Crow-AMSAA Confidence Bounds: Difference between revisions

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===Bounds on Beta===
===Bounds on Beta===
====Fisher Matrix Bounds====<!-- THIS SECTION HEADER IS LINKED TO: Crow-AMSAA - NHPP. IF YOU RENAME THE SECTION, YOU MUST UPDATE THE LINK. -->
====Fisher Matrix Bounds====<!-- THIS SECTION HEADER IS LINKED TO: Crow-AMSAA - NHPP. IF YOU RENAME THE SECTION, YOU MUST UPDATE THE LINK. -->
The parameter <math>\beta \,\!</math> must be positive, thus <math>\ln \beta \,\!</math> is treated as being normally distributed as well.   
The parameter <math>\beta \,\!</math> must be positive, thus <math>\ln \beta \,\!</math> is treated as being normally distributed as well.   


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


The approximate confidence bounds are given as:  
The approximate confidence bounds are given as:  


::<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>\alpha \,\!</math> in <math>{{z}_{\alpha }}\,\!</math> is different ( <math>\alpha /2\,\!</math> , <math>\alpha \,\!</math> ) according to a 2-sided confidence interval or a 1-sided confidence interval, and variances can be calculated using the Fisher Matrix.  
<math>\alpha \,\!</math> in <math>{{z}_{\alpha }}\,\!</math> is different ( <math>\alpha /2\,\!</math>, <math>\alpha \,\!</math> ) according to a 2-sided confidence interval or a 1-sided confidence interval, and variances can be calculated using the Fisher Matrix.  


::<math>\left[ \begin{matrix}
::<math>\left[ \begin{matrix}
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   Var(\widehat{\lambda }) & Cov(\widehat{\beta },\widehat{\lambda })  \\
   Var(\widehat{\lambda }) & Cov(\widehat{\beta },\widehat{\lambda })  \\
   Cov(\widehat{\beta },\widehat{\lambda }) & Var(\widehat{\beta })  \\
   Cov(\widehat{\beta },\widehat{\lambda }) & Var(\widehat{\beta })  \\
\end{matrix} \right]</math>
\end{matrix} \right]\,\!</math>


::<math>\Lambda \,\!</math> is the natural log-likelihood function:  
::<math>\Lambda \,\!</math> is the natural log-likelihood function:  


::<math>\Lambda =N\ln \lambda +N\ln \beta -\lambda {{T}^{\beta }}+(\beta -1)\underset{i=1}{\overset{N}{\mathop \sum }}\,\ln {{T}_{i}}</math>
::<math>\Lambda =N\ln \lambda +N\ln \beta -\lambda {{T}^{\beta }}+(\beta -1)\underset{i=1}{\overset{N}{\mathop \sum }}\,\ln {{T}_{i}}\,\!</math>


::<math>\frac{{{\partial }^{2}}\Lambda }{\partial {{\lambda }^{2}}}=-\frac{N}{{{\lambda }^{2}}}</math>
::<math>\frac{{{\partial }^{2}}\Lambda }{\partial {{\lambda }^{2}}}=-\frac{N}{{{\lambda }^{2}}}\,\!</math>


:and:  
:and:  


::<math>\frac{{{\partial }^{2}}\Lambda }{\partial {{\beta }^{2}}}=-\frac{N}{{{\beta }^{2}}}-\lambda {{T}^{\beta }}{{(\ln T)}^{2}}</math>
::<math>\frac{{{\partial }^{2}}\Lambda }{\partial {{\beta }^{2}}}=-\frac{N}{{{\beta }^{2}}}-\lambda {{T}^{\beta }}{{(\ln T)}^{2}}\,\!</math>


:also:  
:also:  


::<math>\frac{{{\partial }^{2}}\Lambda }{\partial \lambda \partial \beta }=-{{T}^{\beta }}\ln T</math>
::<math>\frac{{{\partial }^{2}}\Lambda }{\partial \lambda \partial \beta }=-{{T}^{\beta }}\ln T\,\!</math>


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


For the 2-sided <math>(1-\alpha )\,\!</math> 100-percent confidence interval on <math>\beta \,\!</math> , calculate:
For the 2-sided <math>(1-\alpha )\,\!</math> 100-percent confidence interval on <math>\beta \,\!</math>, calculate:


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


The fractiles can be found in the tables of the <math>{{\chi }^{2}}\,\!</math> distribution. Thus the confidence bounds on <math>\beta \,\!</math> are:
The fractiles can be found in the tables of the <math>{{\chi }^{2}}\,\!</math> distribution. Thus the confidence bounds on <math>\beta \,\!</math> are:


::<math>\begin{align}
::<math>\begin{align}
   {{\beta }_{L}}= & {{D}_{L}}\cdot \hat{\beta } \\  
   {{\beta }_{L}}= & {{D}_{L}}\cdot \hat{\beta } \\  
   {{\beta }_{U}}= & {{D}_{U}}\cdot \hat{\beta }   
   {{\beta }_{U}}= & {{D}_{U}}\cdot \hat{\beta }   
\end{align}</math>
\end{align}\,\!</math>


'''Failure Terminated Data'''
'''Failure Terminated Data'''


For the 2-sided <math>(1-\alpha )\,\!</math> 100-percent confidence interval on <math>\beta \,\!</math> , calculate:
For the 2-sided <math>(1-\alpha )\,\!</math> 100-percent confidence interval on <math>\beta \,\!</math>, calculate:


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


Thus the confidence bounds on <math>\beta \,\!</math> are:
Thus the confidence bounds on <math>\beta \,\!</math> are:


::<math>\begin{align}
::<math>\begin{align}
   {{\beta }_{L}}= & {{D}_{L}}\cdot \hat{\beta } \\  
   {{\beta }_{L}}= & {{D}_{L}}\cdot \hat{\beta } \\  
   {{\beta }_{U}}= & {{D}_{U}}\cdot \hat{\beta }   
   {{\beta }_{U}}= & {{D}_{U}}\cdot \hat{\beta }   
\end{align}</math>
\end{align}\,\!</math>


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


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


::<math>\hat{\lambda }=\frac{n}{{{T}^{*\hat{\beta }}}}</math>
::<math>\hat{\lambda }=\frac{n}{{{T}^{*\hat{\beta }}}}\,\!</math>


The variance calculation is the same as given above in the [[Crow-AMSAA_Confidence_Bounds#Fisher_Matrix_Bounds|Bounds on Beta]] section.
The variance calculation is the same as given above in the [[Crow-AMSAA_Confidence_Bounds#Fisher_Matrix_Bounds|Bounds on Beta]] section.
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'''Time Terminated Data'''
'''Time Terminated Data'''


For the 2-sided <math>(1-\alpha )\,\!</math> 100-percent confidence interval, the confidence bounds on <math>\lambda \,\!</math> are:
For the 2-sided <math>(1-\alpha )\,\!</math> 100-percent confidence interval, the confidence bounds on <math>\lambda \,\!</math> are:


::<math>\begin{align}
::<math>\begin{align}
   {{\lambda }_{L}}= & \frac{\chi _{\tfrac{\alpha }{2},2N}^{2}}{2{{T}^{{\hat{\beta }}}}} \\  
   {{\lambda }_{L}}= & \frac{\chi _{\tfrac{\alpha }{2},2N}^{2}}{2{{T}^{{\hat{\beta }}}}} \\  
   {{\lambda }_{U}}= & \frac{\chi _{1-\tfrac{\alpha }{2},2N+2}^{2}}{2{{T}^{{\hat{\beta }}}}}   
   {{\lambda }_{U}}= & \frac{\chi _{1-\tfrac{\alpha }{2},2N+2}^{2}}{2{{T}^{{\hat{\beta }}}}}   
\end{align}</math>
\end{align}\,\!</math>


The fractiles can be found in the tables of the <math>{{\chi }^{2}}\,\!</math> distribution.
The fractiles can be found in the tables of the <math>{{\chi }^{2}}\,\!</math> distribution.


'''Failure Terminated Data'''
'''Failure Terminated Data'''


For the 2-sided <math>(1-\alpha )\,\!</math> 100-percent confidence interval, the confidence bounds on <math>\lambda \,\!</math> are:
For the 2-sided <math>(1-\alpha )\,\!</math> 100-percent confidence interval, the confidence bounds on <math>\lambda \,\!</math> are:


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


===Bounds on Growth Rate===
===Bounds on Growth Rate===
Since the growth rate is equal to <math>1-\beta \,\!</math> , the confidence bounds for both the Fisher Matrix and Crow methods are:
Since the growth rate is equal to <math>1-\beta \,\!</math>, the confidence bounds for both the Fisher Matrix and Crow methods are:
<br>
<br>
::<math>G\text{row}th Rate_L=1-\beta_U\,\!</math>
::<math>G\text{row}th Rate_L=1-\beta_U\,\!</math>
::<math>G\text{row}th Rate_U=1-\beta_L\,\!</math>
::<math>G\text{row}th Rate_U=1-\beta_L\,\!</math>


<math>{{\beta }_{L}}\,\!</math> and <math>{{\beta }_{U}}\,\!</math> are obtained using the methods described above in the [[Crow-AMSAA_Confidence_Bounds#Bounds_on_Beta|Bounds on Beta]] section.  <PER LISA: ASK SME TO CONFIRM THAT THIS IS ADEQUATE.>>
<math>{{\beta }_{L}}\,\!</math> and <math>{{\beta }_{U}}\,\!</math> are obtained using the methods described above in the [[Crow-AMSAA_Confidence_Bounds#Bounds_on_Beta|Bounds on Beta]] section.  <PER LISA: ASK SME TO CONFIRM THAT THIS IS ADEQUATE.>>


===Bounds on Cumulative MTBF===
===Bounds on Cumulative MTBF===
====Fisher Matrix Bounds====
====Fisher Matrix Bounds====
The cumulative MTBF, <math>{{m}_{c}}(t)\,\!</math> , must be positive, thus <math>\ln {{m}_{c}}(t)\,\!</math> is treated as being normally distributed as well.
The cumulative MTBF, <math>{{m}_{c}}(t)\,\!</math>, must be positive, thus <math>\ln {{m}_{c}}(t)\,\!</math> is treated as being normally distributed as well.


