{{{(t+\hat{d})}^{{\hat{\beta }}}}}-\ln (t+\hat{d}) \right]\cdot \frac{t+\hat{d}}{{\hat{\beta }}} \\  
   \frac{\partial d}{\partial \lambda }= & \frac{{{t}^{{\hat{\beta }}}}-{{(t+\hat{d})}^{{\hat{\beta }}}}}{\hat{\lambda }\hat{\beta }{{(t+\hat{d})}^{\hat{\beta }-1}}}   
   \frac{\partial d}{\partial \lambda }= & \frac{{{t}^{{\hat{\beta }}}}-{{(t+\hat{d})}^{{\hat{\beta }}}}}{\hat{\lambda }\hat{\beta }{{(t+\hat{d})}^{\hat{\beta }-1}}}   
\end{align}\,\!</math>
\end{align}\,\!</math>


=====Crow Bounds=====
====Crow Bounds====
'''Failure Terminated Data'''
'''Failure Terminated'''


Step 1: Calculate <math>({{\hat{R}}_{lower}},{{\hat{R}}_{upper}})=({{R}^{\tfrac{1}{{{p}_{1}}}}},{{R}^{\tfrac{1}{{{p}_{2}}}}})\,\!</math>.
Step 1: Calculate <math>({{\hat{R}}_{lower}},{{\hat{R}}_{upper}})=({{R}^{\tfrac{1}{{{p}_{1}}}}},{{R}^{\tfrac{1}{{{p}_{2}}}}})\,\!</math>.
Line 552: Line 573:
Step 2: Let <math>R={{\hat{R}}_{lower}}\,\!</math> and solve for <math>{{d}_{1}}\,\!</math> such that:
Step 2: Let <math>R={{\hat{R}}_{lower}}\,\!</math> and solve for <math>{{d}_{1}}\,\!</math> such that:


::<math>{{d}_{1}}={{\left( {{t}^{{\hat{\beta }}}}-\frac{\ln ({{R}_{lower}})}{{\hat{\lambda }}} \right)}^{\tfrac{1}{{\hat{\beta }}}}}-t\,\!</math>
:<math>{{d}_{1}}={{\left( {{t}^{{\hat{\beta }}}}-\frac{\ln ({{R}_{lower}})}{{\hat{\lambda }}} \right)}^{\tfrac{1}{{\hat{\beta }}}}}-t\,\!</math>


Step 3: Let <math>R={{\hat{R}}_{upper}}\,\!</math> and solve for <math>{{d}_{2}}\,\!</math> such that:
Step 3: Let <math>R={{\hat{R}}_{upper}}\,\!</math> and solve for <math>{{d}_{2}}\,\!</math> such that:


::<math>{{d}_{2}}={{\left( {{t}^{{\hat{\beta }}}}-\frac{\ln ({{R}_{upper}})}{{\hat{\lambda }}} \right)}^{\tfrac{1}{{\hat{\beta }}}}}-t\,\!</math>
:<math>{{d}_{2}}={{\left( {{t}^{{\hat{\beta }}}}-\frac{\ln ({{R}_{upper}})}{{\hat{\lambda }}} \right)}^{\tfrac{1}{{\hat{\beta }}}}}-t\,\!</math>


Step 4: If <math>{{d}_{1}}<{{d}_{2}}\,\!</math>. then <math>{{d}_{lower}}={{d}_{1}}\,\!</math> and <math>{{d}_{upper}}={{d}_{2}}\,\!</math>. If <math>{{d}_{1}}>{{d}_{2}}\,\!</math>. then <math>{{d}_{lower}}={{d}_{2}}\,\!</math> and <math>{{d}_{upper}}={{d}_{1}}\,\!</math>.
Step 4: If <math>{{d}_{1}}<{{d}_{2}}\,\!</math>. then <math>{{d}_{lower}}={{d}_{1}}\,\!</math> and <math>{{d}_{upper}}={{d}_{2}}\,\!</math>. If <math>{{d}_{1}}>{{d}_{2}}\,\!</math>. then <math>{{d}_{lower}}={{d}_{2}}\,\!</math> and <math>{{d}_{upper}}={{d}_{1}}\,\!</math>.


 
'''Time Terminated'''
'''Time Terminated Data'''


Step 1: Calculate <math>({{\hat{R}}_{lower}},{{\hat{R}}_{upper}})=({{R}^{\tfrac{1}{{{\Pi }_{1}}}}},{{R}^{\tfrac{1}{{{\Pi }_{2}}}}})\,\!</math>.
Step 1: Calculate <math>({{\hat{R}}_{lower}},{{\hat{R}}_{upper}})=({{R}^{\tfrac{1}{{{\Pi }_{1}}}}},{{R}^{\tfrac{1}{{{\Pi }_{2}}}}})\,\!</math>.
Line 570: Line 590:


Step 4: If <math>{{d}_{1}}<{{d}_{2}}\,\!</math>. then <math>{{d}_{lower}}={{d}_{1}}\,\!</math> and <math>{{d}_{upper}}={{d}_{2}}\,\!</math>. If <math>{{d}_{1}}>{{d}_{2}}\,\!</math>. then <math>{{d}_{lower}}={{d}_{2}}\,\!</math> and <math>{{d}_{upper}}={{d}_{1}}\,\!</math>.
Step 4: If <math>{{d}_{1}}<{{d}_{2}}\,\!</math>. then <math>{{d}_{lower}}={{d}_{1}}\,\!</math> and <math>{{d}_{upper}}={{d}_{2}}\,\!</math>. If <math>{{d}_{1}}>{{d}_{2}}\,\!</math>. then <math>{{d}_{lower}}={{d}_{2}}\,\!</math> and <math>{{d}_{upper}}={{d}_{1}}\,\!</math>.
====Bounds on Cumulative Number of Failures====
=====Fisher Matrix Bounds=====
The cumulative number of failures, <math>N(t)\,\!</math>. must be positive, thus <math>\ln \left( N(t) \right)\,\!</math> is approximately treated as being normally distributed.
::<math>\frac{\ln (\hat{N}(t))-\ln (N(t))}{\sqrt{Var\left[ \ln \hat{N}(t) \right]}}\sim N(0,1)\,\!</math>
::<math>N(t)=\hat{N}(t){{e}^{\pm {{z}_{\alpha }}\sqrt{Var(\hat{N}(t))}/\hat{N}(t)}}\,\!</math>
:where:
::<math>\hat{N}(t)=\hat{\lambda }{{t}^{\hat{\beta }}}\,\!</math>
::<math>\begin{align}
  Var(\hat{N}(t))= & {{\left( \frac{\partial N(t)}{\partial \beta } \right)}^{2}}Var(\hat{\beta })+{{\left( \frac{\partial N(t)}{\partial \lambda } \right)}^{2}}Var(\hat{\lambda }) \\
  & +2\left( \frac{\partial N(t)}{\partial \beta } \right)\left( \frac{\partial N(t)}{\partial \lambda } \right)cov(\hat{\beta },\hat{\lambda }) 
\end{align}\,\!</math>
The variance calculation is the same as the calculations in the [[Confidence_Bounds_for_Repairable_Systems_Analysis#Bounds_on_Beta|Bounds on Beta]] section.
::<math>\begin{align}
  \frac{\partial N(t)}{\partial \beta }= & \hat{\lambda }{{t}^{\hat{\beta }}}\ln (t) \\
  \frac{\partial N(t)}{\partial \lambda }= & t\hat{\beta } 
\end{align}\,\!</math>
=====Crow Bounds=====
::<math>\begin{array}{*{35}{l}}
  {{N}_{L}}(T)=\tfrac{T}{\hat{\beta }}{{\lambda }_{i}}{{(T)}_{L}}  \\
  {{N}_{U}}(T)=\tfrac{T}{\hat{\beta }}{{\lambda }_{i}}{{(T)}_{U}}  \\
\end{array}\,\!</math>
where <math>{{\lambda }_{i}}{{(T)}_{L}}\,\!</math> and <math>{{\lambda }_{i}}{{(T)}_{U}}\,\!</math> can be obtained using the equation given in the [[Confidence_Bounds_for_Repairable_Systems_Analysis#Bounds_on_Instantaneous_Failure_Intensity|Bounds on Instantaneous Failure Intensity]] section.

