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{{Template:RGA_BOOK_SUB|Appendix C|Crow AMSAA Confidence Bounds}}
{{Template:RGA_BOOK|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|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.
In this appendix, we will present the two methods used in the RGA software to estimate the confidence bounds for the [[Crow-AMSAA (NHPP)|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.
 
''Note regarding the Crow Bounds calculations: The equations that involve the use of the chi-squared distribution assume left-tail probability.''


==Individual (Non-Grouped) Data==
==Individual (Non-Grouped) Data==
===Bounds on Beta===
This section presents the confidence bounds for the Crow-AMSAA model under developmental testing when the failure times are known. The confidence bounds for when the failure times are not known are presented in the [[Crow-AMSAA_Confidence_Bounds#Grouped_Data|Grouped Data]] section.
====Fisher Matrix Bounds====
===Beta===<!-- THIS SECTION HEADER IS LINKED FROM SEVERAL SECTIONS IN THIS PAGE. IF YOU RENAME THE SECTION, YOU MUST UPDATE THE LINK(S). -->
The parameter <math>\beta \,\!</math> must be positive, thus <math>\ln \beta \,\!</math> is treated as being normally distributed as well.   
====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.   


::<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}
   -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\lambda }^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial \lambda \partial \beta }  \\
   -\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}}}  \\
   -\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}
\end{matrix} \right]_{\beta =\hat{\beta },\lambda =\hat{\lambda }}^{-1}=\left[ \begin{matrix}
   Var(\widehat{\lambda }) & Cov(\widehat{\beta },\widehat{\lambda })  \\
   Var(\hat{\lambda }) & Cov(\hat{\beta },\hat{\lambda })  \\
   Cov(\widehat{\beta },\widehat{\lambda }) & Var(\widehat{\beta })  \\
   Cov(\hat{\beta },\hat{\lambda }) & Var(\hat{\beta })  \\
\end{matrix} \right]</math>
\end{matrix} \right]\,\!</math>
 
::<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 \,\!</math> is the natural log-likelihood function:


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


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


:also:
:<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 \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'''
'''Failure Terminated'''


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% 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{N\cdot \chi _{\tfrac{\alpha }{2},2(N-1)}^{2}}{2(N-1)(N-2)} \\  
& {{D}_{U}}= & \frac{\chi _{1-\tfrac{\alpha }{2},2N}^{2}}{2(N-1)}   
  {{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>


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


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


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


Thus the confidence bounds on <math>\beta \,\!</math> are:
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>
 
===Growth Rate===
Since the growth rate, <math>\alpha \,\!</math>, is equal to <math>1-\beta \,\!</math>, the confidence bounds for both the Fisher matrix and Crow methods are:
<br>
 
:<math>\alpha_L=1-\beta_U\,\!</math>
:<math>\alpha_U=1-\beta_L\,\!</math>
 
<math>{{\beta }_{L}}\,\!</math> and <math>{{\beta }_{U}}\,\!</math> are obtained using the methods described above in the confidence bounds on [[Crow-AMSAA_Confidence_Bounds#Beta|Beta]].


===Bounds on Lambda===
===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 confidence bounds on [[Crow-AMSAA_Confidence_Bounds#Beta|Beta]].


====Crow Bounds====
====Crow Bounds====
'''Time Terminated Data'''
'''Failure Terminated'''
 
For the 2-sided <math>(1-\alpha )\,\!</math> 100% confidence interval, the confidence bounds on <math>\lambda \,\!</math> are:
 
:<math>\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>
 
where:
*<math>N\,\!</math> = total number of failures.
*<math>T\,\!</math> = termination time.
 
'''Time Terminated'''


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% 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>
 
where:
*<math>N\,\!</math> = total number of failures.
*<math>T\,\!</math> = termination time.
 
===Cumulative Number of Failures===
====Fisher Matrix Bounds====
The cumulative number of failures, <math>N(t)\,\!</math>, must be positive, thus <math>\ln N(t)\,\!</math> is treated as being normally distributed. 
 
:<math>\frac{\ln \hat{N}(t)-\ln N(t)}{\sqrt{Var(\ln \hat{N}(t)})}\ \tilde{\ }\ N(0,1)\,\!</math>
 
:<math>N(t)=\hat{N}(t){{e}^{\pm {{z}_{\alpha }}\sqrt{Var(\hat{N}(t))}/\hat{N}(t)}}\,\!</math>
 
where:


The fractiles can be found in the tables of the  <math>{{\chi }^{2}}\,\!</math> distribution.
:<math>\hat{N}(t)=\hat{\lambda }{{t}^{{\hat{\beta }}}}\,\!</math>


'''Failure Terminated Data'''
:<math>\begin{align}
  Var(\hat{N}(t))= & {{\left( \frac{\partial \hat{N}(t)}{\partial \beta } \right)}^{2}}Var(\hat{\beta })+{{\left( \frac{\partial \hat{N}(t)}{\partial \lambda } \right)}^{2}}Var(\hat{\lambda }) \\
  & +2\left( \frac{\partial \hat{N}(t)}{\partial \beta } \right)\left( \frac{\partial \hat{N}(t)}{\partial \lambda } \right)cov(\hat{\beta },\hat{\lambda }) 
\end{align}\,\!</math>


For the 2-sided  <math>(1-\alpha )\,\!</math> 100-percent confidence interval, the confidence bounds on <math>\lambda \,\!</math>  are:
The variance calculation is the same as given above in the confidence bounds on [[Crow-AMSAA_Confidence_Bounds#Beta|Beta]]. And:


::<math>\begin{align}
:<math>\begin{align}
   {{\lambda }_{L}}= & \frac{\chi _{\tfrac{\alpha }{2},2N}^{2}}{2{{T}^{{\hat{\beta }}}}} \\  
   \frac{\partial \hat{N}(t)}{\partial \beta }= & \hat{\lambda }{{t}^{{\hat{\beta }}}}\ln t \\  
   {{\lambda }_{U}}= & \frac{\chi _{1-\tfrac{\alpha }{2},2N}^{2}}{2{{T}^{{\hat{\beta }}}}}   
   \frac{\partial \hat{N}(t)}{\partial \lambda }= & {{t}^{{\hat{\beta }}}}   
\end{align}</math>
\end{align}\,\!</math>
 
====Crow Bounds====
The Crow cumulative number of failure confidence bounds are:


===Bounds on Growth Rate===
:<math>\begin{align}
Since the growth rate is equal to  <math>1-\beta \,\!</math> , the confidence bounds for both the Fisher Matrix and Crow methods are:
  {N(t)_{L}}= & \frac{t}{{\hat{\beta }}}{IFI}{{(t)}_{L}} \\  
<br>
  {N(t)_{U}}= & \frac{t}{{\hat{\beta }}}{IFI}{{(t)}_{U}} 
::<math>G\text{row}th Rate_L=1-\beta_U\,\!</math>
\end{align}\,\!</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.>>
where <math>IFI{{(t)}_{L}}\,\!</math> and <math>IFI{{(t)}_{U}}\,\!</math> are calculated using the process for calculating the confidence bounds on [[Crow-AMSAA_Confidence_Bounds#Crow_Bounds_7|instantaneous failure intensity]].