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


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


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


:where:  
:where:  


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


::<math>\begin{align}
::<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 }) \\  
   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 })\,   
   & +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>
\end{align}\,\!</math>


The variance calculation is the same as given above in the [[Crow-AMSAA_Confidence_Bounds#Fisher_Matrix_Bounds|Bounds on Beta]] section and:
The variance calculation is the same as given above in the [[Crow-AMSAA_Confidence_Bounds#Fisher_Matrix_Bounds|Bounds on Beta]] section and:
Line 139: Line 139:
   \frac{\partial {{m}_{c}}(t)}{\partial \beta }= & -\frac{1}{{\hat{\lambda }}}{{t}^{1-\hat{\beta }}}\ln t \\  
   \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 }}}   
   \frac{\partial {{m}_{c}}(t)}{\partial \lambda }= & -\frac{1}{{{{\hat{\lambda }}}^{2}}}{{t}^{1-\hat{\beta }}}   
\end{align}</math>
\end{align}\,\!</math>


====Crow Bounds====
====Crow Bounds====
To calculate the Crow confidence bounds on cumulative MTBF, first calculate the Crow cumulative failure intensity confidence bounds:  
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>
::<math>C{{(t)}_{L}}=\frac{\chi _{\tfrac{\alpha }{2},2N}^{2}}{2\cdot t}\,\!</math>


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


:Then:
:Then:
Line 153: Line 153:
   & {{[MTB{{F}_{c}}]}_{L}}= & \frac{1}{C{{(t)}_{U}}} \\  
   & {{[MTB{{F}_{c}}]}_{L}}= & \frac{1}{C{{(t)}_{U}}} \\  
  & {{[MTB{{F}_{c}}]}_{U}}= & \frac{1}{C{{(t)}_{L}}}   
  & {{[MTB{{F}_{c}}]}_{U}}= & \frac{1}{C{{(t)}_{L}}}   
\end{align}</math>
\end{align}\,\!</math>




===Bounds on Instantaneous MTBF===
===Bounds on Instantaneous MTBF===
====Fisher Matrix Bounds====
====Fisher Matrix Bounds====
The instantaneous MTBF, <math>{{m}_{i}}(t)\,\!</math> , must be positive, thus <math>\ln {{m}_{i}}(t)\,\!</math> is treated as being normally distributed as well.  
The instantaneous MTBF, <math>{{m}_{i}}(t)\,\!</math>, must be positive, thus <math>\ln {{m}_{i}}(t)\,\!</math> is treated as being normally distributed as well.  


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


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


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


:where:  
:where:  


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


::<math>\begin{align}
::<math>\begin{align}
   Var({{{\hat{m}}}_{i}}(t))= & {{\left( \frac{\partial {{m}_{i}}(t)}{\partial \beta } \right)}^{2}}Var(\hat{\beta })+{{\left( \frac{\partial {{m}_{i}}(t)}{\partial \lambda } \right)}^{2}}Var(\hat{\lambda }) \\  
   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 }).   
   & +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>


The variance calculation is the same as given above in the [[Crow-AMSAA_Confidence_Bounds#Fisher_Matrix_Bounds|Bounds on Beta]] section and:
The variance calculation is the same as given above in the [[Crow-AMSAA_Confidence_Bounds#Fisher_Matrix_Bounds|Bounds on Beta]] section and:
Line 180: Line 180:
   \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 \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 }}}   
   \frac{\partial {{m}_{i}}(t)}{\partial \lambda }= & -\frac{1}{{{{\hat{\lambda }}}^{2}}\hat{\beta }}{{t}^{1-\hat{\beta }}}   
\end{align}</math>
\end{align}\,\!</math>


====Crow Bounds====
====Crow Bounds====
Line 188: Line 188:
Consider the following equation:  
Consider the following equation:  


::<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>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>{{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:
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:


::<math>\begin{align}
::<math>\begin{align}
   {{[MTB{{F}_{i}}]}_{L}}= & MTB{{F}_{i}}\cdot {{p}_{1}} \\  
   {{[MTB{{F}_{i}}]}_{L}}= & MTB{{F}_{i}}\cdot {{p}_{1}} \\  
   {{[MTB{{F}_{i}}]}_{U}}= & MTB{{F}_{i}}\cdot {{p}_{2}}   
   {{[MTB{{F}_{i}}]}_{U}}= & MTB{{F}_{i}}\cdot {{p}_{2}}   
\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:
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:


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


where <math>MTB{{F}_{i}}=\tfrac{1}{\hat{\lambda }\hat{\beta }{{t}^{\hat{\beta }-1}}}\,\!</math> .
where <math>MTB{{F}_{i}}=\tfrac{1}{\hat{\lambda }\hat{\beta }{{t}^{\hat{\beta }-1}}}\,\!</math>.


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


Consider the following equation where <math>{{I}_{1}}(.)\,\!</math> is the modified Bessel function of order one:  
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>
::<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:
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}
::<math>\begin{align}
   {{[MTB{{F}_{i}}]}_{L}}= & MTB{{F}_{i}}\cdot {{\Pi }_{1}} \\  
   {{[MTB{{F}_{i}}]}_{L}}= & MTB{{F}_{i}}\cdot {{\Pi }_{1}} \\  
   {{[MTB{{F}_{i}}]}_{U}}= & MTB{{F}_{i}}\cdot {{\Pi }_{2}}   
   {{[MTB{{F}_{i}}]}_{U}}= & MTB{{F}_{i}}\cdot {{\Pi }_{2}}   
\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:
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:


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


where <math>MTB{{F}_{i}}=\tfrac{1}{\hat{\lambda }\hat{\beta }{{t}^{\hat{\beta }-1}}}\,\!</math> .
where <math>MTB{{F}_{i}}=\tfrac{1}{\hat{\lambda }\hat{\beta }{{t}^{\hat{\beta }-1}}}\,\!</math>.


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


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


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


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


:where:  
:where:  


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


:and:  
:and:  
Line 247: Line 247:
   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 }) \\  
   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 })   
   & +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>
\end{align}\,\!</math>


The variance calculation is the same as given above in the [[Crow-AMSAA_Confidence_Bounds#Fisher_Matrix_Bounds|Bounds on Beta]] section and:  
The variance calculation is the same as given above in the [[Crow-AMSAA_Confidence_Bounds#Fisher_Matrix_Bounds|Bounds on Beta]] section and:  
Line 254: Line 254:
   \frac{\partial {{\lambda }_{c}}(t)}{\partial \beta }= & \hat{\lambda }{{t}^{\hat{\beta }-1}}\ln t \\  
   \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}}   
   \frac{\partial {{\lambda }_{c}}(t)}{\partial \lambda }= & {{t}^{\hat{\beta }-1}}   
\end{align}</math>
\end{align}\,\!</math>


====Crow Bounds====
====Crow Bounds====
Line 262: Line 262:
   C{{(t)}_{L}}= & \frac{\chi _{\tfrac{\alpha }{2},2N}^{2}}{2\cdot t} \\  
   C{{(t)}_{L}}= & \frac{\chi _{\tfrac{\alpha }{2},2N}^{2}}{2\cdot t} \\  
   C{{(t)}_{U}}= & \frac{\chi _{1-\tfrac{\alpha }{2},2N+2}^{2}}{2\cdot t}   
   C{{(t)}_{U}}= & \frac{\chi _{1-\tfrac{\alpha }{2},2N+2}^{2}}{2\cdot t}   
\end{align}</math>
\end{align}\,\!</math>


===Bounds on Instantaneous Failure Intensity===
===Bounds on Instantaneous Failure Intensity===
====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 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 treated as being normally distributed.  


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


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


::<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 given above in the [[Crow-AMSAA_Confidence_Bounds#Fisher_Matrix_Bounds|Bounds on Beta]] section and:  
The variance calculation is the same as given above in the [[Crow-AMSAA_Confidence_Bounds#Fisher_Matrix_Bounds|Bounds on Beta]] section and:  
Line 288: Line 288:
   \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====
Line 296: Line 296:
   {{\lambda }_{i}}{{(t)}_{L}}= & \frac{1}{{{[MTB{{F}_{i}}]}_{U}}} \\  
   {{\lambda }_{i}}{{(t)}_{L}}= & \frac{1}{{{[MTB{{F}_{i}}]}_{U}}} \\  
   {{\lambda }_{i}}{{(t)}_{U}}= & \frac{1}{{{[MTB{{F}_{i}}]}_{L}}}   
   {{\lambda }_{i}}{{(t)}_{U}}= & \frac{1}{{{[MTB{{F}_{i}}]}_{L}}}   
\end{align}</math>
\end{align}\,\!</math>


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


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


Confidence bounds on the time are given by:  
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:  
Line 313: Line 313:
   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 given above in the [[Crow-AMSAA_Confidence_Bounds#Fisher_Matrix_Bounds|Bounds on Beta]] section and:  
The variance calculation is the same as given above in the [[Crow-AMSAA_Confidence_Bounds#Fisher_Matrix_Bounds|Bounds on Beta]] section and:  
Line 320: Line 320:
   \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 \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 )}   
   \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====
====Crow Bounds====
:Step 1: Calculate:
:Step 1: Calculate:


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


:Step 2: Estimate the number of failures:
:Step 2: Estimate the number of failures:


::<math>N(\hat{T})=\hat{\lambda }{{\hat{T}}^{{\hat{\beta }}}}</math>
::<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:  
: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}
::<math>\begin{align}
   {{t}_{l}}= & \frac{\chi _{\tfrac{\alpha }{2},2N}^{2}}{2\cdot {{\lambda }_{c}}(T)} \\  
   {{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)}   
   {{t}_{u}}= & \frac{\chi _{1-\tfrac{\alpha }{2},2N+2}^{2}}{2\cdot {{\lambda }_{c}}(T)}   
\end{align}</math>
\end{align}\,\!</math>


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


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


Confidence bounds on the time are given by:  
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:  
Line 353: Line 353:
   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 given above in the [[Crow-AMSAA_Confidence_Bounds#Fisher_Matrix_Bounds|Bounds on Beta]] section and:  
The variance calculation is the same as given above in the [[Crow-AMSAA_Confidence_Bounds#Fisher_Matrix_Bounds|Bounds on Beta]] section and:  


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


====Crow Bounds====
====Crow Bounds====
:Step 1: Calculate <math>{{\lambda }_{c}}(T)=\tfrac{1}{MTB{{F}_{c}}}\,\!</math> .
:Step 1: Calculate <math>{{\lambda }_{c}}(T)=\tfrac{1}{MTB{{F}_{c}}}\,\!</math>.
:Step 2: Use the equations from 5.2.8.2 to calculate the bounds on time given the cumulative failure intensity.
:Step 2: Use the equations from 5.2.8.2 to calculate the bounds on time given the cumulative failure intensity.