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Chapter Appendix E: Confidence Bounds for Repairable Systems Analysis


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Chapter Appendix E  
Confidence Bounds for Repairable Systems Analysis  

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Available Software:
RGA

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More Resources:
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In this appendix, we will present the two methods used in the RGA software to estimate the confidence bounds for Repairable Systems Analysis. The Fisher Matrix approach is based on the Fisher Information Matrix and is commonly employed in the reliability field. The Crow bounds were developed by Dr. Larry Crow. 

Beta

Fisher Matrix Bounds

The parameter [math]\displaystyle{ \beta \,\! }[/math] must be positive, thus [math]\displaystyle{ \ln \beta \,\! }[/math] is approximately treated as being normally distributed.

[math]\displaystyle{ \frac{\ln (\hat{\beta })-\ln (\beta )}{\sqrt{Var\left[ \ln (\hat{\beta }) \right]}}\ \tilde{\ }\ N(0,1)\,\! }[/math]
[math]\displaystyle{ C{{B}_{\beta }}=\hat{\beta }{{e}^{\pm {{z}_{\alpha }}\sqrt{Var(\hat{\beta })}/\hat{\beta }}}\,\! }[/math]
[math]\displaystyle{ \hat{\beta }=\frac{\underset{q=1}{\overset{K}{\mathop{\sum }}}\,{{N}_{q}}}{\hat{\lambda }\underset{q=1}{\overset{K}{\mathop{\sum }}}\,\left[ (T_{q}^{\hat{\beta }}\ln ({{T}_{q}})-S_{q}^{\hat{\beta }}\ln ({{S}_{q}}) \right]-\underset{q=1}{\overset{K}{\mathop{\sum }}}\,\underset{i=1}{\overset{{{N}_{q}}}{\mathop{\sum }}}\,\ln ({{X}_{i}}{{}_{q}})}\,\! }[/math]

All variance can be calculated using the Fisher Information Matrix. [math]\displaystyle{ \Lambda \,\! }[/math] is the natural log-likelihood function.

[math]\displaystyle{ \Lambda =\underset{q=1}{\overset{K}{\mathop \sum }}\,\left[ {{N}_{q}}(\ln (\lambda )+\ln (\beta ))-\lambda (T_{q}^{\beta }-S_{q}^{\beta })+(\beta -1)\underset{i=1}{\overset{{{N}_{q}}}{\mathop \sum }}\,\ln ({{x}_{iq}}) \right]\,\! }[/math]
[math]\displaystyle{ \frac{{{\partial }^{2}}\Lambda }{\partial {{\lambda }^{2}}}=-\frac{\underset{q=1}{\overset{K}{\mathop{\sum }}}\,{{N}_{q}}}{{{\lambda }^{2}}}\,\! }[/math]
[math]\displaystyle{ \frac{{{\partial }^{2}}\Lambda }{\partial \lambda \partial \beta }=-\underset{q=1}{\overset{K}{\mathop \sum }}\,\left[ T_{q}^{\beta }\ln ({{T}_{q}})-S_{q}^{\beta }\ln ({{S}_{q}}) \right]\,\! }[/math]
[math]\displaystyle{ \frac{{{\partial }^{2}}\Lambda }{\partial {{\beta }^{2}}}=-\frac{\underset{q=1}{\overset{K}{\mathop{\sum }}}\,{{N}_{q}}}{{{\beta }^{2}}}-\lambda \underset{q=1}{\overset{K}{\mathop \sum }}\,\left[ T_{q}^{\beta }{{(\ln ({{T}_{q}}))}^{2}}-S_{q}^{\beta }{{(\ln ({{S}_{q}}))}^{2}} \right]\,\! }[/math]

Crow Bounds

Calculate the conditional maximum likelihood estimate of [math]\displaystyle{ \tilde{\beta \,\!}\,\! }[/math] :

[math]\displaystyle{ \tilde{\beta }=\frac{\underset{q=1}{\overset{K}{\mathop{\sum }}}\,{{M}_{q}}}{\underset{q=1}{\overset{K}{\mathop{\sum }}}\,\underset{i=1}{\overset{M}{\mathop{\sum }}}\,\ln \left( \tfrac{{{T}_{q}}}{{{X}_{iq}}} \right)}\,\! }[/math]

The Crow 2-sided [math]\displaystyle{ (1-a)\,\! }[/math] 100% confidence bounds on [math]\displaystyle{ \beta \,\! }[/math] are:

[math]\displaystyle{ \begin{align} {{\beta }_{L}}= & \tilde{\beta }\frac{\chi _{\tfrac{\alpha }{2},2M}^{2}}{2M} \\ {{\beta }_{U}}= & \tilde{\beta }\frac{\chi _{1-\tfrac{\alpha }{2},2M}^{2}}{2M} \end{align}\,\! }[/math]

Lambda

Fisher Matrix Bounds

The parameter [math]\displaystyle{ \lambda \,\! }[/math] must be positive, thus [math]\displaystyle{ \ln \lambda \,\! }[/math] is approximately treated as being normally distributed. These bounds are based on:

[math]\displaystyle{ \frac{\ln (\hat{\lambda })-\ln (\lambda )}{\sqrt{Var\left[ \ln (\hat{\lambda }) \right]}}\ \tilde{\ }\ N(0,1)\,\! }[/math]

The approximate confidence bounds on [math]\displaystyle{ \lambda \,\! }[/math] are given as:

[math]\displaystyle{ C{{B}_{\lambda }}=\hat{\lambda }{{e}^{\pm {{z}_{\alpha }}\sqrt{Var(\hat{\lambda })}/\hat{\lambda }}}\,\! }[/math]

where [math]\displaystyle{ \hat{\lambda }=\tfrac{n}{T_{K}^{{\hat{\beta }}}}\,\! }[/math].

The variance calculation is the same the equations given in the confidence bounds on Beta.

Crow Bounds

Failure Terminated

The confidence bounds on [math]\displaystyle{ \lambda \,\! }[/math] for failure terminated data are calculated using:

[math]\displaystyle{ \begin{align} {{\lambda }_{L}}= & \frac{\chi _{\tfrac{\alpha }{2},2N}^{2}}{2\cdot \underset{q=1}{\overset{K}{\mathop{\sum }}}\,T_{q}^{^{\beta }}} \\ {{\lambda }_{U}}= & \frac{\chi _{1-\tfrac{\alpha }{2},2N}^{2}}{2\cdot \underset{q=1}{\overset{K}{\mathop{\sum }}}\,T_{q}^{^{\beta }}} \end{align}\,\! }[/math]

where:

  • [math]\displaystyle{ N\,\! }[/math] = total number of failures.
  • [math]\displaystyle{ K\,\! }[/math] = number of systems.
  • [math]\displaystyle{ {{T}_{q}}\,\! }[/math] = end time for the [math]\displaystyle{ {{q}^{th}} }[/math] system.