===Bounds on Cumulative MTBF===
===Cumulative Failure Intensity===
====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 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{m}}}_{c}}(t)-\ln {{m}_{c}}(t)}{\sqrt{Var(\ln {{{\hat{m}}}_{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 MTBF 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{m}}_{c}}(t){{e}^{\pm {{z}_{\alpha }}\sqrt{Var({{{\hat{m}}}_{c}}(t))}/{{{\hat{m}}}_{c}}(t)}}</math>
where:  


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


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


::<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{\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 {{m}_{c}}(t)}{\partial \beta } \right)\left( \frac{\partial {{m}_{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 confidence bounds on [[Crow-AMSAA_Confidence_Bounds#Beta|Beta]]. And:  


::<math>\begin{align}
:<math>\begin{align}
   \frac{\partial {{m}_{c}}(t)}{\partial \beta }= & -\frac{1}{{\hat{\lambda }}}{{t}^{1-\hat{\beta }}}\ln t \\  
   \frac{\partial {{\lambda }_{c}}(t)}{\partial \beta }= & \hat{\lambda }{{t}^{\hat{\beta }-1}}\ln t \\  
   \frac{\partial {{m}_{c}}(t)}{\partial \lambda }= & -\frac{1}{{{{\hat{\lambda }}}^{2}}}{{t}^{1-\hat{\beta }}}   
   \frac{\partial {{\lambda }_{c}}(t)}{\partial \lambda }= & {{t}^{\hat{\beta }-1}}   
\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:  
The Crow bounds on the cumulative failure intensity <math>(CFI)\,\!</math> are given below. Let:


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


::<math>C{{(t)}_{U}}=\frac{\chi _{1-\tfrac{\alpha }{2},2N+2}^{2}}{2\cdot t}</math>
'''Failure Terminated'''
:<math>\begin{align}
  CFI{_{L}}= & \frac{\chi _{\tfrac{\alpha }{2},2N}^{2}}{2\cdot t} \\
\end{align}\,\!</math>


:Then:
:<math>\begin{align}
  CFI{_{U}}= & \frac{\chi _{1-\tfrac{\alpha }{2},2N}^{2}}{2\cdot t} 
\end{align}\,\!</math>


::<math>\begin{align}
'''Time Terminated'''
  & {{[MTB{{F}_{c}}]}_{L}}= & \frac{1}{C{{(t)}_{U}}} \\
& {{[MTB{{F}_{c}}]}_{U}}= & \frac{1}{C{{(t)}_{L}}} 
\end{align}</math>


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


===Bounds on Instantaneous MTBF===
===Cumulative 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 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}}}_{i}}(t)-\ln {{m}_{i}}(t)}{\sqrt{Var(\ln {{{\hat{m}}}_{i}}(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 instantaneous MTBF are then estimated from:  
The approximate confidence bounds on the cumulative 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}}_{c}}(t){{e}^{\pm {{z}_{\alpha }}\sqrt{Var({{{\hat{m}}}_{c}}(t))}/{{{\hat{m}}}_{c}}(t)}}\,\!</math>


:where:  
where:  


::<math>{{\hat{m}}_{i}}(t)=\frac{1}{\lambda \beta {{t}^{\beta -1}}}</math>
:<math>{{\hat{m}}_{c}}(t)=\frac{1}{{\hat{\lambda }}}{{t}^{1-\hat{\beta }}}\,\!</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}}}_{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}_{i}}(t)}{\partial \beta } \right)\left( \frac{\partial {{m}_{i}}(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 confidence bounds on [[Crow-AMSAA_Confidence_Bounds#Beta|Beta]]. And:


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


====Crow Bounds====
====Crow Bounds====
The 2-sided confidence bounds on the cumulative MTBF <math>(CMTBF)\,\!</math> are given by:
:<math>\begin{align}
& CMTBF_{L}=\frac{1}{CFI_{U}} \\
& CMTBF_{U}=\frac{1}{CFI_{L}} 
\end{align}\,\!</math>
where <math>CFI_L\,\!</math> and <math>CFI_U\,\!</math> are calculated using the process for calculating the confidence bounds on [[Crow-AMSAA_Confidence_Bounds#Crow_Bounds_4|cumulative failure intensity]].


'''Failure Terminated Data'''
===Instantaneous MTBF===
====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.


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


::<math>G(\mu |n)=\mathop{}_{0}^{\infty }\frac{{{e}^{-x}}{{x}^{n-2}}}{(n-2)!}\underset{i=0}{\overset{n-1}{\mathop \sum }}\,\frac{1}{i!}{{\left( \frac{\mu }{x} \right)}^{i}}\exp (-\frac{\mu }{x})\,dx</math>
The approximate confidence bounds on the instantaneous MTBF are then estimated from:  


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


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


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


where  <math>MTB{{F}_{i}}=\tfrac{1}{\hat{\lambda }\hat{\beta }{{t}^{\hat{\beta }-1}}}\,\!</math> .
The variance calculation is the same as given above in the confidence bounds on [[Crow-AMSAA_Confidence_Bounds#Beta|Beta]]. And:
 
:<math>\begin{align}
  \frac{\partial {{m}_{i}}(t)}{\partial \beta }= & -\frac{1}{\hat{\lambda }{{{\hat{\beta }}}^{2}}}{{t}^{1-\hat{\beta }}}-\frac{1}{\hat{\lambda }\hat{\beta }}{{t}^{1-\hat{\beta }}}\ln t \\
  \frac{\partial {{m}_{i}}(t)}{\partial \lambda }= & -\frac{1}{{{{\hat{\lambda }}}^{2}}\hat{\beta }}{{t}^{1-\hat{\beta }}} 
\end{align}\,\!</math>


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


Consider the following equation where  <math>{{I}_{1}}(.)\,\!</math> is the modified Bessel function of order one:  
For failure terminated data and the 2-sided confidence bounds on instantaneous MTBF <math>(IMTBF)\,\!</math>, consider the following equation:  


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


::<math>\begin{align}
If using the biased parameters, <math>\hat{\beta }\,\!</math> and <math>\hat{\lambda }\,\!</math>, then the upper and lower confidence bounds are:
  {{[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>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}
  {{IMTBF}_{L}}= & IMTBF\cdot {{p}_{1}} \\  
  {{IMTBF}_{U}}= & IMTBF\cdot {{p}_{2}}   
\end{align}\,\!</math>


::<math>\begin{align}
where <math>IMTBF=\tfrac{1}{\hat{\lambda }\hat{\beta }{{t}^{\hat{\beta }-1}}}\,\!</math>.
  {{[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>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:


===Bounds on Cumulative Failure Intensity===
:<math>\begin{align}
====Fisher Matrix Bounds====
  {{IMTBF}_{L}}= & IMTBF\cdot \left( \frac{N-2}{N} \right)\cdot {{p}_{1}} \\  
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.
  {{IMTBF}_{U}}= & IMTBF\cdot \left( \frac{N-2}{N} \right)\cdot {{p}_{2}} 
\end{align}\,\!</math>


::<math>\frac{\ln {{{\hat{\lambda }}}_{c}}(t)-\ln {{\lambda }_{c}}(t)}{\sqrt{Var(\ln {{{\hat{\lambda }}}_{c}}(t)})}\ \tilde{\ }\ N(0,1)</math>
where <math>IMTBF=\tfrac{1}{\bar{\lambda }\bar{\beta }{{t}^{\bar{\beta }-1}}}\,\!</math>.