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


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


Confidence bounds on the time are given by:  
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:  
Line 383: Line 383:
   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 given above in the [[Crow-AMSAA_Confidence_Bounds#Fisher_Matrix_Bounds|Bounds on Beta]] section and:
The variance calculation is the same as given above in the [[Crow-AMSAA_Confidence_Bounds#Fisher_Matrix_Bounds|Bounds on Beta]] section and:


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


====Crow Bounds====
====Crow Bounds====
Line 400: Line 400:
====Failure Terminated Data====
====Failure Terminated Data====


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


So the lower an upper bounds on time are:
So the lower an upper bounds on time are:


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


::<math>{{\hat{T}}_{U}}={{(\frac{\lambda \beta \cdot MTB{{F}_{i}}}{{{c}_{2}}})}^{1/(1-\beta )}}</math>
::<math>{{\hat{T}}_{U}}={{(\frac{\lambda \beta \cdot MTB{{F}_{i}}}{{{c}_{2}}})}^{1/(1-\beta )}}\,\!</math>


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


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


So the lower and upper bounds on time are:
So the lower and upper bounds on time are:


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


::<math>{{\hat{T}}_{U}}={{(\frac{\lambda \beta \cdot MTB{{F}_{i}}}{{{\Pi }_{2}}})}^{1/(1-\beta )}}</math>
::<math>{{\hat{T}}_{U}}={{(\frac{\lambda \beta \cdot MTB{{F}_{i}}}{{{\Pi }_{2}}})}^{1/(1-\beta )}}\,\!</math>


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


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


Confidence bounds on the time are given by:  
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:  
Line 433: Line 433:
   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 given above in the [[Crow-AMSAA_Confidence_Bounds#Fisher_Matrix_Bounds|Bounds on Beta]] section and:   
The variance calculation is the same as given above in the [[Crow-AMSAA_Confidence_Bounds#Fisher_Matrix_Bounds|Bounds on Beta]] section and:   


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


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


====Crow Bounds====
====Crow Bounds====
:Step 1: Calculate <math>MTB{{F}_{i}}=\tfrac{1}{{{\lambda }_{i}}(T)}\,\!</math> .
:Step 1: Calculate <math>MTB{{F}_{i}}=\tfrac{1}{{{\lambda }_{i}}(T)}\,\!</math>.
:Step 2: Use the equations from 5.2.10.2 to calculate the bounds on time given the instantaneous failure intensity.
:Step 2: Use the equations from 5.2.10.2 to calculate the bounds on time given the instantaneous failure intensity.


===Bounds on Cumulative Number of Failures===
===Bounds on Cumulative Number of Failures===
====Fisher Matrix Bounds====
====Fisher Matrix Bounds====
The cumulative number of failures, <math>N(t)\,\!</math> , must be positive, thus <math>\ln N(t)\,\!</math> is treated as being normally distributed.   
The cumulative number of failures, <math>N(t)\,\!</math>, must be positive, thus <math>\ln N(t)\,\!</math> is treated as being normally distributed.   


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


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


:where:  
:where:  


::<math>\hat{N}(t)=\hat{\lambda }{{t}^{{\hat{\beta }}}}</math>
::<math>\hat{N}(t)=\hat{\lambda }{{t}^{{\hat{\beta }}}}\,\!</math>


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


The variance calculation is the same as given above in the [[Crow-AMSAA_Confidence_Bounds#Fisher_Matrix_Bounds|Bounds on Beta]] section and:  
The variance calculation is the same as given above in the [[Crow-AMSAA_Confidence_Bounds#Fisher_Matrix_Bounds|Bounds on Beta]] section and:  
Line 470: Line 470:
   \frac{\partial \hat{N}(t)}{\partial \beta }= & \hat{\lambda }{{t}^{{\hat{\beta }}}}\ln t \\  
   \frac{\partial \hat{N}(t)}{\partial \beta }= & \hat{\lambda }{{t}^{{\hat{\beta }}}}\ln t \\  
   \frac{\partial \hat{N}(t)}{\partial \lambda }= & {{t}^{{\hat{\beta }}}}   
   \frac{\partial \hat{N}(t)}{\partial \lambda }= & {{t}^{{\hat{\beta }}}}   
\end{align}</math>
\end{align}\,\!</math>


====Crow Bounds====
====Crow Bounds====
Line 479: Line 479:
   {{N}_{L}}(T)= & \frac{T}{{\hat{\beta }}}{{\lambda }_{i}}{{(T)}_{L}} \\  
   {{N}_{L}}(T)= & \frac{T}{{\hat{\beta }}}{{\lambda }_{i}}{{(T)}_{L}} \\  
   {{N}_{U}}(T)= & \frac{T}{{\hat{\beta }}}{{\lambda }_{i}}{{(T)}_{U}}   
   {{N}_{U}}(T)= & \frac{T}{{\hat{\beta }}}{{\lambda }_{i}}{{(T)}_{U}}   
\end{align}</math>
\end{align}\,\!</math>


where <math>{{\lambda }_{i}}{{(T)}_{L}}\,\!</math> and <math>{{\lambda }_{i}}{{(T)}_{U}}\,\!</math> can be obtained from the [[Crow-AMSAA_Confidence_Bounds#Bounds_on_Instantaneous_Failure_Intensity|Crow instantaneous failure intensity confidence bounds equations]] given above.
where <math>{{\lambda }_{i}}{{(T)}_{L}}\,\!</math> and <math>{{\lambda }_{i}}{{(T)}_{U}}\,\!</math> can be obtained from the [[Crow-AMSAA_Confidence_Bounds#Bounds_on_Instantaneous_Failure_Intensity|Crow instantaneous failure intensity confidence bounds equations]] given above.


==Grouped Data==
==Grouped Data==
====Bounds on Beta (Grouped)====<!-- THIS SECTION HEADER IS LINKED TO: Crow-AMSAA - NHPP. IF YOU RENAME THE SECTION, YOU MUST UPDATE THE LINK. -->
====Bounds on Beta (Grouped)====<!-- THIS SECTION HEADER IS LINKED TO: Crow-AMSAA - NHPP. IF YOU RENAME THE SECTION, YOU MUST UPDATE THE LINK. -->
=====Fisher Matrix Bounds=====
=====Fisher Matrix Bounds=====
The parameter <math>\beta \,\!</math> must be positive, thus <math>\ln \beta \,\!</math> is treated as being normally distributed as well.   
The parameter <math>\beta \,\!</math> must be positive, thus <math>\ln \beta \,\!</math> is treated as being normally distributed as well.   


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


The approximate confidence bounds are given as:  
The approximate confidence bounds are given as:  


::<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>\widehat{\beta }\,\!</math> can be obtained by <math>\underset{i=1}{\overset{K}{\mathop{\sum }}}\,{{n}_{i}}\left( \tfrac{T_{i}^{{\hat{\beta }}}\ln {{T}_{i}}-T_{i-1}^{{\hat{\beta }}}\ln \,{{T}_{i-1}}}{T_{i}^{{\hat{\beta }}}-T_{i-1}^{{\hat{\beta }}}}-\ln {{T}_{k}} \right)=0\,\!</math> .
::<math>\widehat{\beta }\,\!</math> can be obtained by <math>\underset{i=1}{\overset{K}{\mathop{\sum }}}\,{{n}_{i}}\left( \tfrac{T_{i}^{{\hat{\beta }}}\ln {{T}_{i}}-T_{i-1}^{{\hat{\beta }}}\ln \,{{T}_{i-1}}}{T_{i}^{{\hat{\beta }}}-T_{i-1}^{{\hat{\beta }}}}-\ln {{T}_{k}} \right)=0\,\!</math>.


All variance can be calculated using the Fisher Matrix:  
All variance can be calculated using the Fisher Matrix:  
Line 504: Line 504:
   Var(\widehat{\lambda }) & Cov(\widehat{\beta },\widehat{\lambda })  \\
   Var(\widehat{\lambda }) & Cov(\widehat{\beta },\widehat{\lambda })  \\
   Cov(\widehat{\beta },\widehat{\lambda }) & Var(\widehat{\beta })  \\
   Cov(\widehat{\beta },\widehat{\lambda }) & Var(\widehat{\beta })  \\
\end{matrix} \right]</math>
\end{matrix} \right]\,\!</math>


<math>\Lambda \,\!</math> is the natural log-likelihood function where ln <math>^{2}T={{\left( \ln T \right)}^{2}}\,\!</math> and:
<math>\Lambda \,\!</math> is the natural log-likelihood function where ln <math>^{2}T={{\left( \ln T \right)}^{2}}\,\!</math> and:


::<math>\Lambda =\underset{i=1}{\overset{k}{\mathop \sum }}\,\left[ {{n}_{i}}\ln (\lambda T_{i}^{\beta }-\lambda T_{i-1}^{\beta })-(\lambda T_{i}^{\beta }-\lambda T_{i-1}^{\beta })-\ln {{n}_{i}}! \right]</math>
::<math>\Lambda =\underset{i=1}{\overset{k}{\mathop \sum }}\,\left[ {{n}_{i}}\ln (\lambda T_{i}^{\beta }-\lambda T_{i-1}^{\beta })-(\lambda T_{i}^{\beta }-\lambda T_{i-1}^{\beta })-\ln {{n}_{i}}! \right]\,\!</math>


::<math>\begin{align}
::<math>\begin{align}
Line 517: Line 517:
\end{matrix} \right] \\  
\end{matrix} \right] \\  
   \frac{{{\partial }^{2}}\Lambda }{\partial \lambda \partial \beta }= & -T_{K}^{\beta }\ln {{T}_{k}}   
   \frac{{{\partial }^{2}}\Lambda }{\partial \lambda \partial \beta }= & -T_{K}^{\beta }\ln {{T}_{k}}   
\end{align}</math>
\end{align}\,\!</math>