Time Terminated

The confidence bounds on [math]\displaystyle{ \lambda \,\! }[/math] for time terminated data are calculated using:

[math]\displaystyle{ \begin{align} {{\lambda }_{L}}= & \frac{\chi _{\tfrac{\alpha }{2},2N}^{2}}{2\cdot \underset{q=1}{\overset{K}{\mathop{\sum }}}\,T_{q}^{^{\beta }}} \\ {{\lambda }_{U}}= & \frac{\chi _{1-\tfrac{\alpha }{2},2N+2}^{2}}{2\cdot \underset{q=1}{\overset{K}{\mathop{\sum }}}\,T_{q}^{^{\beta }}} \end{align}\,\! }[/math]

where:

  • [math]\displaystyle{ N\,\! }[/math] = total number of failures.
  • [math]\displaystyle{ K\,\! }[/math] = number of systems.
  • [math]\displaystyle{ {{T}_{q}}\,\! }[/math] = end time for the [math]\displaystyle{ {{q}^{th}} }[/math] system.

Cumulative Number of Failures

Fisher Matrix Bounds

The cumulative number of failures, [math]\displaystyle{ N(t)\,\! }[/math]. must be positive, thus [math]\displaystyle{ \ln \left( N(t) \right)\,\! }[/math] is approximately treated as being normally distributed.

[math]\displaystyle{ \frac{\ln (\hat{N}(t))-\ln (N(t))}{\sqrt{Var\left[ \ln \hat{N}(t) \right]}}\sim N(0,1)\,\! }[/math]
[math]\displaystyle{ N(t)=\hat{N}(t){{e}^{\pm {{z}_{\alpha }}\sqrt{Var(\hat{N}(t))}/\hat{N}(t)}}\,\! }[/math]

where:

[math]\displaystyle{ \hat{N}(t)=\hat{\lambda }{{t}^{\hat{\beta }}}\,\! }[/math]
[math]\displaystyle{ \begin{align} Var(\hat{N}(t))= & {{\left( \frac{\partial N(t)}{\partial \beta } \right)}^{2}}Var(\hat{\beta })+{{\left( \frac{\partial N(t)}{\partial \lambda } \right)}^{2}}Var(\hat{\lambda }) \\ & +2\left( \frac{\partial N(t)}{\partial \beta } \right)\left( \frac{\partial N(t)}{\partial \lambda } \right)cov(\hat{\beta },\hat{\lambda }) \end{align}\,\! }[/math]

The variance calculation is the same as the calculations in the confidence bounds on Beta.

[math]\displaystyle{ \begin{align} \frac{\partial N(t)}{\partial \beta }= & \hat{\lambda }{{t}^{\hat{\beta }}}\ln (t) \\ \frac{\partial N(t)}{\partial \lambda }= & t\hat{\beta } \end{align}\,\! }[/math]

Crow Bounds

The 2-sided confidence bounds on the cumulative number of failures are given by:

[math]\displaystyle{ N{{\left( t \right)}_{L}}=\frac{\chi _{\tfrac{\alpha }{2},2N}^{2}}{2\cdot S}\,\! }[/math]
[math]\displaystyle{ N{{\left( t \right)}_{U}}=\frac{\chi _{1-\tfrac{\alpha }{2},2N+2}^{2}}{2\cdot S}\,\! }[/math]

where:

  • [math]\displaystyle{ N\,\! }[/math] = total number of failures across all systems. This is not the number of failures up to time [math]\displaystyle{ t\,\! }[/math].
  • [math]\displaystyle{ S=\frac{\left( \frac{N}{{\hat{\lambda }}} \right)}{{{t}^{{\hat{\beta }}}}}\,\! }[/math]
  • [math]\displaystyle{ t\,\! }[/math] = time at which calculations are being conducted.

Cumulative Failure Intensity

Fisher Matrix Bounds

The cumulative failure intensity, [math]\displaystyle{ {{\lambda }_{c}}(t)\,\! }[/math] must be positive, thus [math]\displaystyle{ \ln {{\lambda }_{c}}(t)\,\! }[/math] is approximately treated as being normally distributed.

[math]\displaystyle{ \frac{\ln ({{\hat{\lambda }}_{c}}(t))-\ln ({{\lambda }_{c}}(t))}{\sqrt{Var\left[ \ln ({{\hat{\lambda }}_{c}}(t)) \right]}}\ \tilde{\ }\ N(0,1)\,\! }[/math]

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

[math]\displaystyle{ CB={{\hat{\lambda }}_{c}}(t){{e}^{\pm {{z}_{\alpha }}\sqrt{Var({{\hat{\lambda }}_{c}}(t))}/{{\hat{\lambda }}_{c}}(t)}}\,\! }[/math]

where:

[math]\displaystyle{ {{\hat{\lambda }}_{c}}(t)=\hat{\lambda }{{t}^{\hat{\beta }-1}}\,\! }[/math]

and:

[math]\displaystyle{ \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 the calculations in the confidence bounds on Beta.

[math]\displaystyle{ \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 2-sided confidence bounds on the cumulative failure intensity are given by:

[math]\displaystyle{ CFI_L=\frac{\chi _{\tfrac{\alpha }{2},2N}^{2}}{2\cdot t \cdot S}\,\! }[/math]
[math]\displaystyle{ CFI_U=\frac{\chi _{1-\tfrac{\alpha }{2},2N+2}^{2}}{2\cdot t \cdot S}\,\! }[/math]

where:

  • [math]\displaystyle{ N\,\! }[/math] = total number of failures across all systems. This is not the number of failures up to time [math]\displaystyle{ t\,\! }[/math].
  • [math]\displaystyle{ S=\frac{\left( \frac{N}{{\hat{\lambda }}} \right)}{{{t}^{{\hat{\beta }}}}}\,\! }[/math]
  • [math]\displaystyle{ t\,\! }[/math] = time at which calculations are being conducted.

Cumulative MTBF

Fisher Matrix Bounds

The cumulative MTBF, [math]\displaystyle{ {{m}_{c}}(t)\,\! }[/math]. must be positive, thus [math]\displaystyle{ \ln {{m}_{c}}(t)\,\! }[/math] is approximately treated as being normally distributed.

[math]\displaystyle{ \frac{\ln ({{\hat{m}}_{c}}(t))-\ln ({{m}_{c}}(t))}{\sqrt{Var\left[ \ln ({{\hat{m}}_{c}}(t)) \right]}}\ \tilde{\ }\ N(0,1)\,\! }[/math]

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

[math]\displaystyle{ CB={{\hat{m}}_{c}}(t){{e}^{\pm {{z}_{\alpha }}\sqrt{Var({{\hat{m}}_{c}}(t))}/{{\hat{m}}_{c}}(t)}}\,\! }[/math]

where:

[math]\displaystyle{ {{\hat{m}}_{c}}(t)=\frac{1}{\hat{\lambda }}{{t}^{1-\hat{\beta }}}\,\! }[/math]
[math]\displaystyle{ \begin{align} Var({{\hat{m}}_{c}}(t))= & {{\left( \frac{\partial {{m}_{c}}(t)}{\partial \beta } \right)}^{2}}Var(\hat{\beta })+{{\left( \frac{\partial {{m}_{c}}(t)}{\partial \lambda } \right)}^{2}}Var(\hat{\lambda }) \\ & +2\left( \frac{\partial {{m}_{c}}(t)}{\partial \beta } \right)\left( \frac{\partial {{m}_{c}}(t)}{\partial \lambda } \right)cov(\hat{\beta },\hat{\lambda })\, \end{align}\,\! }[/math]

The variance calculation is the same as the calculations given in the confidence bounds on Beta.