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


::<math>CB={{\hat{\lambda }}_{c}}(t){{e}^{\pm {{z}_{\alpha }}\sqrt{Var({{{\hat{\lambda }}}_{c}}(t))}/{{{\hat{\lambda }}}_{c}}(t)}}</math>
Consider the following equation where <math>{{I}_{1}}(.)\,\!</math> is the modified Bessel function of order one:


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


:and:  
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}
   Var({{{\hat{\lambda }}}_{c}}(t))= & {{\left( \frac{\partial {{\lambda }_{c}}(t)}{\partial \beta } \right)}^{2}}Var(\hat{\beta })+{{\left( \frac{\partial {{\lambda }_{c}}(t)}{\partial \lambda } \right)}^{2}}Var(\hat{\lambda }) \\
   {{IMTBF}_{L}}= & IMTBF\cdot {{\Pi }_{1}} \\  
  & +2\left( \frac{\partial {{\lambda }_{c}}(t)}{\partial \beta } \right)\left( \frac{\partial {{\lambda }_{c}}(t)}{\partial \lambda } \right)cov(\hat{\beta },\,\,\,\hat{\lambda })  
  {{IMTBF}_{U}}= & IMTBF\cdot {{\Pi }_{2}}   
\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:
where <math>IMTBF=\tfrac{1}{\hat{\lambda }\hat{\beta }{{t}^{\hat{\beta }-1}}}\,\!</math>.


::<math>\begin{align}
If using the unbiased parameters, <math>\bar{\beta }\,\!</math> and <math>\bar{\lambda }\,\!</math>, then the upper and lower confidence bounds are:
  \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====
:<math>\begin{align}
The Crow cumulative failure intensity confidence bounds are given as:
  {{IMTBF}_{L}}= & IMTBF\cdot \left( \frac{N-1}{N} \right)\cdot {{\Pi }_{1}} \\
  {{IMTBF}_{U}}= & IMTBF\cdot \left( \frac{N-1}{N} \right)\cdot {{\Pi }_{2}} 
\end{align}\,\!</math>


::<math>\begin{align}
where <math>IMTBF=\tfrac{1}{\bar{\lambda }\bar{\beta }{{t}^{\bar{\beta }-1}}}\,\!</math>.
  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===
===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 confidence bounds on [[Crow-AMSAA_Confidence_Bounds#Beta|Beta]]. And:  


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


====Crow Bounds====
====Crow Bounds====
The Crow instantaneous failure intensity confidence bounds are given as:  
The 2-sided confidence bounds on the instantaneous failure intensity <math>(IFI)\,\!</math> are given by:
 
:<math>\begin{align}
  {IFI_{L}}= & \frac{1}{{IMTBF}_{U}} \\
  {IFI_{U}}= & \frac{1}{{IMTBF}_{L}} 
\end{align}\,\!</math>


::<math>\begin{align}
where <math>IMTB{{F}_{L}}\,\!</math> and <math>IMTB{{F}_{U}}\,\!</math> are calculated using the process presented for the confidence bounds on the [[Crow-AMSAA_Confidence_Bounds#Crow_Bounds_6|instantaneous MTBF]].
  {{\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===
===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:  


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


The variance calculation is the same as 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 confidence bounds on [[Crow-AMSAA_Confidence_Bounds#Beta|Beta]]. And:  
   
   
::<math>\begin{align}
:<math>\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 \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:
The 2-sided confidence bounds on time given cumulative failure intensity <math>(CFI)\,\!</math> are given by:


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


:Step 2: Estimate the number of failures:
Then estimate the number of failures, <math>N\,\!</math>, such that:


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


::<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 CFI} \\  
   {{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 CFI}   
\end{align}</math>
\end{align}\,\!</math>


===Bounds on Time Given Cumulative MTBF===
===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:  


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


The variance calculation is the same as 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 confidence bounds on [[Crow-AMSAA_Confidence_Bounds#Beta|Beta]]. 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> .
The 2-sided confidence bounds on time given cumulative MTBF <math>(CMTBF)\,\!</math> are estimated using the process for calculating the confidence bounds on [[Crow-AMSAA_Confidence_Bounds#Crow_Bounds_8|time given cumulative failure intensity]] <math>(CFI)\,\!</math> where <math>CFI=\frac{1}{CMTBF}\,\!</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===
===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:  


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


The variance calculation is the same as 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 confidence bounds on [[Crow-AMSAA_Confidence_Bounds#Bounds_on_Beta|Beta]]. 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====
:Step 1: Calculate the confidence bounds on the instantaneous MTBF as presented in Section 5.5.2.
'''Failure Terminated'''
:Step 2: Calculate the bounds on time as follows.


====Failure Terminated Data====
If the unbiased value <math>\bar{\beta }\,\!</math> is used then:


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


So the lower an upper bounds on time are:
where:
*<math>IMTBF\,\!</math> = instantaneous MTBF.
*<math>N\,\!</math> = total number of failures.


::<math>{{\hat{T}}_{L}}={{(\frac{\lambda \beta \cdot MTB{{F}_{i}}}{{{c}_{1}}})}^{1/(1-\beta )}}</math>
Calculate the constants <math>p_1\,\!</math> and <math>p_2\,\!</math> using procedures described for the confidence bounds on [[Crow-AMSAA_Confidence_Bounds#Crow_Bounds_6|instantaneous MTBF]]. The lower and upper confidence bounds on time are then given by:


::<math>{{\hat{T}}_{U}}={{(\frac{\lambda \beta \cdot MTB{{F}_{i}}}{{{c}_{2}}})}^{1/(1-\beta )}}</math>
:<math>{{\hat{t}}_{L}}={{\left( \frac{\lambda \beta \cdot IMTBF}{{{p}_{1}}} \right)}^{\tfrac{1}{1-\beta }}}</math>


'''Time Terminated Data'''
:<math>{{\hat{t}}_{U}}={{\left( \frac{\lambda \beta \cdot IMTBF}{{{p}_{2}}} \right)}^{\tfrac{1}{1-\beta }}}</math>


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


So the lower and upper bounds on time are:
If the unbiased value <math>\bar{\beta }\,\!</math> is used then:


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


::<math>{{\hat{T}}_{U}}={{(\frac{\lambda \beta \cdot MTB{{F}_{i}}}{{{\Pi }_{2}}})}^{1/(1-\beta )}}</math>
where:
*<math>IMTBF\,\!</math> = instantaneous MTBF.
*<math>N\,\!</math> = total number of failures.