=====Crow Bounds=====
=====Crow Bounds=====
:Step 1: Calculate <math>P(i)=\tfrac{{{T}_{i}}}{{{T}_{K}}},\,\,i=1,2,\ldots ,K\,\!</math> .
:Step 1: Calculate <math>P(i)=\tfrac{{{T}_{i}}}{{{T}_{K}}},\,\,i=1,2,\ldots ,K\,\!</math>.
:Step 2: Calculate:  
:Step 2: Calculate:  


::<math>A=\underset{i=1}{\overset{K}{\mathop \sum }}\,\frac{{{[P{{(i)}^{{\hat{\beta }}}}\ln P{{(i)}^{{\hat{\beta }}}}-P{{(i-1)}^{\widehat{\beta }}}\ln P{{(i-1)}^{{\hat{\beta }}}}]}^{2}}}{[P{{(i)}^{{\hat{\beta }}}}-P{{(i-1)}^{{\hat{\beta }}}}]}</math>
::<math>A=\underset{i=1}{\overset{K}{\mathop \sum }}\,\frac{{{[P{{(i)}^{{\hat{\beta }}}}\ln P{{(i)}^{{\hat{\beta }}}}-P{{(i-1)}^{\widehat{\beta }}}\ln P{{(i-1)}^{{\hat{\beta }}}}]}^{2}}}{[P{{(i)}^{{\hat{\beta }}}}-P{{(i-1)}^{{\hat{\beta }}}}]}\,\!</math>


:Step 3: Calculate <math>c=\tfrac{1}{\sqrt{A}}\,\!</math> and <math>S=\tfrac{({{z}_{1-\alpha /2}})\cdot C}{\sqrt{N}}\,\!</math> . Thus an approximate 2-sided <math>(1-\alpha )\,\!</math> 100-percent confidence interval on <math>\widehat{\beta }\,\!</math>   is:
:Step 3: Calculate <math>c=\tfrac{1}{\sqrt{A}}\,\!</math> and <math>S=\tfrac{({{z}_{1-\alpha /2}})\cdot C}{\sqrt{N}}\,\!</math>. Thus an approximate 2-sided <math>(1-\alpha )\,\!</math> 100-percent confidence interval on <math>\widehat{\beta }\,\!</math> is:


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


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


::<math>\hat{\lambda }=\frac{n}{T_{k}^{{\hat{\beta }}}}</math>
::<math>\hat{\lambda }=\frac{n}{T_{k}^{{\hat{\beta }}}}\,\!</math>


The variance calculation is the same as given above in the [[Crow-AMSAA_Confidence_Bounds#Bounds_on_Beta_.28Grouped.29|Bounds on Beta (Grouped)]] section.
The variance calculation is the same as given above in the [[Crow-AMSAA_Confidence_Bounds#Bounds_on_Beta_.28Grouped.29|Bounds on Beta (Grouped)]] section.
Line 547: Line 547:
'''Time Terminated Data'''
'''Time Terminated Data'''


For the 2-sided <math>(1-\alpha )\,\!</math> 100-percent confidence interval, the confidence bounds on <math>\lambda \,\!</math> are:
For the 2-sided <math>(1-\alpha )\,\!</math> 100-percent confidence interval, the confidence bounds on <math>\lambda \,\!</math> are:


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


'''Failure Terminated Data'''
'''Failure Terminated Data'''


For the 2-sided <math>(1-\alpha )\,\!</math> 100-percent confidence interval, the confidence bounds on <math>\lambda \,\!</math> are:
For the 2-sided <math>(1-\alpha )\,\!</math> 100-percent confidence interval, the confidence bounds on <math>\lambda \,\!</math> are:


::<math>\begin{align}
::<math>\begin{align}
   {{\lambda }_{L}}= & \frac{\chi _{\tfrac{\alpha }{2},2N}^{2}}{2\cdot T_{k}^{\beta }} \\  
   {{\lambda }_{L}}= & \frac{\chi _{\tfrac{\alpha }{2},2N}^{2}}{2\cdot T_{k}^{\beta }} \\  
   {{\lambda }_{U}}= & \frac{\chi _{1-\tfrac{\alpha }{2},2N}^{2}}{2\cdot T_{k}^{\beta }}   
   {{\lambda }_{U}}= & \frac{\chi _{1-\tfrac{\alpha }{2},2N}^{2}}{2\cdot T_{k}^{\beta }}   
\end{align}</math>
\end{align}\,\!</math>


===Bounds on Growth Rate (Grouped)===
===Bounds on Growth Rate (Grouped)===
====Fisher Matrix Bounds====
====Fisher Matrix Bounds====
Since the growth rate is equal to <math>1-\beta \,\!</math> , the confidence bounds are calculated from:
Since the growth rate is equal to <math>1-\beta \,\!</math>, the confidence bounds are calculated from:


::<math>\begin{align}
::<math>\begin{align}
   G\operatorname{row}th\text{ }Rat{{e}_{L}}= & 1-{{\beta }_{U}} \\  
   G\operatorname{row}th\text{ }Rat{{e}_{L}}= & 1-{{\beta }_{U}} \\  
   G\operatorname{row}th\text{ }Rat{{e}_{U}}= & 1-{{\beta }_{L}}   
   G\operatorname{row}th\text{ }Rat{{e}_{U}}= & 1-{{\beta }_{L}}   
\end{align}</math>
\end{align}\,\!</math>


<math>{{\beta }_{L}}\,\!</math> and <math>{{\beta }_{U}}\,\!</math> are obtained using the methods described above in the [[Crow-AMSAA_Confidence_Bounds#Bounds_on_Beta_.28Grouped.29|Bounds on Beta (Grouped)]] section.  <PER LISA: ASK SME TO CONFIRM THAT THIS IS ADEQUATE.>>
<math>{{\beta }_{L}}\,\!</math> and <math>{{\beta }_{U}}\,\!</math> are obtained using the methods described above in the [[Crow-AMSAA_Confidence_Bounds#Bounds_on_Beta_.28Grouped.29|Bounds on Beta (Grouped)]] section.  <PER LISA: ASK SME TO CONFIRM THAT THIS IS ADEQUATE.>>


===Bounds on Cumulative MTBF===
===Bounds on Cumulative MTBF===
====Fisher Matrix Bounds====
====Fisher Matrix Bounds====
The cumulative MTBF, <math>{{m}_{c}}(t)\,\!</math> , must be positive, thus <math>\ln {{m}_{c}}(t)\,\!</math> is treated as being normally distributed as well.  
The cumulative MTBF, <math>{{m}_{c}}(t)\,\!</math>, must be positive, thus <math>\ln {{m}_{c}}(t)\,\!</math> is treated as being normally distributed as well.  


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


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


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


:where:  
:where:  


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


::<math>\begin{align}
::<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 }) \\  
   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 })\,   
   & +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>
\end{align}\,\!</math>


The variance calculation is the same as given above in the [[Crow-AMSAA_Confidence_Bounds#Bounds_on_Beta_.28Grouped.29|Bounds on Beta (Grouped)]] section and:  
The variance calculation is the same as given above in the [[Crow-AMSAA_Confidence_Bounds#Bounds_on_Beta_.28Grouped.29|Bounds on Beta (Grouped)]] section and:  
Line 598: Line 598:
   \frac{\partial {{m}_{c}}(t)}{\partial \beta }= & -\frac{1}{{\hat{\lambda }}}{{t}^{1-\hat{\beta }}}\ln t \\  
   \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 }}}   
   \frac{\partial {{m}_{c}}(t)}{\partial \lambda }= & -\frac{1}{{{{\hat{\lambda }}}^{2}}}{{t}^{1-\hat{\beta }}}   
\end{align}</math>
\end{align}\,\!</math>


====Crow Bounds====
====Crow Bounds====
Calculate the Crow cumulative failure intensity confidence bounds:  
Calculate the Crow cumulative failure intensity confidence bounds:  


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


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


:Then:
:Then:
Line 612: Line 612:
   {{[MTB{{F}_{c}}]}_{L}}= & \frac{1}{C{{(t)}_{U}}} \\  
   {{[MTB{{F}_{c}}]}_{L}}= & \frac{1}{C{{(t)}_{U}}} \\  
   {{[MTB{{F}_{c}}]}_{U}}= & \frac{1}{C{{(t)}_{L}}}   
   {{[MTB{{F}_{c}}]}_{U}}= & \frac{1}{C{{(t)}_{L}}}   
\end{align}</math>
\end{align}\,\!</math>


===Bounds on Instantaneous MTBF (Grouped)===
===Bounds on Instantaneous MTBF (Grouped)===
====Fisher Matrix Bounds====
====Fisher Matrix Bounds====
The instantaneous MTBF, <math>{{m}_{i}}(t)\,\!</math> , must be positive, thus <math>\ln {{m}_{i}}(t)\,\!</math> is approximately treated as being normally distributed as well.  
The instantaneous MTBF, <math>{{m}_{i}}(t)\,\!</math>, must be positive, thus <math>\ln {{m}_{i}}(t)\,\!</math> is approximately treated as being normally distributed as well.  