[math]\displaystyle{ \begin{align} \frac{\partial {{m}_{c}}(t)}{\partial \beta }= & -\frac{1}{\hat{\lambda }}{{t}^{1-\hat{\beta }}}\ln (t) \\ \frac{\partial {{m}_{c}}(t)}{\partial \lambda }= & -\frac{1}{{{\hat{\lambda }}^{2}}}{{t}^{1-\hat{\beta }}} \end{align}\,\! }[/math]

Crow Bounds

The 2-sided confidence bounds on the cumulative MTBF [math]\displaystyle{ (CMTBF)\,\! }[/math] are given by:

[math]\displaystyle{ \begin{align} & CMTBF_{L}=\frac{1}{CFI_{U}} \\ & CMTBF_{U}=\frac{1}{CFI_{L}} \end{align}\,\! }[/math]

where [math]\displaystyle{ CFI_L\,\! }[/math] and [math]\displaystyle{ CFI_U\,\! }[/math] are calculated using the process for the confidence bounds on cumulative failure intensity.

Instantaneous MTBF

Fisher Matrix Bounds

The instantaneous MTBF, [math]\displaystyle{ {{m}_{i}}(t)\,\! }[/math]. must be positive, thus [math]\displaystyle{ \ln {{m}_{i}}(t)\,\! }[/math] is approximately treated as being normally distributed.

[math]\displaystyle{ \frac{\ln ({{\hat{m}}_{i}}(t))-\ln ({{m}_{i}}(t))}{\sqrt{Var\left[ \ln ({{\hat{m}}_{i}}(t)) \right]}}\ \tilde{\ }\ N(0,1)\,\! }[/math]

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

[math]\displaystyle{ CB={{\hat{m}}_{i}}(t){{e}^{\pm {{z}_{\alpha }}\sqrt{Var({{\hat{m}}_{i}}(t))}/{{\hat{m}}_{i}}(t)}}\,\! }[/math]

where:

[math]\displaystyle{ {{\hat{m}}_{i}}(t)=\frac{1}{\lambda \beta {{t}^{\beta -1}}}\,\! }[/math]
[math]\displaystyle{ \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 the calculations given in the confidence bounds on Beta.

[math]\displaystyle{ \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

Failure Terminated

For failure terminated data and the 2-sided confidence bounds on instantaneous MTBF [math]\displaystyle{ (IMTBF)\,\! }[/math], consider the following equation:

[math]\displaystyle{ G(\mu |n)=\mathop{}_{0}^{\infty }\frac{{{e}^{-x}}{{x}^{n-2}}}{(n-2)!}\underset{i=0}{\overset{n-1}{\mathop \sum }}\,\frac{1}{i!}{{\left( \frac{\mu }{x} \right)}^{i}}\exp (-\frac{\mu }{x})\,dx\,\! }[/math]

Find the values [math]\displaystyle{ {{p}_{1}}\,\! }[/math] and [math]\displaystyle{ {{p}_{2}}\,\! }[/math] by finding the solution [math]\displaystyle{ G\left( \left. \frac{{{n}^{2}}}{c} \right|n \right)=\frac{\alpha }{2} }[/math] and [math]\displaystyle{ G\left( \left. \frac{{{n}^{2}}}{c} \right|n \right)=1-\frac{\alpha }{2} }[/math] for the lower and upper bounds, respectively.

If using the biased parameters, [math]\displaystyle{ \hat{\beta }\,\! }[/math] and [math]\displaystyle{ \hat{\lambda }\,\! }[/math], then the upper and lower confidence bounds are:

[math]\displaystyle{ \begin{align} {{IMTBF}_{L}}= & IMTBF\cdot {{p}_{1}} \\ {{IMTBF}_{U}}= & IMTBF\cdot {{p}_{2}} \end{align}\,\! }[/math]

where [math]\displaystyle{ IMTBF=\tfrac{1}{\hat{\lambda }\hat{\beta }{{t}^{\hat{\beta }-1}}}\,\! }[/math].

If using the unbiased parameters, [math]\displaystyle{ \bar{\beta }\,\! }[/math] and [math]\displaystyle{ \bar{\lambda }\,\! }[/math], then the upper and lower confidence bounds are:

[math]\displaystyle{ \begin{align} {{IMTBF}_{L}}= & IMTBF\cdot \left( \frac{N-2}{N} \right)\cdot {{p}_{1}} \\ {{IMTBF}_{U}}= & IMTBF\cdot \left( \frac{N-2}{N} \right)\cdot {{p}_{2}} \end{align}\,\! }[/math]

where [math]\displaystyle{ IMTBF=\tfrac{1}{\bar{\lambda }\bar{\beta }{{t}^{\bar{\beta }-1}}}\,\! }[/math].

Time Terminated

Consider the following equation where [math]\displaystyle{ {{I}_{1}}(.)\,\! }[/math] is the modified Bessel function of order one:

[math]\displaystyle{ H(x|k)=\underset{j=1}{\overset{k}{\mathop \sum }}\,\frac{{{x}^{2j-1}}}{{{2}^{2j-1}}(j-1)!j!{{I}_{1}}(x)}\,\! }[/math]

Find the values [math]\displaystyle{ {{\Pi }_{1}}\,\! }[/math] and [math]\displaystyle{ {{\Pi }_{2}}\,\! }[/math] by finding the solution [math]\displaystyle{ x\,\! }[/math] to [math]\displaystyle{ H(x|k)=\tfrac{\alpha }{2}\,\! }[/math] and [math]\displaystyle{ H(x|k)=1-\tfrac{\alpha }{2}\,\! }[/math] in the cases corresponding to the lower and upper bounds, respectively. Calculate [math]\displaystyle{ \Pi =\tfrac{4{{n}^{2}}}{{{x}^{2}}}\,\! }[/math] for each case.

If using the biased parameters, [math]\displaystyle{ \hat{\beta }\,\! }[/math] and [math]\displaystyle{ \hat{\lambda }\,\! }[/math], then the upper and lower confidence bounds are:

[math]\displaystyle{ \begin{align} {{IMTBF}_{L}}= & IMTBF\cdot {{\Pi }_{1}} \\ {{IMTBF}_{U}}= & IMTBF\cdot {{\Pi }_{2}} \end{align}\,\! }[/math]

where [math]\displaystyle{ IMTBF=\tfrac{1}{\hat{\lambda }\hat{\beta }{{t}^{\hat{\beta }-1}}}\,\! }[/math].

If using the unbiased parameters, [math]\displaystyle{ \bar{\beta }\,\! }[/math] and [math]\displaystyle{ \bar{\lambda }\,\! }[/math], then the upper and lower confidence bounds are:

[math]\displaystyle{ \begin{align} {{IMTBF}_{L}}= & IMTBF\cdot \left( \frac{N-1}{N} \right)\cdot {{\Pi }_{1}} \\ {{IMTBF}_{U}}= & IMTBF\cdot \left( \frac{N-1}{N} \right)\cdot {{\Pi }_{2}} \end{align}\,\! }[/math]

where [math]\displaystyle{ IMTBF=\tfrac{1}{\bar{\lambda }\bar{\beta }{{t}^{\bar{\beta }-1}}}\,\! }[/math].

Instantaneous Failure Intensity

Fisher Matrix Bounds

The instantaneous failure intensity, [math]\displaystyle{ {{\lambda }_{i}}(t)\,\! }[/math]. must be positive, thus [math]\displaystyle{ \ln {{\lambda }_{i}}(t)\,\! }[/math] is approximately treated as being normally distributed.