===Bounds on Time Given Instantaneous Failure Intensity===
Calculate the constants <math>{{\Pi }_{1}}\,\!</math> and <math>{{\Pi }_{2}}\,\!</math> using procedures described for the confidence bounds on [[Crow-AMSAA_Confidence_Bounds#Crow_Bounds_6|instantaneous MTBF]]. The lower and upper confidence bounds on time are then given by:
 
:<math>{{\hat{t}}_{L}}={{\left( \frac{\lambda \beta \cdot IMTBF}{{{\Pi }_{1}}} \right)}^{\tfrac{1}{1-\beta }}}\,\!</math>
 
:<math>{{\hat{t}}_{U}}={{\left( \frac{\lambda \beta \cdot IMTBF}{{{\Pi }_{2}}} \right)}^{\tfrac{1}{1-\beta }}}\,\!</math>
 
===Time Given Instantaneous Failure Intensity===<!-- THIS SECTION HEADER IS LINKED FROM ANOTHER SECTION IN THIS PAGE. IF YOU RENAME THE SECTION, YOU MUST UPDATE THE LINK(S). -->
====Fisher Matrix Bounds====
====Fisher Matrix Bounds====
The 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:  


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


The variance calculation is the same as 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 confidence bounds on [[Crow-AMSAA_Confidence_Bounds#Beta|Beta]]. 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> .
The 2-sided confidence bounds on time given instantaneous failure intensity <math>(IFI)\,\!</math> are estimated using the process for calculating the confidence bounds on [[Crow-AMSAA_Confidence_Bounds#Crow_Bounds_10|time given instantaneous MTBF]] where <math>IMTBF=\frac{1}{IFI}\,\!</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===
==Grouped Data==
====Fisher Matrix Bounds====
This section presents the confidence bounds for the Crow-AMSAA model when using Grouped data.
The cumulative number of failures,  <math>N(t)\,\!</math> , must be positive, thus <math>\ln N(t)\,\!</math> is treated as being normally distributed.   
====Beta (Grouped)====<!-- THIS SECTION HEADER IS LINKED TO: Crow-AMSAA (NHPP) and to several sections in this page. IF YOU RENAME THE SECTION, YOU MUST UPDATE THE LINK. -->
=====Fisher Matrix Bounds=====
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{N}(t)-\ln N(t)}{\sqrt{Var(\ln \hat{N}(t)})}\ \tilde{\ }\ N(0,1)</math>
:<math>\frac{\ln \hat{\beta }-\ln \beta }{\sqrt{Var(\ln \hat{\beta }})}\ \tilde{\ }\ N(0,1)\,\!</math>


::<math>N(t)=\hat{N}(t){{e}^{\pm {{z}_{\alpha }}\sqrt{Var(\hat{N}(t))}/\hat{N}(t)}}</math>
The approximate confidence bounds are given as:  


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


::<math>\hat{N}(t)=\hat{\lambda }{{t}^{{\hat{\beta }}}}</math>
:<math>\hat{\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>\begin{align}
All variance can be calculated using the Fisher matrix:  
  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 [[Crow-AMSAA_Confidence_Bounds#Fisher_Matrix_Bounds|Bounds on Beta]] section and:
:<math>\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 =\hat{\beta },\lambda =\hat{\lambda }}^{-1}=\left[ \begin{matrix}
  Var(\hat{\lambda }) & Cov(\hat{\beta },\hat{\lambda })  \\
  Cov(\hat{\beta },\hat{\lambda }) & Var(\hat{\beta })  \\
\end{matrix} \right]\,\!</math>


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


The Crow cumulative number of failure confidence bounds are:  
:<math>\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>


::<math>\begin{align}
=====Crow Bounds=====
  {{N}_{L}}(T)= & \frac{T}{{\hat{\beta }}}{{\lambda }_{i}}{{(T)}_{L}} \\
The 2-sided confidence bounds on <math>\hat{\beta }\,\!</math> are given by first calculating:
  {{N}_{U}}(T)= & \frac{T}{{\hat{\beta }}}{{\lambda }_{i}}{{(T)}_{U}} 
\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.
:<math>P\left( i \right)=\frac{{{T}_{i}}}{{{T}_{K}}};\text{ }i=1,2,...,K</math>


==Grouped Data==
where:
====Bounds on Beta (Grouped)====
=====Fisher Matrix Bounds=====
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>T_i\,\!</math> = interval end time for the <math>{{i}^{th}}\,\!</math> interval.
*<math>K\,\!</math> = number of intervals.
*<math>T_K\,\!</math> = end time for the last interval.


The approximate confidence bounds are given as:  
Next:  


::<math>C{{B}_{\beta }}=\hat{\beta }{{e}^{\pm {{z}_{\alpha }}\sqrt{Var(\hat{\beta })}/\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)}^{\hat{\beta }}}\ln P{{(i-1)}^{{\hat{\beta }}}}]}^{2}}}{[P{{(i)}^{{\hat{\beta }}}}-P{{(i-1)}^{{\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> .
And:


All variance can be calculated using the Fisher Matrix:  
:<math>c=\frac{1}{\sqrt{A}}</math>


::<math>\left[ \begin{matrix}
Then:
  -\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>\Lambda \,\!</math>  is the natural log-likelihood function where ln <math>^{2}T={{\left( \ln T \right)}^{2}}\,\!</math> and:
:<math>S=\frac{\left( {{z}_{1-\tfrac{\alpha }{2}}} \right)\cdot c}{\sqrt{N}}</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>
where:


::<math>\begin{align}
*<math>{{z}_{1-\tfrac{\alpha }{2}}}\,\!</math> = inverse standard normal.
  \frac{{{\partial }^{2}}\Lambda }{\partial {{\lambda }^{2}}}= & -\frac{n}{{{\lambda }^{2}}} \\
*<math>N\,\!</math> = number of failures.
  \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=====
The 2-sided confidence bounds on <math>\beta\,\!</math> are then <math>\hat{\beta }\left( 1\pm S \right)\,\!</math>.
:Step 1: Calculate  <math>P(i)=\tfrac{{{T}_{i}}}{{{T}_{K}}},\,\,i=1,2,\ldots ,K\,\!</math> .
:Step 2: Calculate:


::<math>A=\underset{i=1}{\overset{K}{\mathop \sum }}\,\frac{{{[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>
===Growth Rate (Grouped)===
Since the growth rate, <math>\alpha \,\!</math>, is equal to <math>1-\beta \,\!</math>, the confidence bounds for both the Fisher matrix and Crow methods are:
<br>


: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:
:<math>\alpha_L=1-\beta_U\,\!</math>
:<math>\alpha_U=1-\beta_L\,\!</math>


===Bounds on Lambda (Grouped)===
<math>{{\beta }_{L}}\,\!</math> and <math>{{\beta }_{U}}\,\!</math> are obtained using the methods described above in the confidence bounds on [[Crow-AMSAA_Confidence_Bounds#Beta_.28Grouped.29|Beta]].
===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 confidence bounds on [[Crow-AMSAA_Confidence_Bounds#Beta_.28Grouped.29|Beta]].


====Crow Bounds====
====Crow Bounds====
'''Failure Terminated'''
For failure terminated data, the 2-sided <math>(1-\alpha )\,\!</math> 100% confidence interval, the confidence bounds on <math>\lambda \,\!</math> are:


'''Time Terminated Data'''
:<math>\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>


For the 2-sided  <math>(1-\alpha )\,\!</math> 100-percent confidence interval, the confidence bounds on  <math>\lambda \,\!</math> are:
where:
*<math>N\,\!</math> = total number of failures.
*<math>T_K\,\!</math> = end time of last interval.


::<math>\begin{align}
'''Time Terminated'''
 
For time terminated data, the 2-sided <math>(1-\alpha )\,\!</math> 100% confidence interval, the confidence bounds on <math>\lambda \,\!</math> are:
 
:<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>
 
where:
*<math>N\,\!</math> = total number of failures.
*<math>T_K\,\!</math> = end time of last interval.
 