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


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


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


:where:  
:where:  


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


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


The variance calculation is the same as given above in the [[Crow-AMSAA_Confidence_Bounds#Bounds_on_Beta_.28Grouped.29|Bounds on Beta (Grouped)]] section and:
The variance calculation is the same as given above in the [[Crow-AMSAA_Confidence_Bounds#Bounds_on_Beta_.28Grouped.29|Bounds on Beta (Grouped)]] section and:
Line 638: Line 638:
   \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 \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 }}}   
   \frac{\partial {{m}_{i}}(t)}{\partial \lambda }= & -\frac{1}{{{{\hat{\lambda }}}^{2}}\hat{\beta }}{{t}^{1-\hat{\beta }}}   
\end{align}</math>
\end{align}\,\!</math>


====Crow Bounds====
====Crow Bounds====
:Step 1: Calculate <math>P(i)=\tfrac{{{T}_{i}}}{{{T}_{K}}},\,\,i=1,2,\ldots ,K\,\!</math> .
:Step 1: Calculate <math>P(i)=\tfrac{{{T}_{i}}}{{{T}_{K}}},\,\,i=1,2,\ldots ,K\,\!</math>.
:Step 2: Calculate:  
:Step 2: Calculate:  


::<math>A=\underset{i=1}{\overset{K}{\mathop \sum }}\,\frac{{{\left[ P{{(i)}^{{\hat{\beta }}}}\ln P{{(i)}^{{\hat{\beta }}}}-P{{(i-1)}^{\widehat{\beta }}}\ln P{{(i-1)}^{{\hat{\beta }}}} \right]}^{2}}}{\left[ P{{(i)}^{{\hat{\beta }}}}-P{{(i-1)}^{{\hat{\beta }}}} \right]}</math>
::<math>A=\underset{i=1}{\overset{K}{\mathop \sum }}\,\frac{{{\left[ P{{(i)}^{{\hat{\beta }}}}\ln P{{(i)}^{{\hat{\beta }}}}-P{{(i-1)}^{\widehat{\beta }}}\ln P{{(i-1)}^{{\hat{\beta }}}} \right]}^{2}}}{\left[ P{{(i)}^{{\hat{\beta }}}}-P{{(i-1)}^{{\hat{\beta }}}} \right]}\,\!</math>


:Step 3: Calculate <math>D=\sqrt{\tfrac{1}{A}+1}\,\!</math> and <math>W=\tfrac{({{z}_{1-\alpha /2}})\cdot D}{\sqrt{N}}\,\!</math> . Thus an approximate 2-sided <math>(1-\alpha )\,\!</math> 100-percent confidence interval on <math>{{\hat{m}}_{i}}(t)\,\!</math> is:
:Step 3: Calculate <math>D=\sqrt{\tfrac{1}{A}+1}\,\!</math> and <math>W=\tfrac{({{z}_{1-\alpha /2}})\cdot D}{\sqrt{N}}\,\!</math>. Thus an approximate 2-sided <math>(1-\alpha )\,\!</math> 100-percent confidence interval on <math>{{\hat{m}}_{i}}(t)\,\!</math> is:


::<math>MTB{{F}_{i}}={{\widehat{m}}_{i}}(1\pm W)</math>
::<math>MTB{{F}_{i}}={{\widehat{m}}_{i}}(1\pm W)\,\!</math>


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


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


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


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


:where:  
:where:  


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


:and:  
:and:  
Line 669: Line 669:
   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 }) \\  
   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 })   
   & +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>
\end{align}\,\!</math>


The variance calculation is the same as given above in the [[Crow-AMSAA_Confidence_Bounds#Bounds_on_Beta_.28Grouped.29|Bounds on Beta (Grouped)]] section and:  
The variance calculation is the same as given above in the [[Crow-AMSAA_Confidence_Bounds#Bounds_on_Beta_.28Grouped.29|Bounds on Beta (Grouped)]] section and:  
Line 676: Line 676:
   \frac{\partial {{\lambda }_{c}}(t)}{\partial \beta }= & \hat{\lambda }{{t}^{\hat{\beta }-1}}\ln t \\  
   \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}}   
   \frac{\partial {{\lambda }_{c}}(t)}{\partial \lambda }= & {{t}^{\hat{\beta }-1}}   
\end{align}</math>
\end{align}\,\!</math>


====Crow Bounds====
====Crow Bounds====
Line 684: Line 684:
   C{{(t)}_{L}}= & \frac{\chi _{\tfrac{\alpha }{2},2N}^{2}}{2\cdot t} \\  
   C{{(t)}_{L}}= & \frac{\chi _{\tfrac{\alpha }{2},2N}^{2}}{2\cdot t} \\  
   C{{(t)}_{U}}= & \frac{\chi _{1-\tfrac{\alpha }{2},2N+2}^{2}}{2\cdot t}   
   C{{(t)}_{U}}= & \frac{\chi _{1-\tfrac{\alpha }{2},2N+2}^{2}}{2\cdot t}   
\end{align}</math>
\end{align}\,\!</math>


===Bounds on Instantaneous Failure Intensity (Grouped)===
===Bounds on Instantaneous Failure Intensity (Grouped)===
====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 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 treated as being normally distributed.  


::<math>\frac{\ln {{{\hat{\lambda }}}_{i}}(t)-\ln {{\lambda }_{i}}(t)}{\sqrt{Var(\ln {{{\hat{\lambda }}}_{i}}(t)})}\tilde{\ }N(0,1)</math>
::<math>\frac{\ln {{{\hat{\lambda }}}_{i}}(t)-\ln {{\lambda }_{i}}(t)}{\sqrt{Var(\ln {{{\hat{\lambda }}}_{i}}(t)})}\tilde{\ }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 given above in the [[Crow-AMSAA_Confidence_Bounds#Bounds_on_Beta_.28Grouped.29|Bounds on Beta (Grouped)]] section and:  
The variance calculation is the same as given above in the [[Crow-AMSAA_Confidence_Bounds#Bounds_on_Beta_.28Grouped.29|Bounds on Beta (Grouped)]] section and:  
Line 708: Line 708:
   \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====
Line 716: Line 716:
   {{[{{\lambda }_{i}}(t)]}_{L}}= & \frac{1}{{{[MTB{{F}_{i}}]}_{U}}} \\  
   {{[{{\lambda }_{i}}(t)]}_{L}}= & \frac{1}{{{[MTB{{F}_{i}}]}_{U}}} \\  
   {{[{{\lambda }_{i}}(t)]}_{U}}= & \frac{1}{{{[MTB{{F}_{i}}]}_{L}}}   
   {{[{{\lambda }_{i}}(t)]}_{U}}= & \frac{1}{{{[MTB{{F}_{i}}]}_{L}}}   
\end{align}</math>
\end{align}\,\!</math>


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


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


Confidence bounds on the time are given by:  
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:  
Line 733: Line 733:
   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 given above in the [[Crow-AMSAA_Confidence_Bounds#Bounds_on_Beta_.28Grouped.29|Bounds on Beta (Grouped)]] section and:  
The variance calculation is the same as given above in the [[Crow-AMSAA_Confidence_Bounds#Bounds_on_Beta_.28Grouped.29|Bounds on Beta (Grouped)]] section and:  


::<math>\hat{T}={{(\lambda \cdot {{m}_{c}})}^{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 }= & \frac{{{(\lambda \cdot \,{{m}_{c}})}^{1/(1-\beta )}}\ln (\lambda \cdot \text{ }{{m}_{c}})}{{{(1-\beta )}^{2}}} \\  
   \frac{\partial T}{\partial \beta }= & \frac{{{(\lambda \cdot \,{{m}_{c}})}^{1/(1-\beta )}}\ln (\lambda \cdot \text{ }{{m}_{c}})}{{{(1-\beta )}^{2}}} \\  
   \frac{\partial T}{\partial \lambda }= & \frac{{{(\lambda \cdot {{m}_{c}})}^{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 <math>{{\lambda }_{c}}(T)=\tfrac{1}{MTB{{F}_{c}}}\,\!</math> .
:Step 1: Calculate <math>{{\lambda }_{c}}(T)=\tfrac{1}{MTB{{F}_{c}}}\,\!</math>.
:Step 2: Use equations in 5.4.10.1 to calculate the bounds on time given the cumulative failure intensity.
:Step 2: Use equations in 5.4.10.1 to calculate the bounds on time given the cumulative failure intensity.


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


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


Confidence bounds on the time are given by:  
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:  
Line 763: Line 763:
   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 given above in the [[Crow-AMSAA_Confidence_Bounds#Bounds_on_Beta_.28Grouped.29|Bounds on Beta (Grouped)]] section and:
The variance calculation is the same as given above in the [[Crow-AMSAA_Confidence_Bounds#Bounds_on_Beta_.28Grouped.29|Bounds on Beta (Grouped)]] section and:


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


::<math>\begin{align}
::<math>\begin{align}
   \frac{\partial T}{\partial \beta }= & {{\left( \lambda \beta \cdot \text{ }{{m}_{i}}(T) \right)}^{1/(1-\beta )}}\left[ \frac{1}{{{(1-\beta )}^{2}}}\ln (\lambda \beta \cdot {{m}_{i}}(T))+\frac{1}{\beta (1-\beta )} \right] \\  
   \frac{\partial T}{\partial \beta }= & {{\left( \lambda \beta \cdot \text{ }{{m}_{i}}(T) \right)}^{1/(1-\beta )}}\left[ \frac{1}{{{(1-\beta )}^{2}}}\ln (\lambda \beta \cdot {{m}_{i}}(T))+\frac{1}{\beta (1-\beta )} \right] \\  
   \frac{\partial T}{\partial \lambda }= & \frac{{{(\lambda \beta \cdot \text{ }{{m}_{i}}(T))}^{1/(1-\beta )}}}{\lambda (1-\beta )}   
   \frac{\partial T}{\partial \lambda }= & \frac{{{(\lambda \beta \cdot \text{ }{{m}_{i}}(T))}^{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:
:Step 1: Calculate the confidence bounds on the instantaneous MTBF:


::<math>MTB{{F}_{i}}={{\widehat{m}}_{i}}(1\pm W)</math>
::<math>MTB{{F}_{i}}={{\widehat{m}}_{i}}(1\pm W)\,\!</math>


:Step 2: Use equations in 5.4.5.2 to calculate the time given the instantaneous MTBF.
:Step 2: Use equations in 5.4.5.2 to calculate the time given the instantaneous MTBF.
Line 783: Line 783:
===Bounds on Time Given Cumulative Failure Intensity (Grouped)===
===Bounds on Time Given Cumulative Failure Intensity (Grouped)===
====Fisher Matrix Bounds====
====Fisher Matrix Bounds====
The time, <math>T\,\!</math> , must be positive, thus <math>\ln T\,\!</math> is treated as being normally distributed.  
The time, <math>T\,\!</math>, must be positive, thus <math>\ln T\,\!</math> is treated as being normally distributed.  