[math]\displaystyle{ \frac{\ln ({{\hat{\lambda }}_{i}}(t))-\ln ({{\lambda }_{i}}(t))}{\sqrt{Var\left[ \ln ({{\hat{\lambda }}_{i}}(t)) \right]}}\sim N(0,1)\,\! }[/math]

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

[math]\displaystyle{ CB={{\hat{\lambda }}_{i}}(t){{e}^{\pm {{z}_{\alpha }}\sqrt{Var({{\hat{\lambda }}_{i}}(t))}/{{\hat{\lambda }}_{i}}(t)}}\,\! }[/math]

where [math]\displaystyle{ {{\lambda }_{i}}(t)=\lambda \beta {{t}^{\beta -1}}\,\! }[/math] and:

[math]\displaystyle{ \begin{align} Var({{\hat{\lambda }}_{i}}(t))= & {{\left( \frac{\partial {{\lambda }_{i}}(t)}{\partial \beta } \right)}^{2}}Var(\hat{\beta })+{{\left( \frac{\partial {{\lambda }_{i}}(t)}{\partial \lambda } \right)}^{2}}Var(\hat{\lambda }) \\ & +2\left( \frac{\partial {{\lambda }_{i}}(t)}{\partial \beta } \right)\left( \frac{\partial {{\lambda }_{i}}(t)}{\partial \lambda } \right)cov(\hat{\beta },\hat{\lambda }) \end{align}\,\! }[/math]

The variance calculation is the same as the calculations in the confidence bounds on Beta.

[math]\displaystyle{ \begin{align} \frac{\partial {{\lambda }_{i}}(t)}{\partial \beta }= & \hat{\lambda }{{t}^{\hat{\beta }-1}}+\hat{\lambda }\hat{\beta }{{t}^{\hat{\beta }-1}}\ln (t) \\ \frac{\partial {{\lambda }_{i}}(t)}{\partial \lambda }= & \hat{\beta }{{t}^{\hat{\beta }-1}} \end{align}\,\! }[/math]

Crow Bounds

The 2-sided confidence bounds on the instantaneous failure intensity [math]\displaystyle{ (IFI)\,\! }[/math] are given by:

[math]\displaystyle{ \begin{align} {IFI_{L}}= & \frac{1}{{IMTBF}_{U}} \\ {IFI_{U}}= & \frac{1}{{IMTBF}_{L}} \end{align}\,\! }[/math]

where [math]\displaystyle{ IMTB{{F}_{L}}\,\! }[/math] and [math]\displaystyle{ IMTB{{F}_{U}}\,\! }[/math] are calculated using the process presented for the confidence bounds on the instantaneous MTBF.

Time Given Cumulative Failure Intensity

Fisher Matrix Bounds

The time, [math]\displaystyle{ T\,\! }[/math]. must be positive, thus [math]\displaystyle{ \ln T\,\! }[/math] is approximately treated as being normally distributed.

[math]\displaystyle{ \frac{\ln (\hat{T})-\ln (T)}{\sqrt{Var\left[ \ln \hat{T} \right]}}\ \tilde{\ }\ N(0,1)\,\! }[/math]

The confidence bounds on the time are given by:

[math]\displaystyle{ CB=\hat{T}{{e}^{\pm {{z}_{\alpha }}\sqrt{Var(\hat{T})}/\hat{T}}}\,\! }[/math]

where:

[math]\displaystyle{ Var(\hat{T})={{\left( \frac{\partial T}{\partial \beta } \right)}^{2}}Var(\hat{\beta })+{{\left( \frac{\partial T}{\partial \lambda } \right)}^{2}}Var(\hat{\lambda })+2\left( \frac{\partial T}{\partial \beta } \right)\left( \frac{\partial T}{\partial \lambda } \right)cov(\hat{\beta },\hat{\lambda })\,\! }[/math]

The variance calculation is the same as the calculations given in the confidence bounds on Beta.

[math]\displaystyle{ \hat{T}={{\left( \frac{{{\lambda }_{c}}(T)}{\lambda } \right)}^{1/(\beta -1)}}\,\! }[/math]
[math]\displaystyle{ \begin{align} \frac{\partial T}{\partial \beta }= & \frac{-{{\left( \tfrac{{{\lambda }_{c}}(T)}{\lambda } \right)}^{1/(\beta -1)}}\ln \left( \tfrac{{{\lambda }_{c}}(T)}{\lambda } \right)}{{{(1-\beta )}^{2}}} \\ \frac{\partial T}{\partial \lambda }= & {{\left( \frac{{{\lambda }_{c}}(T)}{\lambda } \right)}^{1/(\beta -1)}}\frac{1}{\lambda (1-\beta )} \end{align}\,\! }[/math]

Crow Bounds

The 2-sided confidence bounds on time given cumulative failure intensity [math]\displaystyle{ (CFI)\,\! }[/math] are given by:

[math]\displaystyle{ \hat{t}={{\left( \frac{CFI}{{\hat{\lambda }}} \right)}^{\tfrac{1}{\hat{\beta }-1}}}\,\! }[/math]

Then estimate, the number of failures, [math]\displaystyle{ N\,\! }[/math], such that:

[math]\displaystyle{ N=\hat{\lambda }{{\hat{t}}^{{\hat{\beta }}}}\,\! }[/math]

The lower and upper confidence bounds on time are then estimated using:

[math]\displaystyle{ \begin{align} {{t}_{L}}= & \frac{\chi _{\tfrac{\alpha }{2},2N}^{2}}{2\cdot CFI} \\ {{t}_{U}}= & \frac{\chi _{1-\tfrac{\alpha }{2},2N+2}^{2}}{2\cdot CFI} \end{align}\,\! }[/math]

Time Given Cumulative MTBF

Fisher Matrix Bounds

The time, [math]\displaystyle{ T\,\! }[/math]. must be positive, thus [math]\displaystyle{ \ln T\,\! }[/math] is approximately treated as being normally distributed.

[math]\displaystyle{ \frac{\ln (\hat{T})-\ln (T)}{\sqrt{Var\left[ \ln (\hat{T}) \right]}}\ \tilde{\ }\ N(0,1)\,\! }[/math]

The confidence bounds on the time are given by:

[math]\displaystyle{ CB=\hat{T}{{e}^{\pm {{z}_{\alpha }}\sqrt{Var(\hat{T})}/\hat{T}}}\,\! }[/math]

where:

[math]\displaystyle{ Var(\hat{T})={{\left( \frac{\partial T}{\partial \beta } \right)}^{2}}Var(\hat{\beta })+{{\left( \frac{\partial T}{\partial \lambda } \right)}^{2}}Var(\hat{\lambda })+2\left( \frac{\partial T}{\partial \beta } \right)\left( \frac{\partial T}{\partial \lambda } \right)cov(\hat{\beta },\hat{\lambda })\,\! }[/math]

The variance calculation is the same as the calculations in the confidence bounds on Beta.

[math]\displaystyle{ \hat{T}={{(\lambda \cdot {{m}_{c}})}^{1/(1-\beta )}}\,\! }[/math]
[math]\displaystyle{ \begin{align} \frac{\partial T}{\partial \beta }= & \frac{{{(\lambda \cdot {{m}_{c}})}^{1/(1-\beta )}}\ln (\lambda \cdot {{m}_{c}})}{{{(1-\beta )}^{2}}} \\ \frac{\partial T}{\partial \lambda }= & \frac{{{(\lambda \cdot {{m}_{c}})}^{1/(1-\beta )}}}{\lambda (1-\beta )} \end{align}\,\! }[/math]

Crow Bounds

The 2-sided confidence bounds on time given cumulative MTBF [math]\displaystyle{ (CMTBF)\,\! }[/math] are estimated using the process for the confidence bounds on time given cumulative failure intensity [math]\displaystyle{ (CFI)\,\! }[/math] where [math]\displaystyle{ CFI=\frac{1}{CMTBF}\,\! }[/math].