===Cumulative Number of Failures (Grouped)===
====Fisher Matrix Bounds====
The cumulative number of failures, <math>N(t)\,\!</math>, must be positive, thus <math>\ln N(t)\,\!</math> is treated as being normally distributed. 
 
:<math>\frac{\ln \hat{N}(t)-\ln N(t)}{\sqrt{Var(\ln \hat{N}(t)})}\ \tilde{\ }\ N(0,1)\,\!</math>
 
:<math>N(t)=\hat{N}(t){{e}^{\pm {{z}_{\alpha }}\sqrt{Var(\hat{N}(t))}/\hat{N}(t)}}\,\!</math>
 
where:
 
:<math>\hat{N}(t)=\hat{\lambda }{{t}^{{\hat{\beta }}}}\,\!</math>
 
:<math>\begin{align}
  Var(\hat{N}(t))= & {{\left( \frac{\partial \hat{N}(t)}{\partial \beta } \right)}^{2}}Var(\hat{\beta })+{{\left( \frac{\partial \hat{N}(t)}{\partial \lambda } \right)}^{2}}Var(\hat{\lambda }) \\
  & +2\left( \frac{\partial \hat{N}(t)}{\partial \beta } \right)\left( \frac{\partial \hat{N}(t)}{\partial \lambda } \right)cov(\hat{\beta },\hat{\lambda }) 
\end{align}\,\!</math>


'''Failure Terminated Data'''
The variance calculation is the same as given above in the confidence bounds on [[Crow-AMSAA_Confidence_Bounds#Beta_.28Grouped.29|Beta]]. And: 


For the 2-sided  <math>(1-\alpha )\,\!</math> 100-percent confidence interval, the confidence bounds on <math>\lambda \,\!</math> are:
:<math>\begin{align}
  \frac{\partial \hat{N}(t)}{\partial \beta }= & \hat{\lambda }{{t}^{{\hat{\beta }}}}\ln t \\
  \frac{\partial \hat{N}(t)}{\partial \lambda }= & {{t}^{{\hat{\beta }}}}  
\end{align}\,\!</math>


::<math>\begin{align}
====Crow Bounds====
  {{\lambda }_{L}}= & \frac{\chi _{\tfrac{\alpha }{2},2N}^{2}}{2\cdot T_{k}^{\beta }} \\  
The 2-sided confidence bounds on the cumulative number of failures are given by:
  {{\lambda }_{U}}= & \frac{\chi _{1-\tfrac{\alpha }{2},2N}^{2}}{2\cdot T_{k}^{\beta }}
 
\end{align}</math>
:<math>N{{(t)}_{L}}=\frac{t}{{\hat{\beta }}}IF{{I}_{L}}\,\!</math>
 
:<math>N{{(t)}_{U}}=\frac{t}{{\hat{\beta }}}IF{{I}_{U}}\,\!</math>
 
where <math>IFI_L\,\!</math> and <math>IFI_U\,\!</math> are calculated based on the procedures for the confidence bounds on the [[Crow-AMSAA_Confidence_Bounds#Crow_Bounds_18|instantaneous failure intensity]].


===Bounds on Growth Rate (Grouped)===
===Cumulative Failure Intensity (Grouped)===
====Fisher Matrix Bounds====
====Fisher Matrix Bounds====
Since the growth rate is equal to <math>1-\beta \,\!</math> , the confidence bounds are calculated from:
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>
 
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>
 
where:
 
:<math>{{\hat{\lambda }}_{c}}(t)=\hat{\lambda }{{t}^{\hat{\beta }-1}}\,\!</math>
 
and:  


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


<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.>>
The variance calculation is the same as given above in the confidence bounds on [[Crow-AMSAA_Confidence_Bounds#Beta_.28Grouped.29|Beta]]. And:  


===Bounds on Cumulative MTBF===
:<math>\begin{align}
  \frac{\partial {{\lambda }_{c}}(t)}{\partial \beta }= & \hat{\lambda }{{t}^{\hat{\beta }-1}}\ln t \\
  \frac{\partial {{\lambda }_{c}}(t)}{\partial \lambda }= & {{t}^{\hat{\beta }-1}} 
\end{align}\,\!</math>
 
====Crow Bounds====
The 2-sided confidence bounds on the cumulative failure intensity <math>(CFI\,\!)</math> are given below. Let:
 
:<math>N=\hat{\lambda }{{t}^{{\hat{\beta }}}}</math>
 
Then:
 
:<math>\begin{align}
CFI_{L}= & \frac{\chi _{\tfrac{\alpha }{2},2N}^{2}}{2\cdot t} \\
CFI_{U}= & \frac{\chi _{1-\tfrac{\alpha }{2},2N+2}^{2}}{2\cdot t} 
\end{align}\,\!</math>
 
===Cumulative MTBF (Grouped)===
====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 confidence bounds on [[Crow-AMSAA_Confidence_Bounds#Beta_.28Grouped.29|Beta]]. And:  


::<math>\begin{align}
:<math>\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 \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:  
The 2-sided confidence bounds on cumulative MTBF <math>(CMTBF)\,\!</math> are given by:


::<math>C{{(t)}_{L}}=\frac{\chi _{\tfrac{\alpha }{2},2N}^{2}}{2\cdot t}</math>
:<math>CMTB{{F}_{L}}=\frac{1}{CF{{I}_{U}}}\,\!</math>


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


:Then:
where <math>CFI_{L}\,\!</math> and <math>CFI_{U}\,\!</math> are calculating using the process for calculating the confidence bounds on the [[Crow-AMSAA_Confidence_Bounds#Cumulative_Failure_Intensity_.28Grouped.29|cumulative failure intensity]].


::<math>\begin{align}
===Instantaneous MTBF (Grouped)===
  {{[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====
====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 confidence bounds on [[Crow-AMSAA_Confidence_Bounds#Beta_.28Grouped.29|Beta]]. And:


::<math>\begin{align}
:<math>\begin{align}
   \frac{\partial {{m}_{i}}(t)}{\partial \beta }= & -\frac{1}{\hat{\lambda }{{{\hat{\beta }}}^{2}}}{{t}^{1-\hat{\beta }}}-\frac{1}{\hat{\lambda }\hat{\beta }}{{t}^{1-\hat{\beta }}}\ln t \\  
   \frac{\partial {{m}_{i}}(t)}{\partial \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> .
The 2-sided confidence bounds on instantaneous MTBF <math>(IMTBF)\,\!</math> are given by first calculating:
:Step 2: Calculate:
 
::<math>A=\underset{i=1}{\overset{K}{\mathop \sum }}\,\frac{{{\left[ P{{(i)}^{{\hat{\beta }}}}\ln P{{(i)}^{{\hat{\beta }}}}-P{{(i-1)}^{\widehat{\beta }}}\ln P{{(i-1)}^{{\hat{\beta }}}} \right]}^{2}}}{\left[ P{{(i)}^{{\hat{\beta }}}}-P{{(i-1)}^{{\hat{\beta }}}} \right]}</math>
 
:Step 3: Calculate  <math>D=\sqrt{\tfrac{1}{A}+1}\,\!</math>  and  <math>W=\tfrac{({{z}_{1-\alpha /2}})\cdot D}{\sqrt{N}}\,\!</math> . Thus an approximate 2-sided <math>(1-\alpha )\,\!</math> 100-percent confidence interval on <math>{{\hat{m}}_{i}}(t)\,\!</math> is:
 
::<math>MTB{{F}_{i}}={{\widehat{m}}_{i}}(1\pm W)</math>


===Bounds on Cumulative Failure Intensity (Grouped)===
:<math>P\left( i \right)=\frac{{{T}_{i}}}{{{T}_{K}}};\text{ }i=1,2,...,K</math>
====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. 