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


Confidence bounds on the time are given by:  
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:  
Line 796: Line 796:
   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 given above in the [[Crow-AMSAA_Confidence_Bounds#Bounds_on_Beta_.28Grouped.29|Bounds on Beta (Grouped)]] section and:  
The variance calculation is the same as given above in the [[Crow-AMSAA_Confidence_Bounds#Bounds_on_Beta_.28Grouped.29|Bounds on Beta (Grouped)]] section and:  
Line 803: Line 803:
   \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 \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 )}   
   \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>




Line 809: Line 809:
:Step 1: Calculate:
:Step 1: Calculate:


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


:Step 2: Estimate the number of failures:
:Step 2: Estimate the number of failures:


::<math>N(\hat{T})=\hat{\lambda }{{\hat{T}}^{{\hat{\beta }}}}</math>
::<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:  
: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}
::<math>\begin{align}
   {{t}_{l}}= & \frac{\chi _{\tfrac{\alpha }{2},2N}^{2}}{2\cdot {{\lambda }_{c}}(T)} \\  
   {{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)}   
   {{t}_{u}}= & \frac{\chi _{1-\tfrac{\alpha }{2},2N+2}^{2}}{2\cdot {{\lambda }_{c}}(T)}   
\end{align}</math>
\end{align}\,\!</math>


===Bounds on Time Given Instantaneous Failure Intensity (Grouped)===
===Bounds on Time Given Instantaneous Failure Intensity (Grouped)===
====Fisher Matrix Bounds====
====Fisher Matrix Bounds====
The time, <math>T\,\!</math> , must be positive, thus <math>\ln T\,\!</math> is treated as being normally distributed.  
The time, <math>T\,\!</math>, must be positive, thus <math>\ln T\,\!</math> is treated as being normally distributed.  


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


Confidence bounds on the time are given by:  
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:  
Line 837: Line 837:
   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 given above in the [[Crow-AMSAA_Confidence_Bounds#Bounds_on_Beta_.28Grouped.29|Bounds on Beta (Grouped)]] section and:   
The variance calculation is the same as given above in the [[Crow-AMSAA_Confidence_Bounds#Bounds_on_Beta_.28Grouped.29|Bounds on Beta (Grouped)]] section and:   


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


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




====Crow Bounds====
====Crow Bounds====
:Step 1: Calculate <math>MTB{{F}_{i}}=\tfrac{1}{{{\lambda }_{i}}(T)}\,\!</math> .
:Step 1: Calculate <math>MTB{{F}_{i}}=\tfrac{1}{{{\lambda }_{i}}(T)}\,\!</math>.
:Step 2: Follow the same process as in 5.4.9.2 to calculate the bounds on time given the instantaneous failure intensity.
:Step 2: Follow the same process as in 5.4.9.2 to calculate the bounds on time given the instantaneous failure intensity.


===Bounds on Cumulative Number of Failures (Grouped)===
===Bounds on Cumulative Number of Failures (Grouped)===
====Fisher Matrix Bounds====
====Fisher Matrix Bounds====
The cumulative number of failures, <math>N(t)\,\!</math> , must be positive, thus <math>\ln N(t)\,\!</math> is treated as being normally distributed.   
The cumulative number of failures, <math>N(t)\,\!</math>, must be positive, thus <math>\ln N(t)\,\!</math> is treated as being normally distributed.   


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


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


:where:  
:where:  


::<math>\hat{N}(t)=\hat{\lambda }{{t}^{{\hat{\beta }}}}</math>
::<math>\hat{N}(t)=\hat{\lambda }{{t}^{{\hat{\beta }}}}\,\!</math>


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


The variance calculation is the same as given above in the [[Crow-AMSAA_Confidence_Bounds#Bounds_on_Beta_.28Grouped.29|Bounds on Beta (Grouped)]] section and:   
The variance calculation is the same as given above in the [[Crow-AMSAA_Confidence_Bounds#Bounds_on_Beta_.28Grouped.29|Bounds on Beta (Grouped)]] section and:   
Line 875: Line 875:
   \frac{\partial \hat{N}(t)}{\partial \beta }= & \hat{\lambda }{{t}^{{\hat{\beta }}}}\ln t \\  
   \frac{\partial \hat{N}(t)}{\partial \beta }= & \hat{\lambda }{{t}^{{\hat{\beta }}}}\ln t \\  
   \frac{\partial \hat{N}(t)}{\partial \lambda }= & {{t}^{{\hat{\beta }}}}   
   \frac{\partial \hat{N}(t)}{\partial \lambda }= & {{t}^{{\hat{\beta }}}}   
\end{align}</math>
\end{align}\,\!</math>




Line 884: Line 884:
   {{N}_{L}}(T)= & \frac{T}{{\hat{\beta }}}{{\lambda }_{i}}{{(T)}_{L}} \\  
   {{N}_{L}}(T)= & \frac{T}{{\hat{\beta }}}{{\lambda }_{i}}{{(T)}_{L}} \\  
   {{N}_{U}}(T)= & \frac{T}{{\hat{\beta }}}{{\lambda }_{i}}{{(T)}_{U}}   
   {{N}_{U}}(T)= & \frac{T}{{\hat{\beta }}}{{\lambda }_{i}}{{(T)}_{U}}   
\end{align}</math>
\end{align}\,\!</math>


where <math>{{\lambda }_{i}}{{(T)}_{L}}\,\!</math> and <math>{{\lambda }_{i}}{{(T)}_{U}}\,\!</math> can be obtained from the equations given above for [[Crow-AMSAA_Confidence_Bounds#Bounds_on_Instantaneous_Failure_Intensity_.28Grouped.29|Crow instantaneous failure intensity confidence bounds with grouped data]].
where <math>{{\lambda }_{i}}{{(T)}_{L}}\,\!</math> and <math>{{\lambda }_{i}}{{(T)}_{U}}\,\!</math> can be obtained from the equations given above for [[Crow-AMSAA_Confidence_Bounds#Bounds_on_Instantaneous_Failure_Intensity_.28Grouped.29|Crow instantaneous failure intensity confidence bounds with grouped data]].

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RGAbox.png

Appendix C  
Crow-AMSAA Confidence Bounds  


In this appendix, we will present the two methods used in the RGA software to estimate the confidence bounds for the Crow-AMSAA (NHPP) model when applied to developmental testing data. 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.

Individual (Non-Grouped) Data

Bounds on Beta

Fisher Matrix Bounds

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

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

The approximate confidence bounds are given as:

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

[math]\displaystyle{ \alpha \,\! }[/math] in [math]\displaystyle{ {{z}_{\alpha }}\,\! }[/math] is different ( [math]\displaystyle{ \alpha /2\,\! }[/math], [math]\displaystyle{ \alpha \,\! }[/math] ) according to a 2-sided confidence interval or a 1-sided confidence interval, and variances can be calculated using the Fisher Matrix.

[math]\displaystyle{ \left[ \begin{matrix} -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\lambda }^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial \lambda \partial \beta } \\ -\tfrac{{{\partial }^{2}}\Lambda }{\partial \lambda \partial \beta } & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\beta }^{2}}} \\ \end{matrix} \right]_{\beta =\widehat{\beta },\lambda =\widehat{\lambda }}^{-1}=\left[ \begin{matrix} Var(\widehat{\lambda }) & Cov(\widehat{\beta },\widehat{\lambda }) \\ Cov(\widehat{\beta },\widehat{\lambda }) & Var(\widehat{\beta }) \\ \end{matrix} \right]\,\! }[/math]
[math]\displaystyle{ \Lambda \,\! }[/math] is the natural log-likelihood function:
[math]\displaystyle{ \Lambda =N\ln \lambda +N\ln \beta -\lambda {{T}^{\beta }}+(\beta -1)\underset{i=1}{\overset{N}{\mathop \sum }}\,\ln {{T}_{i}}\,\! }[/math]
[math]\displaystyle{ \frac{{{\partial }^{2}}\Lambda }{\partial {{\lambda }^{2}}}=-\frac{N}{{{\lambda }^{2}}}\,\! }[/math]
and:
[math]\displaystyle{ \frac{{{\partial }^{2}}\Lambda }{\partial {{\beta }^{2}}}=-\frac{N}{{{\beta }^{2}}}-\lambda {{T}^{\beta }}{{(\ln T)}^{2}}\,\! }[/math]
also:
[math]\displaystyle{ \frac{{{\partial }^{2}}\Lambda }{\partial \lambda \partial \beta }=-{{T}^{\beta }}\ln T\,\! }[/math]

Crow Bounds

Time Terminated Data

For the 2-sided [math]\displaystyle{ (1-\alpha )\,\! }[/math] 100-percent confidence interval on [math]\displaystyle{ \beta \,\! }[/math], calculate:

[math]\displaystyle{ \begin{align} & {{D}_{L}}= & \frac{\chi _{\tfrac{\alpha }{2},2N}^{2}}{2(N-1)} \\ & {{D}_{U}}= & \frac{\chi _{1-\tfrac{\alpha }{2},2N}^{2}}{2(N-1)} \end{align}\,\! }[/math]

The fractiles can be found in the tables of the [math]\displaystyle{ {{\chi }^{2}}\,\! }[/math] distribution. Thus the confidence bounds on [math]\displaystyle{ \beta \,\! }[/math] are:

[math]\displaystyle{ \begin{align} {{\beta }_{L}}= & {{D}_{L}}\cdot \hat{\beta } \\ {{\beta }_{U}}= & {{D}_{U}}\cdot \hat{\beta } \end{align}\,\! }[/math]

Failure Terminated Data

For the 2-sided [math]\displaystyle{ (1-\alpha )\,\! }[/math] 100-percent confidence interval on [math]\displaystyle{ \beta \,\! }[/math], calculate:

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

Thus the confidence bounds on [math]\displaystyle{ \beta \,\! }[/math] are:

[math]\displaystyle{ \begin{align} {{\beta }_{L}}= & {{D}_{L}}\cdot \hat{\beta } \\ {{\beta }_{U}}= & {{D}_{U}}\cdot \hat{\beta } \end{align}\,\! }[/math]

Bounds on Lambda

Fisher Matrix Bounds

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

[math]\displaystyle{ \frac{\ln \hat{\lambda }-\ln \lambda }{\sqrt{Var(\ln \hat{\lambda }})}\ \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 }=\frac{n}{{{T}^{*\hat{\beta }}}}\,\! }[/math]

The variance calculation is the same as given above in the Bounds on Beta section.

Crow Bounds

Time Terminated Data

For the 2-sided [math]\displaystyle{ (1-\alpha )\,\! }[/math] 100-percent confidence interval, the confidence bounds on [math]\displaystyle{ \lambda \,\! }[/math] are:

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

The fractiles can be found in the tables of the [math]\displaystyle{ {{\chi }^{2}}\,\! }[/math] distribution.