Time Given Instantaneous MTBF

Fisher Matrix Bounds

The time, [math]\displaystyle{ T\,\! }[/math]. must be positive, thus [math]\displaystyle{ \ln T\,\! }[/math] is approximately treated as being normally distributed.

[math]\displaystyle{ \frac{\ln (\hat{T})-\ln (T)}{\sqrt{Var\left[ \ln (\hat{T}) \right]}}\ \tilde{\ }\ N(0,1)\,\! }[/math]

The confidence bounds on the time are given by:

[math]\displaystyle{ CB=\hat{T}{{e}^{\pm {{z}_{\alpha }}\sqrt{Var(\hat{T})}/\hat{T}}}\,\! }[/math]

where:

[math]\displaystyle{ Var(\hat{T})={{\left( \frac{\partial T}{\partial \beta } \right)}^{2}}Var(\hat{\beta })+{{\left( \frac{\partial T}{\partial \lambda } \right)}^{2}}Var(\hat{\lambda })+2\left( \frac{\partial T}{\partial \beta } \right)\left( \frac{\partial T}{\partial \lambda } \right)cov(\hat{\beta },\hat{\lambda })\,\! }[/math]

The variance calculation is the same as the calculations in the confidence bounds on Beta.

[math]\displaystyle{ \hat{T}={{(\lambda \beta \cdot MTB{{F}_{i}})}^{1/(1-\beta )}}\,\! }[/math]
[math]\displaystyle{ \begin{align} \frac{\partial T}{\partial \beta }= & {{\left( \lambda \beta \cdot MTB{{F}_{i}} \right)}^{1/(1-\beta )}}[\frac{1}{{{(1-\beta )}^{2}}}\ln (\lambda \beta \cdot MTB{{F}_{i}})+\frac{1}{\beta (1-\beta )}] \\ \frac{\partial T}{\partial \lambda }= & \frac{{{(\lambda \beta \cdot MTB{{F}_{i}})}^{1/(1-\beta )}}}{\lambda (1-\beta )} \end{align}\,\! }[/math]

Crow Bounds

Failure Terminated

Calculate the constants [math]\displaystyle{ p_1\,\! }[/math] and [math]\displaystyle{ p_2\,\! }[/math] using procedures described for the confidence bounds on instantaneous MTBF. The lower and upper confidence bounds on time are then given by:

[math]\displaystyle{ {{\hat{t}}_{L}}={{\left( \frac{\lambda \beta \cdot IMTBF}{{{p}_{1}}} \right)}^{\tfrac{1}{1-\beta }}} }[/math]
[math]\displaystyle{ {{\hat{t}}_{U}}={{\left( \frac{\lambda \beta \cdot IMTBF}{{{p}_{2}}} \right)}^{\tfrac{1}{1-\beta }}} }[/math]

Time Terminated

Calculate the constants [math]\displaystyle{ {{\Pi }_{1}}\,\! }[/math] and [math]\displaystyle{ {{\Pi }_{2}}\,\! }[/math] using procedures described for the confidence bounds on instantaneous MTBF. The lower and upper confidence bounds on time are then given by:

[math]\displaystyle{ {{\hat{t}}_{L}}={{\left( \frac{\lambda \beta \cdot IMTBF}{{{\Pi }_{1}}} \right)}^{\tfrac{1}{1-\beta }}}\,\! }[/math]
[math]\displaystyle{ {{\hat{t}}_{U}}={{\left( \frac{\lambda \beta \cdot IMTBF}{{{\Pi }_{2}}} \right)}^{\tfrac{1}{1-\beta }}}\,\! }[/math]

Time Given Instantaneous Failure Intensity

Fisher Matrix Bounds

These bounds are based on:

[math]\displaystyle{ \frac{\ln (\hat{T})-\ln (T)}{\sqrt{Var\left[ \ln (\hat{T}) \right]}}\sim N(0,1)\,\! }[/math]

The confidence bounds on the time are given by:

[math]\displaystyle{ CB=\hat{T}{{e}^{\pm {{z}_{\alpha }}\sqrt{Var(\hat{T})}/\hat{T}}}\,\! }[/math]

where:

[math]\displaystyle{ \begin{align} Var(\hat{T})= & {{\left( \frac{\partial T}{\partial \beta } \right)}^{2}}Var(\hat{\beta })+{{\left( \frac{\partial T}{\partial \lambda } \right)}^{2}}Var(\hat{\lambda }) \\ & +2\left( \frac{\partial T}{\partial \beta } \right)\left( \frac{\partial T}{\partial \lambda } \right)cov(\hat{\beta },\hat{\lambda }) \end{align}\,\! }[/math]

The variance calculation is the same as the calculations given in the confidence bounds on Beta.

[math]\displaystyle{ \hat{T}={{\left( \frac{{{\lambda }_{i}}(T)}{\lambda \cdot \beta } \right)}^{1/(\beta -1)}}\,\! }[/math]
[math]\displaystyle{ \begin{align} \frac{\partial T}{\partial \beta }= & {{\left( \frac{{{\lambda }_{i}}(T)}{\lambda \cdot \beta } \right)}^{1/(\beta -1)}}[-\frac{\ln (\tfrac{{{\lambda }_{i}}(T)}{\lambda \cdot \beta })}{{{(\beta -1)}^{2}}}+\frac{1}{\beta (1-\beta )}] \\ \frac{\partial T}{\partial \lambda }= & {{\left( \frac{{{\lambda }_{i}}(T)}{\lambda \cdot \beta } \right)}^{1/(\beta -1)}}\frac{1}{\lambda (1-\beta )} \end{align}\,\! }[/math]

Crow Bounds

The 2-sided confidence bounds on time given instantaneous failure intensity [math]\displaystyle{ (IFI)\,\! }[/math] are estimated using the process for the confidence bounds on time given instantaneous MTBF where [math]\displaystyle{ IMTBF=\frac{1}{IFI}\,\! }[/math].

Reliability

Fisher Matrix Bounds

These bounds are based on:

[math]\displaystyle{ \log it(\hat{R}(t))\sim N(0,1)\,\! }[/math]
[math]\displaystyle{ \log it(\hat{R}(t))=\ln \left\{ \frac{\hat{R}(t)}{1-\hat{R}(t)} \right\}\,\! }[/math]

The confidence bounds on reliability are given by:

[math]\displaystyle{ CB=\frac{\hat{R}(t)}{\hat{R}(t)+(1-\hat{R}(t)){{e}^{\pm {{z}_{\alpha }}\sqrt{Var(\hat{R}(t))}/\left[ \hat{R}(t)(1-\hat{R}(t)) \right]}}}\,\! }[/math]
[math]\displaystyle{ Var(\hat{R}(t))={{\left( \frac{\partial R}{\partial \beta } \right)}^{2}}Var(\hat{\beta })+{{\left( \frac{\partial R}{\partial \lambda } \right)}^{2}}Var(\hat{\lambda })+2\left( \frac{\partial R}{\partial \beta } \right)\left( \frac{\partial R}{\partial \lambda } \right)cov(\hat{\beta },\hat{\lambda })\,\! }[/math]

The variance calculation is the same as the calculations in the confidence bounds on Beta.