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


The approximate confidence bounds on the cumulative failure intensity are then estimated from:
*<math>T_i\,\!</math> = interval end time for the <math>{{i}^{th}}\,\!</math> interval.
*<math>K\,\!</math> = number of intervals.
*<math>T_K\,\!</math> = end time for the last interval.


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


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


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


:and:
:<math>D=\sqrt{\frac{1}{A}+1}</math>


::<math>\begin{align}
And:
  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 [[Crow-AMSAA_Confidence_Bounds#Bounds_on_Beta_.28Grouped.29|Bounds on Beta (Grouped)]] section and:
:<math>W=\frac{\left( {{z}_{1-\tfrac{\alpha }{2}}} \right)\cdot D}{\sqrt{N}}</math>


::<math>\begin{align}
where:
  \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====
*<math>{{z}_{1-\tfrac{\alpha }{2}}}\,\!</math> = inverse standard normal.
The Crow cumulative failure intensity confidence bounds are given as:
*<math>N\,\!</math> = number of failures.


::<math>\begin{align}
The 2-sided confidence bounds on instantaneous MTBF are then <math>IMTBF\left( 1\pm W \right)\,\!</math>.
  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)===
===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 confidence bounds on [[Crow-AMSAA_Confidence_Bounds#Beta_.28Grouped.29|Beta]]. And:  


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


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


::<math>\begin{align}
<math>\begin{align}
  {{[{{\lambda }_{i}}(t)]}_{L}}= & \frac{1}{{{[MTB{{F}_{i}}]}_{U}}} \\  
  IF{{I}_{U}}= & \frac{1}{IMTB{{F}_{L}}} \\
   {{[{{\lambda }_{i}}(t)]}_{U}}= & \frac{1}{{{[MTB{{F}_{i}}]}_{L}}}
   IF{{I}_{L}}= & \frac{1}{IMTB{{F}_{U}}}
\end{align}</math>
\end{align}\,\!</math>
where <math>IMTB{{F}_{L}}\,\!</math>and <math>IMTB{{F}_{U}}\,\!</math> are calculated using the process for calculating the confidence bounds on the [[Crow-AMSAA_Confidence_Bounds#Crow_Bounds_17|instantaneous MTBF]].


===Bounds on Time Given Cumulative MTBF (Grouped)===
===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:  


::<math>\begin{align}
:<math>\begin{align}
   Var(\hat{T})= & {{\left( \frac{\partial T}{\partial \beta } \right)}^{2}}Var(\hat{\beta })+{{\left( \frac{\partial T}{\partial \lambda } \right)}^{2}}Var(\hat{\lambda }) \\  
   Var(\hat{T})= & {{\left( \frac{\partial T}{\partial \beta } \right)}^{2}}Var(\hat{\beta })+{{\left( \frac{\partial T}{\partial \lambda } \right)}^{2}}Var(\hat{\lambda }) \\  
   & +2\left( \frac{\partial T}{\partial \beta } \right)\left( \frac{\partial T}{\partial \lambda } \right)cov(\hat{\beta },\,\,\,\hat{\lambda })   
   & +2\left( \frac{\partial T}{\partial \beta } \right)\left( \frac{\partial T}{\partial \lambda } \right)cov(\hat{\beta },\,\,\,\hat{\lambda })   
\end{align}</math>
\end{align}\,\!</math>
 
The variance calculation is the same as given above in the confidence bounds on [[Crow-AMSAA_Confidence_Bounds#Beta_.28Grouped.29|Beta]]. And:
:<math>\begin{align}
  \frac{\partial T}{\partial \beta }= & \frac{-{{\left( \tfrac{{{\lambda }_{c}}(T)}{\lambda } \right)}^{1/(\beta -1)}}\ln \left( \tfrac{{{\lambda }_{c}}(T)}{\lambda } \right)}{{{(1-\beta )}^{2}}} \\
  \frac{\partial T}{\partial \lambda }= & {{\left( \frac{{{\lambda }_{c}}(T)}{\lambda } \right)}^{1/(\beta -1)}}\frac{1}{\lambda (1-\beta )} 
\end{align}\,\!</math>
 
====Crow Bounds====
The 2-sided confidence bounds on time given cumulative failure intensity <math>(CFI)\,\!</math> are presented below. Let:
 
:<math>\hat{t}={{\left( \frac{CFI}{{\hat{\lambda }}} \right)}^{\tfrac{1}{\hat{\beta }-1}}}\,\!</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:  
Then estimate the number of failures:


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


::<math>\begin{align}
The confidence bounds on time given the cumulative failure intensity are then given by:
  \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====
:<math>\begin{align}
:Step 1: Calculate  <math>{{\lambda }_{c}}(T)=\tfrac{1}{MTB{{F}_{c}}}\,\!</math> .
  {{t}_{L}}= & \frac{\chi _{\tfrac{\alpha }{2},2N}^{2}}{2\cdot {CFI}} \\
:Step 2: Use equations in 5.4.10.1 to calculate the bounds on time given the cumulative failure intensity.
  {{t}_{U}}= & \frac{\chi _{1-\tfrac{\alpha }{2},2N+2}^{2}}{2\cdot {CFI}} 
\end{align}\,\!</math>


===Bounds on Time Given Instantaneous MTBF (Grouped)===
===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:  


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


The variance calculation is the same as 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 confidence bounds on [[Crow-AMSAA_Confidence_Bounds#Beta_.28Grouped.29|Beta]]. And:  


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


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


====Crow Bounds====
====Crow Bounds====
:Step 1: Calculate the confidence bounds on the instantaneous MTBF:
The 2-sided confidence bounds on time given cumulative MTBF <math>(CMTBF)\,\!</math> are estimated using the process for calculating the confidence bounds on [[Crow-AMSAA_Confidence_Bounds#Crow_Bounds_19|time given cumulative failure intensity]] <math>(CFI)\,\!</math> where <math>CFI=\frac{1}{CMTBF}\,\!</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.
===Time Given Instantaneous MTBF (Grouped)===<!-- THIS SECTION HEADER IS LINKED FROM ANOTHER SECTION IN THIS PAGE. IF YOU RENAME THE SECTION, YOU MUST UPDATE THE LINK(S). -->
 
===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:  


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


The variance calculation is the same as 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 confidence bounds on [[Crow-AMSAA_Confidence_Bounds#Beta_.28Grouped.29|Beta]]. And:
 
::<math>\begin{align}
:<math>\hat{T}={{(\lambda \beta \cdot {{m}_{i}}(T))}^{1/(1-\beta )}}\,\!</math>
  \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>


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


====Crow Bounds====
====Crow Bounds====
:Step 1: Calculate:
'''Failure Terminated'''


::<math>\hat{T}={{\left( \frac{{{\lambda }_{c}}(T)}{{\hat{\lambda }}} \right)}^{\tfrac{1}{\beta -1}}}</math>
Calculate the constants <math>p_1\,\!</math> and <math>p_2\,\!</math> using procedures described for the confidence bounds on [[Crow-AMSAA_Confidence_Bounds#Crow_Bounds_17|instantaneous MTBF]]. The lower and upper confidence bounds on time are then given by:


:Step 2: Estimate the number of failures:
:<math>{{\hat{t}}_{L}}={{\left( \frac{\lambda \beta \cdot IMTBF}{{{p}_{1}}} \right)}^{\tfrac{1}{1-\beta }}}</math>


::<math>N(\hat{T})=\hat{\lambda }{{\hat{T}}^{{\hat{\beta }}}}</math>
:<math>{{\hat{t}}_{U}}={{\left( \frac{\lambda \beta \cdot IMTBF}{{{p}_{2}}} \right)}^{\tfrac{1}{1-\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:
'''Time Terminated'''


::<math>\begin{align}
Calculate the constants <math>{{\Pi }_{1}}\,\!</math> and <math>{{\Pi }_{2}}\,\!</math> using procedures described for the confidence bounds on [[Crow-AMSAA_Confidence_Bounds#Crow_Bounds_17|instantaneous MTBF]]. The lower and upper confidence bounds on time are then given by:
  {{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)===
:<math>{{\hat{t}}_{L}}={{\left( \frac{\lambda \beta \cdot IMTBF}{{{\Pi }_{1}}} \right)}^{\tfrac{1}{1-\beta }}}\,\!</math>
 
:<math>{{\hat{t}}_{U}}={{\left( \frac{\lambda \beta \cdot IMTBF}{{{\Pi }_{2}}} \right)}^{\tfrac{1}{1-\beta }}}\,\!</math>
 
===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:  


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


The variance calculation is the same as 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 confidence bounds on [[Crow-AMSAA_Confidence_Bounds#Beta_.28Grouped.29|Beta]]. 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> .
The 2-sided confidence bounds on time given instantaneous failure intensity <math>(IFI)\,\!</math> are estimated using the process for calculating the confidence bounds on [[Crow-AMSAA_Confidence_Bounds#Crow_Bounds_21|time given instantaneous MTBF]] where <math>IMTBF=\frac{1}{IFI}\,\!</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>N(t)\,\!</math> , must be positive, thus  <math>\ln N(t)\,\!</math>  is treated as being normally distributed. 
 
::<math>\frac{\ln \hat{N}(t)-\ln N(t)}{\sqrt{Var(\ln \hat{N}(t)})}\ \tilde{\ }\ N(0,1)</math>
 
::<math>N(t)=\hat{N}(t){{e}^{\pm {{z}_{\alpha }}\sqrt{Var(\hat{N}(t))}/\hat{N}(t)}}</math>
 
:where:
 
::<math>\hat{N}(t)=\hat{\lambda }{{t}^{{\hat{\beta }}}}</math>
 
::<math>\begin{align}
  Var(\hat{N}(t))= & {{\left( \frac{\partial \hat{N}(t)}{\partial \beta } \right)}^{2}}Var(\hat{\beta })+{{\left( \frac{\partial \hat{N}(t)}{\partial \lambda } \right)}^{2}}Var(\hat{\lambda }) \\
  & +2\left( \frac{\partial \hat{N}(t)}{\partial \beta } \right)\left( \frac{\partial \hat{N}(t)}{\partial \lambda } \right)cov(\hat{\beta },\hat{\lambda }) 
\end{align}</math>
 
The variance calculation is the same as given above in the [[Crow-AMSAA_Confidence_Bounds#Bounds_on_Beta_.28Grouped.29|Bounds on Beta (Grouped)]] section and: 
 
::<math>\begin{align}
  \frac{\partial \hat{N}(t)}{\partial \beta }= & \hat{\lambda }{{t}^{{\hat{\beta }}}}\ln t \\
  \frac{\partial \hat{N}(t)}{\partial \lambda }= & {{t}^{{\hat{\beta }}}}
\end{align}</math>
 
 
====Crow Bounds====
The Crow confidence bounds on cumulative number of failures are:
 
::<math>\begin{align}
  {{N}_{L}}(T)= & \frac{T}{{\hat{\beta }}}{{\lambda }_{i}}{{(T)}_{L}} \\
  {{N}_{U}}(T)= & \frac{T}{{\hat{\beta }}}{{\lambda }_{i}}{{(T)}_{U}} 
\end{align}</math>
 
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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Chapter Appendix C: Crow-AMSAA Confidence Bounds


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Chapter Appendix C  
Crow-AMSAA Confidence Bounds  

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Available Software:
RGA

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More Resources:
RGA examples

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.

Note regarding the Crow Bounds calculations: The equations that involve the use of the chi-squared distribution assume left-tail probability.

Individual (Non-Grouped) Data

This section presents the confidence bounds for the Crow-AMSAA model under developmental testing when the failure times are known. The confidence bounds for when the failure times are not known are presented in the Grouped Data section.

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 =\hat{\beta },\lambda =\hat{\lambda }}^{-1}=\left[ \begin{matrix} Var(\hat{\lambda }) & Cov(\hat{\beta },\hat{\lambda }) \\ Cov(\hat{\beta },\hat{\lambda }) & Var(\hat{\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]

And:

[math]\displaystyle{ \frac{{{\partial }^{2}}\Lambda }{\partial {{\lambda }^{2}}}=-\frac{N}{{{\lambda }^{2}}}\,\! }[/math]
[math]\displaystyle{ \frac{{{\partial }^{2}}\Lambda }{\partial {{\beta }^{2}}}=-\frac{N}{{{\beta }^{2}}}-\lambda {{T}^{\beta }}{{(\ln T)}^{2}}\,\! }[/math]
[math]\displaystyle{ \frac{{{\partial }^{2}}\Lambda }{\partial \lambda \partial \beta }=-{{T}^{\beta }}\ln T\,\! }[/math]

Crow Bounds

Failure Terminated

For the 2-sided [math]\displaystyle{ (1-\alpha )\,\! }[/math] 100% 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]

Time Terminated

For the 2-sided [math]\displaystyle{ (1-\alpha )\,\! }[/math] 100% 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 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]

Growth Rate

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

[math]\displaystyle{ \alpha_L=1-\beta_U\,\! }[/math]
[math]\displaystyle{ \alpha_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 confidence bounds on Beta.

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 confidence bounds on Beta.

Crow Bounds

Failure Terminated

For the 2-sided [math]\displaystyle{ (1-\alpha )\,\! }[/math] 100% 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]

where:

  • [math]\displaystyle{ N\,\! }[/math] = total number of failures.
  • [math]\displaystyle{ T\,\! }[/math] = termination time.

Time Terminated

For the 2-sided [math]\displaystyle{ (1-\alpha )\,\! }[/math] 100% 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]

where:

  • [math]\displaystyle{ N\,\! }[/math] = total number of failures.
  • [math]\displaystyle{ T\,\! }[/math] = termination time.

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 confidence bounds on Beta. 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(t)_{L}}= & \frac{t}{{\hat{\beta }}}{IFI}{{(t)}_{L}} \\ {N(t)_{U}}= & \frac{t}{{\hat{\beta }}}{IFI}{{(t)}_{U}} \end{align}\,\! }[/math]

where [math]\displaystyle{ IFI{{(t)}_{L}}\,\! }[/math] and [math]\displaystyle{ IFI{{(t)}_{U}}\,\! }[/math] are calculated using the process for calculating the confidence bounds on instantaneous failure intensity.