Failure Terminated Data

For the 2-sided [math]\displaystyle{ (1-\alpha )\,\! }[/math] 100-percent confidence interval, the confidence bounds on [math]\displaystyle{ \lambda \,\! }[/math] are:

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

Bounds on Growth Rate

Since the growth rate is equal to [math]\displaystyle{ 1-\beta \,\! }[/math], the confidence bounds for both the Fisher Matrix and Crow methods are:

[math]\displaystyle{ G\text{row}th Rate_L=1-\beta_U\,\! }[/math]
[math]\displaystyle{ G\text{row}th Rate_U=1-\beta_L\,\! }[/math]

[math]\displaystyle{ {{\beta }_{L}}\,\! }[/math] and [math]\displaystyle{ {{\beta }_{U}}\,\! }[/math] are obtained using the methods described above in the Bounds on Beta section. <PER LISA: ASK SME TO CONFIRM THAT THIS IS ADEQUATE.>>

Bounds on 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 treated as being normally distributed as well.

[math]\displaystyle{ \frac{\ln {{{\hat{m}}}_{c}}(t)-\ln {{m}_{c}}(t)}{\sqrt{Var(\ln {{{\hat{m}}}_{c}}(t)})}\ \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 given above in the Bounds on Beta section and:

[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

To calculate the Crow confidence bounds on cumulative MTBF, first calculate the Crow cumulative failure intensity confidence bounds:

[math]\displaystyle{ C{{(t)}_{L}}=\frac{\chi _{\tfrac{\alpha }{2},2N}^{2}}{2\cdot t}\,\! }[/math]
[math]\displaystyle{ C{{(t)}_{U}}=\frac{\chi _{1-\tfrac{\alpha }{2},2N+2}^{2}}{2\cdot t}\,\! }[/math]
Then:
[math]\displaystyle{ \begin{align} & {{[MTB{{F}_{c}}]}_{L}}= & \frac{1}{C{{(t)}_{U}}} \\ & {{[MTB{{F}_{c}}]}_{U}}= & \frac{1}{C{{(t)}_{L}}} \end{align}\,\! }[/math]


Bounds on 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 treated as being normally distributed as well.

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

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

[math]\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 given above in the Bounds on Beta section and:

[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 Data

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{ c\,\! }[/math] to [math]\displaystyle{ G({{n}^{2}}/c|n)=\xi \,\! }[/math] for [math]\displaystyle{ \xi =\tfrac{\alpha }{2}\,\! }[/math] and [math]\displaystyle{ \xi =1-\tfrac{\alpha }{2}\,\! }[/math], 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} {{[MTB{{F}_{i}}]}_{L}}= & MTB{{F}_{i}}\cdot {{p}_{1}} \\ {{[MTB{{F}_{i}}]}_{U}}= & MTB{{F}_{i}}\cdot {{p}_{2}} \end{align}\,\! }[/math]

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

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

Time Terminated Data

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} {{[MTB{{F}_{i}}]}_{L}}= & MTB{{F}_{i}}\cdot {{\Pi }_{1}} \\ {{[MTB{{F}_{i}}]}_{U}}= & MTB{{F}_{i}}\cdot {{\Pi }_{2}} \end{align}\,\! }[/math]

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

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

Bounds on 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 treated as being normally distributed.

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

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

[math]\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 given above in the Bounds on Beta section and:

[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 Crow cumulative failure intensity confidence bounds are given as:

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

Bounds on 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 treated as being normally distributed.

[math]\displaystyle{ \frac{\ln {{{\hat{\lambda }}}_{i}}(t)-\ln {{\lambda }_{i}}(t)}{\sqrt{Var(\ln {{{\hat{\lambda }}}_{i}}(t)})}\text{ }\tilde{\ }\text{ }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]
[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 given above in the Bounds on Beta section and:

[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 Crow instantaneous failure intensity confidence bounds are given as:

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

Bounds on Time Given Cumulative Failure Intensity

Fisher Matrix Bounds

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

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

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 given above in the Bounds on Beta section and:

[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

Step 1: Calculate:
[math]\displaystyle{ \hat{T}={{\left( \frac{{{\lambda }_{c}}(T)}{{\hat{\lambda }}} \right)}^{\tfrac{1}{\beta -1}}}\,\! }[/math]
Step 2: Estimate the number of failures:
[math]\displaystyle{ 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]\displaystyle{ {{t}_{l}}\,\! }[/math] and [math]\displaystyle{ {{t}_{u}}\,\! }[/math] in the following equations:
[math]\displaystyle{ \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 Cumulative MTBF

Fisher Matrix Bounds

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

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

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 given above in the Bounds on Beta section and:

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

Crow Bounds

Step 1: Calculate [math]\displaystyle{ {{\lambda }_{c}}(T)=\tfrac{1}{MTB{{F}_{c}}}\,\! }[/math].
Step 2: Use the equations from 5.2.8.2 to calculate the bounds on time given the cumulative failure intensity.

Bounds on Time Given Instantaneous MTBF

Fisher Matrix Bounds

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

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

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 given above in the Bounds on Beta section and:

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

Crow Bounds

Step 1: Calculate the confidence bounds on the instantaneous MTBF as presented in Section 5.5.2.
Step 2: Calculate the bounds on time as follows.

Failure Terminated Data

[math]\displaystyle{ \hat{T}={{(\frac{\lambda \beta \cdot MTB{{F}_{i}}}{c})}^{1/(1-\beta )}}\,\! }[/math]

So the lower an upper bounds on time are:

[math]\displaystyle{ {{\hat{T}}_{L}}={{(\frac{\lambda \beta \cdot MTB{{F}_{i}}}{{{c}_{1}}})}^{1/(1-\beta )}}\,\! }[/math]
[math]\displaystyle{ {{\hat{T}}_{U}}={{(\frac{\lambda \beta \cdot MTB{{F}_{i}}}{{{c}_{2}}})}^{1/(1-\beta )}}\,\! }[/math]

Time Terminated Data

[math]\displaystyle{ \hat{T}={{(\frac{\lambda \beta \cdot MTB{{F}_{i}}}{\Pi })}^{1/(1-\beta )}}\,\! }[/math]

So the lower and upper bounds on time are:

[math]\displaystyle{ {{\hat{T}}_{L}}={{(\frac{\lambda \beta \cdot MTB{{F}_{i}}}{{{\Pi }_{1}}})}^{1/(1-\beta )}}\,\! }[/math]
[math]\displaystyle{ {{\hat{T}}_{U}}={{(\frac{\lambda \beta \cdot MTB{{F}_{i}}}{{{\Pi }_{2}}})}^{1/(1-\beta )}}\,\! }[/math]

Bounds on Time Given Instantaneous Failure Intensity

Fisher Matrix Bounds

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

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

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 given above in the Bounds on Beta section and:

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

Crow Bounds

Step 1: Calculate [math]\displaystyle{ MTB{{F}_{i}}=\tfrac{1}{{{\lambda }_{i}}(T)}\,\! }[/math].
Step 2: Use the equations from 5.2.10.2 to calculate the bounds on time given the instantaneous failure intensity.

Bounds on Cumulative Number of Failures

Fisher Matrix Bounds

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

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

The variance calculation is the same as given above in the Bounds on Beta section and:

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

Crow Bounds

The Crow cumulative number of failure confidence bounds are:

[math]\displaystyle{ \begin{align} {{N}_{L}}(T)= & \frac{T}{{\hat{\beta }}}{{\lambda }_{i}}{{(T)}_{L}} \\ {{N}_{U}}(T)= & \frac{T}{{\hat{\beta }}}{{\lambda }_{i}}{{(T)}_{U}} \end{align}\,\! }[/math]

where [math]\displaystyle{ {{\lambda }_{i}}{{(T)}_{L}}\,\! }[/math] and [math]\displaystyle{ {{\lambda }_{i}}{{(T)}_{U}}\,\! }[/math] can be obtained from the Crow instantaneous failure intensity confidence bounds equations given above.

Grouped Data

Bounds on Beta (Grouped)

Fisher Matrix Bounds

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

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

The approximate confidence bounds are given as:

[math]\displaystyle{ C{{B}_{\beta }}=\hat{\beta }{{e}^{\pm {{z}_{\alpha }}\sqrt{Var(\hat{\beta })}/\hat{\beta }}}\,\! }[/math]
[math]\displaystyle{ \widehat{\beta }\,\! }[/math] can be obtained by [math]\displaystyle{ \underset{i=1}{\overset{K}{\mathop{\sum }}}\,{{n}_{i}}\left( \tfrac{T_{i}^{{\hat{\beta }}}\ln {{T}_{i}}-T_{i-1}^{{\hat{\beta }}}\ln \,{{T}_{i-1}}}{T_{i}^{{\hat{\beta }}}-T_{i-1}^{{\hat{\beta }}}}-\ln {{T}_{k}} \right)=0\,\! }[/math].

All variance can be calculated using the Fisher Matrix:

[math]\displaystyle{ \left[ \begin{matrix} -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\lambda }^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial \lambda \partial \beta } \\ -\tfrac{{{\partial }^{2}}\Lambda }{\partial \lambda \partial \beta } & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\beta }^{2}}} \\ \end{matrix} \right]_{\beta =\widehat{\beta },\lambda =\widehat{\lambda }}^{-1}=\left[ \begin{matrix} Var(\widehat{\lambda }) & Cov(\widehat{\beta },\widehat{\lambda }) \\ Cov(\widehat{\beta },\widehat{\lambda }) & Var(\widehat{\beta }) \\ \end{matrix} \right]\,\! }[/math]

[math]\displaystyle{ \Lambda \,\! }[/math] is the natural log-likelihood function where ln [math]\displaystyle{ ^{2}T={{\left( \ln T \right)}^{2}}\,\! }[/math] and:

[math]\displaystyle{ \Lambda =\underset{i=1}{\overset{k}{\mathop \sum }}\,\left[ {{n}_{i}}\ln (\lambda T_{i}^{\beta }-\lambda T_{i-1}^{\beta })-(\lambda T_{i}^{\beta }-\lambda T_{i-1}^{\beta })-\ln {{n}_{i}}! \right]\,\! }[/math]
[math]\displaystyle{ \begin{align} \frac{{{\partial }^{2}}\Lambda }{\partial {{\lambda }^{2}}}= & -\frac{n}{{{\lambda }^{2}}} \\ \frac{{{\partial }^{2}}\Lambda }{\partial {{\beta }^{2}}}= & \underset{i=1}{\overset{k}{\mathop \sum }}\,\left[ \begin{matrix} {{n}_{i}}\left( \tfrac{(T_{i}^{{\hat{\beta }}}{{\ln }^{2}}{{T}_{i}}-T_{i-1}^{{\hat{\beta }}}{{\ln }^{2}}{{T}_{i-1}})(T_{i}^{{\hat{\beta }}}-T_{i-1}^{{\hat{\beta }}})-{{\left( T_{i}^{{\hat{\beta }}}\ln {{T}_{i}}-T_{i-1}^{{\hat{\beta }}}\ln {{T}_{i-1}} \right)}^{2}}}{{{(T_{i}^{{\hat{\beta }}}-T_{i-1}^{{\hat{\beta }}})}^{2}}} \right) \\ -\left( \lambda T_{i}^{{\hat{\beta }}}{{\ln }^{2}}{{T}_{i}}-\lambda T_{i-1}^{{\hat{\beta }}}{{\ln }^{2}}{{T}_{i-1}} \right) \\ \end{matrix} \right] \\ \frac{{{\partial }^{2}}\Lambda }{\partial \lambda \partial \beta }= & -T_{K}^{\beta }\ln {{T}_{k}} \end{align}\,\! }[/math]
Crow Bounds
Step 1: Calculate [math]\displaystyle{ P(i)=\tfrac{{{T}_{i}}}{{{T}_{K}}},\,\,i=1,2,\ldots ,K\,\! }[/math].
Step 2: Calculate:
[math]\displaystyle{ A=\underset{i=1}{\overset{K}{\mathop \sum }}\,\frac{{{[P{{(i)}^{{\hat{\beta }}}}\ln P{{(i)}^{{\hat{\beta }}}}-P{{(i-1)}^{\widehat{\beta }}}\ln P{{(i-1)}^{{\hat{\beta }}}}]}^{2}}}{[P{{(i)}^{{\hat{\beta }}}}-P{{(i-1)}^{{\hat{\beta }}}}]}\,\! }[/math]
Step 3: Calculate [math]\displaystyle{ c=\tfrac{1}{\sqrt{A}}\,\! }[/math] and [math]\displaystyle{ S=\tfrac{({{z}_{1-\alpha /2}})\cdot C}{\sqrt{N}}\,\! }[/math]. Thus an approximate 2-sided [math]\displaystyle{ (1-\alpha )\,\! }[/math] 100-percent confidence interval on [math]\displaystyle{ \widehat{\beta }\,\! }[/math] is:

Bounds on Lambda (Grouped)

Fisher Matrix Bounds

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

[math]\displaystyle{ \frac{\ln \hat{\lambda }-\ln \lambda }{\sqrt{Var(\ln \hat{\lambda }})}\ \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 }=\frac{n}{T_{k}^{{\hat{\beta }}}}\,\! }[/math]

The variance calculation is the same as given above in the Bounds on Beta (Grouped) section.

Crow Bounds

Time Terminated Data

For the 2-sided [math]\displaystyle{ (1-\alpha )\,\! }[/math] 100-percent confidence interval, the confidence bounds on [math]\displaystyle{ \lambda \,\! }[/math] are:

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

Failure Terminated Data

For the 2-sided [math]\displaystyle{ (1-\alpha )\,\! }[/math] 100-percent confidence interval, the confidence bounds on [math]\displaystyle{ \lambda \,\! }[/math] are:

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

Bounds on Growth Rate (Grouped)

Fisher Matrix Bounds

Since the growth rate is equal to [math]\displaystyle{ 1-\beta \,\! }[/math], the confidence bounds are calculated from:

[math]\displaystyle{ \begin{align} G\operatorname{row}th\text{ }Rat{{e}_{L}}= & 1-{{\beta }_{U}} \\ G\operatorname{row}th\text{ }Rat{{e}_{U}}= & 1-{{\beta }_{L}} \end{align}\,\! }[/math]

[math]\displaystyle{ {{\beta }_{L}}\,\! }[/math] and [math]\displaystyle{ {{\beta }_{U}}\,\! }[/math] are obtained using the methods described above in the Bounds on Beta (Grouped) section. <PER LISA: ASK SME TO CONFIRM THAT THIS IS ADEQUATE.>>

Bounds on 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 treated as being normally distributed as well.

[math]\displaystyle{ \frac{\ln {{{\hat{m}}}_{c}}(t)-\ln {{m}_{c}}(t)}{\sqrt{Var(\ln {{{\hat{m}}}_{c}}(t)})}\ \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 given above in the Bounds on Beta (Grouped) section and:

[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

Calculate the Crow cumulative failure intensity confidence bounds:

[math]\displaystyle{ C{{(t)}_{L}}=\frac{\chi _{\tfrac{\alpha }{2},2N}^{2}}{2\cdot t}\,\! }[/math]
[math]\displaystyle{ C{{(t)}_{U}}=\frac{\chi _{1-\tfrac{\alpha }{2},2N+2}^{2}}{2\cdot t}\,\! }[/math]
Then:
[math]\displaystyle{ \begin{align} {{[MTB{{F}_{c}}]}_{L}}= & \frac{1}{C{{(t)}_{U}}} \\ {{[MTB{{F}_{c}}]}_{U}}= & \frac{1}{C{{(t)}_{L}}} \end{align}\,\! }[/math]

Bounds on Instantaneous MTBF (Grouped)

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 as well.

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

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

[math]\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 given above in the Bounds on Beta (Grouped) section and:

[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

Step 1: Calculate [math]\displaystyle{ P(i)=\tfrac{{{T}_{i}}}{{{T}_{K}}},\,\,i=1,2,\ldots ,K\,\! }[/math].
Step 2: Calculate:
[math]\displaystyle{ A=\underset{i=1}{\overset{K}{\mathop \sum }}\,\frac{{{\left[ P{{(i)}^{{\hat{\beta }}}}\ln P{{(i)}^{{\hat{\beta }}}}-P{{(i-1)}^{\widehat{\beta }}}\ln P{{(i-1)}^{{\hat{\beta }}}} \right]}^{2}}}{\left[ P{{(i)}^{{\hat{\beta }}}}-P{{(i-1)}^{{\hat{\beta }}}} \right]}\,\! }[/math]
Step 3: Calculate [math]\displaystyle{ D=\sqrt{\tfrac{1}{A}+1}\,\! }[/math] and [math]\displaystyle{ W=\tfrac{({{z}_{1-\alpha /2}})\cdot D}{\sqrt{N}}\,\! }[/math]. Thus an approximate 2-sided [math]\displaystyle{ (1-\alpha )\,\! }[/math] 100-percent confidence interval on [math]\displaystyle{ {{\hat{m}}_{i}}(t)\,\! }[/math] is:
[math]\displaystyle{ MTB{{F}_{i}}={{\widehat{m}}_{i}}(1\pm W)\,\! }[/math]

Bounds on Cumulative Failure Intensity (Grouped)

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 treated as being normally distributed.

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

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

[math]\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 given above in the Bounds on Beta (Grouped) section and:

[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 Crow cumulative failure intensity confidence bounds are given as:

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

Bounds on Instantaneous Failure Intensity (Grouped)

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 treated as being normally distributed.

[math]\displaystyle{ \frac{\ln {{{\hat{\lambda }}}_{i}}(t)-\ln {{\lambda }_{i}}(t)}{\sqrt{Var(\ln {{{\hat{\lambda }}}_{i}}(t)})}\tilde{\ }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 given above in the Bounds on Beta (Grouped) section and:

[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 Crow instantaneous failure intensity confidence bounds are given as:

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

Bounds on Time Given Cumulative MTBF (Grouped)

Fisher Matrix Bounds

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

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

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 given above in the Bounds on Beta (Grouped) section and:

[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 \text{ }{{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

Step 1: Calculate [math]\displaystyle{ {{\lambda }_{c}}(T)=\tfrac{1}{MTB{{F}_{c}}}\,\! }[/math].
Step 2: Use equations in 5.4.10.1 to calculate the bounds on time given the cumulative failure intensity.

Bounds on Time Given Instantaneous MTBF (Grouped)

Fisher Matrix Bounds

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

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

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 given above in the Bounds on Beta (Grouped) section and:

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

Crow Bounds

Step 1: Calculate the confidence bounds on the instantaneous MTBF:
[math]\displaystyle{ MTB{{F}_{i}}={{\widehat{m}}_{i}}(1\pm W)\,\! }[/math]
Step 2: Use equations in 5.4.5.2 to calculate the time given the instantaneous MTBF.

Bounds on Time Given Cumulative Failure Intensity (Grouped)

Fisher Matrix Bounds

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

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

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 given above in the Bounds on Beta (Grouped) section and:

[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

Step 1: Calculate:
[math]\displaystyle{ \hat{T}={{\left( \frac{{{\lambda }_{c}}(T)}{{\hat{\lambda }}} \right)}^{\tfrac{1}{\beta -1}}}\,\! }[/math]
Step 2: Estimate the number of failures:
[math]\displaystyle{ 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]\displaystyle{ {{t}_{l}}\,\! }[/math] and [math]\displaystyle{ {{t}_{u}}\,\! }[/math] in the following equations:
[math]\displaystyle{ \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 (Grouped)

Fisher Matrix Bounds

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

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

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 given above in the Bounds on Beta (Grouped) section and:

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


Crow Bounds

Step 1: Calculate [math]\displaystyle{ MTB{{F}_{i}}=\tfrac{1}{{{\lambda }_{i}}(T)}\,\! }[/math].
Step 2: Follow the same process as in 5.4.9.2 to calculate the bounds on time given the instantaneous failure intensity.

Bounds on Cumulative Number of Failures (Grouped)

Fisher Matrix Bounds

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

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

The variance calculation is the same as given above in the Bounds on Beta (Grouped) section and:

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


Crow Bounds

The Crow confidence bounds on cumulative number of failures are:

[math]\displaystyle{ \begin{align} {{N}_{L}}(T)= & \frac{T}{{\hat{\beta }}}{{\lambda }_{i}}{{(T)}_{L}} \\ {{N}_{U}}(T)= & \frac{T}{{\hat{\beta }}}{{\lambda }_{i}}{{(T)}_{U}} \end{align}\,\! }[/math]

where [math]\displaystyle{ {{\lambda }_{i}}{{(T)}_{L}}\,\! }[/math] and [math]\displaystyle{ {{\lambda }_{i}}{{(T)}_{U}}\,\! }[/math] can be obtained from the equations given above for Crow instantaneous failure intensity confidence bounds with grouped data.