[math]\displaystyle{ \begin{align} \frac{\partial R}{\partial \beta }= & {{e}^{-[\hat{\lambda }{{(t+d)}^{\hat{\beta }}}-\hat{\lambda }{{t}^{\hat{\beta }}}]}}[\lambda {{t}^{\hat{\beta }}}\ln (t)-\lambda {{(t+d)}^{\hat{\beta }}}\ln (t+d)] \\ \frac{\partial R}{\partial \lambda }= & {{e}^{-[\hat{\lambda }{{(t+d)}^{\hat{\beta }}}-\hat{\lambda }{{t}^{\hat{\beta }}}]}}[{{t}^{\hat{\beta }}}-{{(t+d)}^{\hat{\beta }}}] \end{align}\,\! }[/math]

Crow Bounds

Failure Terminated

For failure terminated data, the 100( [math]\displaystyle{ 1-\alpha \,\! }[/math] )% confidence interval on the current reliability at time [math]\displaystyle{ t\,\! }[/math] for a specified mission duration [math]\displaystyle{ d\,\! }[/math] is:

[math]\displaystyle{ \left( {{\left[ \hat{R}\left( d \right) \right]}^{\tfrac{1}{{{p}_{1}}}}},{{\left[ \hat{R}\left( d \right) \right]}^{\tfrac{1}{{{p}_{2}}}}} \right)\,\! }[/math]

where:

  • [math]\displaystyle{ \hat{R}\left( d \right)={{e}^{-\left[ \hat{\lambda }{{\left( t+d \right)}^{{\hat{\beta }}}}-\hat{\lambda }{{t}^{{\hat{\beta }}}} \right]}}\,\! }[/math]
  • [math]\displaystyle{ p_1\,\! }[/math] and [math]\displaystyle{ p_2\,\! }[/math] are obtained from the confidence bounds on instantaneous MTBF for failure terminated data.

Time Terminated

For time terminated data, the 100( [math]\displaystyle{ 1-\alpha \,\! }[/math] )% confidence interval on the current reliability at time [math]\displaystyle{ t\,\! }[/math] for a specified mission duration [math]\displaystyle{ d\,\! }[/math] is:

[math]\displaystyle{ \left( {{\left[ \hat{R}\left( d \right) \right]}^{\tfrac{1}{{{\Pi }_{1}}}}},{{\left[ \hat{R}\left( d \right) \right]}^{\tfrac{1}{{{\Pi }_{2}}}}} \right)\,\! }[/math]

where:

  • [math]\displaystyle{ \hat{R}\left( d \right)={{e}^{-\left[ \hat{\lambda }{{\left( t+d \right)}^{{\hat{\beta }}}}-\hat{\lambda }{{t}^{{\hat{\beta }}}} \right]}}\,\! }[/math]
  • [math]\displaystyle{ {{\Pi }_{1}}\,\! }[/math] and [math]\displaystyle{ {{\Pi }_{2}}\,\! }[/math] are obtained from the confidence bounds on instantaneous MTBF for time terminated data.

Time Given Reliability and Mission Time

Fisher Matrix Bounds

The time, [math]\displaystyle{ t\,\! }[/math]. must be positive, thus [math]\displaystyle{ \ln t\,\! }[/math] is approximately treated as being normally distributed.

[math]\displaystyle{ \frac{\ln (\hat{t})-\ln (t)}{\sqrt{Var\left[ \ln (\hat{t}) \right]}}\sim N(0,1)\,\! }[/math]

The confidence bounds on time are calculated by using:

[math]\displaystyle{ CB=\hat{t}{{e}^{\pm {{z}_{\alpha }}\sqrt{Var(\hat{t})}/\hat{t}}}\,\! }[/math]

where:

[math]\displaystyle{ Var(\hat{t})={{\left( \frac{\partial t}{\partial \beta } \right)}^{2}}Var(\hat{\beta })+{{\left( \frac{\partial t}{\partial \lambda } \right)}^{2}}Var(\hat{\lambda })+2\left( \frac{\partial t}{\partial \beta } \right)\left( \frac{\partial t}{\partial \lambda } \right)cov(\hat{\beta },\hat{\lambda })\,\! }[/math]

[math]\displaystyle{ \hat{t}\,\! }[/math] is calculated numerically from:

[math]\displaystyle{ \hat{R}(d)={{e}^{-[\hat{\lambda }{{(\hat{t}+d)}^{\hat{\beta }}}-\hat{\lambda }{{{\hat{t}}}^{\hat{\beta }}}]}}\text{ };\text{ }d\text{ = mission time}\,\! }[/math]

The variance calculations are done by:

[math]\displaystyle{ \begin{align} \frac{\partial t}{\partial \beta }= & \frac{{{{\hat{t}}}^{{\hat{\beta }}}}\ln (\hat{t})-{{(\hat{t}+d)}^{{\hat{\beta }}}}\ln (\hat{t}+d)}{\hat{\beta }{{(\hat{t}+d)}^{\hat{\beta }-1}}-\hat{\beta }{{{\hat{t}}}^{\hat{\beta }-1}}} \\ \frac{\partial t}{\partial \lambda }= & \frac{{{{\hat{t}}}^{{\hat{\beta }}}}-{{(\hat{t}+d)}^{{\hat{\beta }}}}}{\hat{\lambda }\hat{\beta }{{(\hat{t}+d)}^{\hat{\beta }-1}}-\hat{\lambda }\hat{\beta }{{{\hat{t}}}^{\hat{\beta }-1}}} \end{align}\,\! }[/math]

Crow Bounds

Failure Terminated

For failure terminated data, the 2-sided confidence bounds on time given reliability and mission time estimated by calculating:

[math]\displaystyle{ \left( {{{\hat{R}}}_{L}},{{{\hat{R}}}_{U}} \right)=\left( {{R}^{\tfrac{1}{{{p}_{1}}}}},{{R}^{\tfrac{1}{{{p}_{2}}}}}\, \right)\,\! }[/math]

where [math]\displaystyle{ p_1\,\! }[/math] and [math]\displaystyle{ p_2\,\! }[/math] are obtained from the confidence bounds on instantaneous MTBF for failure terminated data.

Let [math]\displaystyle{ R={{\hat{R}}_{L}}\,\! }[/math] and solve numerically for [math]\displaystyle{ {{t}_{1}}\,\! }[/math] using [math]\displaystyle{ R={{e}^{-[\hat{\lambda }{{({{{\hat{t}}}_{1}}+d)}^{\hat{\beta }}}-\hat{\lambda }\hat{t}_{1}^{\hat{\beta }}]}}\,\! }[/math].

Let [math]\displaystyle{ R={{\hat{R}}_{U}}\,\! }[/math] and solve numerically for [math]\displaystyle{ {{t}_{2}}\,\! }[/math] using [math]\displaystyle{ R={{e}^{-[\hat{\lambda }{{({{{\hat{t}}}_{2}}+d)}^{\hat{\beta }}}-\hat{\lambda }\hat{t}_{2}^{\hat{\beta }}]}}\,\! }[/math].

If [math]\displaystyle{ {{t}_{1}}\lt {{t}_{2}}\,\! }[/math] then [math]\displaystyle{ {{t}_{L}}={{t}_{1}}\,\! }[/math] and [math]\displaystyle{ {{t}_{U}}={{t}_{2}}\,\! }[/math]. If [math]\displaystyle{ {{t}_{1}}\gt {{t}_{2}}\,\! }[/math] then [math]\displaystyle{ {{t}_{L}}={{t}_{2}}\,\! }[/math] and [math]\displaystyle{ {{t}_{U}}={{t}_{1}}\,\! }[/math].

Time Terminated

For time terminated data, the 2-sided confidence bounds on time given reliability and mission time estimated by calculating:

[math]\displaystyle{ \left( {{{\hat{R}}}_{L}},{{{\hat{R}}}_{U}} \right)=\left( {{R}^{\tfrac{1}{{{\Pi }_{1}}}}},{{R}^{\tfrac{1}{{{\Pi }_{2}}}}}\, \right) }[/math].

where [math]\displaystyle{ \Pi_1\,\! }[/math] and [math]\displaystyle{ \Pi_2\,\! }[/math] are obtained from the confidence bounds on instantaneous MTBF for time terminated data.

Let [math]\displaystyle{ R={{\hat{R}}_{L}}\,\! }[/math] and solve numerically for [math]\displaystyle{ {{t}_{1}}\,\! }[/math] using [math]\displaystyle{ R={{e}^{-[\hat{\lambda }{{({{{\hat{t}}}_{1}}+d)}^{\hat{\beta }}}-\hat{\lambda }\hat{t}_{1}^{\hat{\beta }}]}}\,\! }[/math].

Let [math]\displaystyle{ R={{\hat{R}}_{U}}\,\! }[/math] and solve numerically for [math]\displaystyle{ {{t}_{2}}\,\! }[/math] using [math]\displaystyle{ R={{e}^{-[\hat{\lambda }{{({{{\hat{t}}}_{2}}+d)}^{\hat{\beta }}}-\hat{\lambda }\hat{t}_{2}^{\hat{\beta }}]}}\,\! }[/math].

If [math]\displaystyle{ {{t}_{1}}\lt {{t}_{2}}\,\! }[/math]. then [math]\displaystyle{ {{t}_{L}}={{t}_{1}}\,\! }[/math] and [math]\displaystyle{ {{t}_{U}}={{t}_{2}}\,\! }[/math]. If [math]\displaystyle{ {{t}_{1}}\gt {{t}_{2}}\,\! }[/math]. then [math]\displaystyle{ {{t}_{L}}={{t}_{2}}\,\! }[/math] and [math]\displaystyle{ {{t}_{U}}={{t}_{1}}\,\! }[/math].

Mission Time Given Reliability and Time

Fisher Matrix Bounds

The mission time, [math]\displaystyle{ d\,\! }[/math]. must be positive, thus [math]\displaystyle{ \ln \left( d \right)\,\! }[/math] is approximately treated as being normally distributed.

[math]\displaystyle{ \frac{\ln (\hat{d})-\ln (d)}{\sqrt{Var\left[ \ln (\hat{d}) \right]}}\sim N(0,1)\,\! }[/math]

The confidence bounds on mission time are given by using:

[math]\displaystyle{ CB=\hat{d}{{e}^{\pm {{z}_{\alpha }}\sqrt{Var(\hat{d})}/\hat{d}}}\,\! }[/math]

where:

[math]\displaystyle{ Var(\hat{d})={{\left( \frac{\partial d}{\partial \beta } \right)}^{2}}Var(\hat{\beta })+{{\left( \frac{\partial d}{\partial \lambda } \right)}^{2}}Var(\hat{\lambda })+2\left( \frac{\partial td}{\partial \beta } \right)\left( \frac{\partial d}{\partial \lambda } \right)cov(\hat{\beta },\hat{\lambda })\,\! }[/math]

Calculate [math]\displaystyle{ \hat{d}\,\! }[/math] from:

[math]\displaystyle{ \hat{d}={{\left[ {{t}^{{\hat{\beta }}}}-\frac{\ln (R)}{{\hat{\lambda }}} \right]}^{\tfrac{1}{{\hat{\beta }}}}}-t\,\! }[/math]

The variance calculations are done by:

[math]\displaystyle{ \begin{align} \frac{\partial d}{\partial \beta }= & \left[ \frac{{{t}^{{\hat{\beta }}}}\ln (t)}{{{(t+\hat{d})}^{{\hat{\beta }}}}}-\ln (t+\hat{d}) \right]\cdot \frac{t+\hat{d}}{{\hat{\beta }}} \\ \frac{\partial d}{\partial \lambda }= & \frac{{{t}^{{\hat{\beta }}}}-{{(t+\hat{d})}^{{\hat{\beta }}}}}{\hat{\lambda }\hat{\beta }{{(t+\hat{d})}^{\hat{\beta }-1}}} \end{align}\,\! }[/math]

Crow Bounds

Failure Terminated

Step 1: Calculate [math]\displaystyle{ ({{\hat{R}}_{lower}},{{\hat{R}}_{upper}})=({{R}^{\tfrac{1}{{{p}_{1}}}}},{{R}^{\tfrac{1}{{{p}_{2}}}}})\,\! }[/math].

Step 2: Let [math]\displaystyle{ R={{\hat{R}}_{lower}}\,\! }[/math] and solve for [math]\displaystyle{ {{d}_{1}}\,\! }[/math] such that:

[math]\displaystyle{ {{d}_{1}}={{\left( {{t}^{{\hat{\beta }}}}-\frac{\ln ({{R}_{lower}})}{{\hat{\lambda }}} \right)}^{\tfrac{1}{{\hat{\beta }}}}}-t\,\! }[/math]

Step 3: Let [math]\displaystyle{ R={{\hat{R}}_{upper}}\,\! }[/math] and solve for [math]\displaystyle{ {{d}_{2}}\,\! }[/math] such that:

[math]\displaystyle{ {{d}_{2}}={{\left( {{t}^{{\hat{\beta }}}}-\frac{\ln ({{R}_{upper}})}{{\hat{\lambda }}} \right)}^{\tfrac{1}{{\hat{\beta }}}}}-t\,\! }[/math]

Step 4: If [math]\displaystyle{ {{d}_{1}}\lt {{d}_{2}}\,\! }[/math]. then [math]\displaystyle{ {{d}_{lower}}={{d}_{1}}\,\! }[/math] and [math]\displaystyle{ {{d}_{upper}}={{d}_{2}}\,\! }[/math]. If [math]\displaystyle{ {{d}_{1}}\gt {{d}_{2}}\,\! }[/math]. then [math]\displaystyle{ {{d}_{lower}}={{d}_{2}}\,\! }[/math] and [math]\displaystyle{ {{d}_{upper}}={{d}_{1}}\,\! }[/math].

Time Terminated

Step 1: Calculate [math]\displaystyle{ ({{\hat{R}}_{lower}},{{\hat{R}}_{upper}})=({{R}^{\tfrac{1}{{{\Pi }_{1}}}}},{{R}^{\tfrac{1}{{{\Pi }_{2}}}}})\,\! }[/math].

Step 2: Let [math]\displaystyle{ R={{\hat{R}}_{lower}}\,\! }[/math] and solve for [math]\displaystyle{ {{d}_{1}}\,\! }[/math] using the same equation given for the failure terminated data.

Step 3: Let [math]\displaystyle{ R={{\hat{R}}_{upper}}\,\! }[/math] and solve for [math]\displaystyle{ {{d}_{2}}\,\! }[/math] using the same equation given for the failure terminated data.

Step 4: If [math]\displaystyle{ {{d}_{1}}\lt {{d}_{2}}\,\! }[/math]. then [math]\displaystyle{ {{d}_{lower}}={{d}_{1}}\,\! }[/math] and [math]\displaystyle{ {{d}_{upper}}={{d}_{2}}\,\! }[/math]. If [math]\displaystyle{ {{d}_{1}}\gt {{d}_{2}}\,\! }[/math]. then [math]\displaystyle{ {{d}_{lower}}={{d}_{2}}\,\! }[/math] and [math]\displaystyle{ {{d}_{upper}}={{d}_{1}}\,\! }[/math].

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