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 confidence bounds on Beta. 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 bounds on the cumulative failure intensity [math]\displaystyle{ (CFI)\,\! }[/math] are given below. Let:

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

Failure Terminated

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

Time Terminated

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

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 confidence bounds on Beta. 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

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

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

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

Instantaneous MTBF

Fisher Matrix Bounds

The instantaneous MTBF, [math]\displaystyle{ {{m}_{i}}(t)\,\! }[/math], must be positive, thus [math]\displaystyle{ \ln {{m}_{i}}(t)\,\! }[/math] is 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 confidence bounds on Beta. 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

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

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

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

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

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

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

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

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

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

Time Terminated

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

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

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

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

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

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

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

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

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

Instantaneous Failure Intensity

Fisher Matrix Bounds

The instantaneous failure intensity, [math]\displaystyle{ {{\lambda }_{i}}(t)\,\! }[/math], must be positive, thus [math]\displaystyle{ \ln {{\lambda }_{i}}(t)\,\! }[/math] is 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 confidence bounds on Beta. 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 2-sided confidence bounds on the instantaneous failure intensity [math]\displaystyle{ (IFI)\,\! }[/math] are given by:

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

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

Time Given Cumulative Failure Intensity

Fisher Matrix Bounds

The time, [math]\displaystyle{ T\,\! }[/math], must be positive, thus [math]\displaystyle{ \ln T\,\! }[/math] is 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 confidence bounds on Beta. 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

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

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

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

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

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

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

Time Given Cumulative MTBF

Fisher Matrix Bounds

The time, [math]\displaystyle{ T\,\! }[/math], must be positive, thus [math]\displaystyle{ \ln T\,\! }[/math] is 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 confidence bounds on Beta. 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

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

Time Given Instantaneous MTBF

Fisher Matrix Bounds

The time, [math]\displaystyle{ T\,\! }[/math], must be positive, thus [math]\displaystyle{ \ln T\,\! }[/math] is 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 confidence bounds on Beta. 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

Failure Terminated

If the unbiased value [math]\displaystyle{ \bar{\beta }\,\! }[/math] is used then:

[math]\displaystyle{ IMTBF=IMTBF\cdot \frac{N-2}{N}\,\! }[/math]

where:

  • [math]\displaystyle{ IMTBF\,\! }[/math] = instantaneous MTBF.
  • [math]\displaystyle{ N\,\! }[/math] = total number of failures.

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

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

Time Terminated

If the unbiased value [math]\displaystyle{ \bar{\beta }\,\! }[/math] is used then:

[math]\displaystyle{ IMTBF=IMTBF\cdot \frac{N-1}{N}\,\! }[/math]

where:

  • [math]\displaystyle{ IMTBF\,\! }[/math] = instantaneous MTBF.
  • [math]\displaystyle{ N\,\! }[/math] = total number of failures.

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

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

Time Given Instantaneous Failure Intensity

Fisher Matrix Bounds

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 confidence bounds on Beta. 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

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

Grouped Data

This section presents the confidence bounds for the Crow-AMSAA model when using Grouped data.

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{ \hat{\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 =\hat{\beta },\lambda =\hat{\lambda }}^{-1}=\left[ \begin{matrix} Var(\hat{\lambda }) & Cov(\hat{\beta },\hat{\lambda }) \\ Cov(\hat{\beta },\hat{\lambda }) & Var(\hat{\beta }) \\ \end{matrix} \right]\,\! }[/math]

[math]\displaystyle{ \Lambda \,\! }[/math] is the natural log-likelihood function where [math]\displaystyle{ \ln^{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

The 2-sided confidence bounds on [math]\displaystyle{ \hat{\beta }\,\! }[/math] are given by first calculating:

[math]\displaystyle{ P\left( i \right)=\frac{{{T}_{i}}}{{{T}_{K}}};\text{ }i=1,2,...,K }[/math]

where:

  • [math]\displaystyle{ T_i\,\! }[/math] = interval end time for the [math]\displaystyle{ {{i}^{th}}\,\! }[/math] interval.
  • [math]\displaystyle{ K\,\! }[/math] = number of intervals.
  • [math]\displaystyle{ T_K\,\! }[/math] = end time for the last interval.

Next:

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

And:

[math]\displaystyle{ c=\frac{1}{\sqrt{A}} }[/math]

Then:

[math]\displaystyle{ S=\frac{\left( {{z}_{1-\tfrac{\alpha }{2}}} \right)\cdot c}{\sqrt{N}} }[/math]

where:

  • [math]\displaystyle{ {{z}_{1-\tfrac{\alpha }{2}}}\,\! }[/math] = inverse standard normal.
  • [math]\displaystyle{ N\,\! }[/math] = number of failures.

The 2-sided confidence bounds on [math]\displaystyle{ \beta\,\! }[/math] are then [math]\displaystyle{ \hat{\beta }\left( 1\pm S \right)\,\! }[/math].

Growth Rate (Grouped)

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

[math]\displaystyle{ \alpha_L=1-\beta_U\,\! }[/math]
[math]\displaystyle{ \alpha_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 confidence bounds on Beta.

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 confidence bounds on Beta.

Crow Bounds

Failure Terminated

For failure terminated data, the 2-sided [math]\displaystyle{ (1-\alpha )\,\! }[/math] 100% 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]

where:

  • [math]\displaystyle{ N\,\! }[/math] = total number of failures.
  • [math]\displaystyle{ T_K\,\! }[/math] = end time of last interval.

Time Terminated

For time terminated data, the 2-sided [math]\displaystyle{ (1-\alpha )\,\! }[/math] 100% 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]

where:

  • [math]\displaystyle{ N\,\! }[/math] = total number of failures.
  • [math]\displaystyle{ T_K\,\! }[/math] = end time of last interval.

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

[math]\displaystyle{ N{{(t)}_{L}}=\frac{t}{{\hat{\beta }}}IF{{I}_{L}}\,\! }[/math]
[math]\displaystyle{ N{{(t)}_{U}}=\frac{t}{{\hat{\beta }}}IF{{I}_{U}}\,\! }[/math]

where [math]\displaystyle{ IFI_L\,\! }[/math] and [math]\displaystyle{ IFI_U\,\! }[/math] are calculated based on the procedures for the confidence bounds on the instantaneous failure intensity.

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 confidence bounds on Beta. 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 2-sided confidence bounds on the cumulative failure intensity [math]\displaystyle{ (CFI\,\!) }[/math] are given below. Let:

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

Then:

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

Cumulative MTBF (Grouped)

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 confidence bounds on Beta. 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

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

[math]\displaystyle{ CMTB{{F}_{L}}=\frac{1}{CF{{I}_{U}}}\,\! }[/math]
[math]\displaystyle{ CMTB{{F}_{U}}=\frac{1}{CF{{I}_{L}}}\,\! }[/math]

where [math]\displaystyle{ CFI_{L}\,\! }[/math] and [math]\displaystyle{ CFI_{U}\,\! }[/math] are calculating using the process for calculating the confidence bounds on the cumulative failure intensity.

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 confidence bounds on Beta. 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

The 2-sided confidence bounds on instantaneous MTBF [math]\displaystyle{ (IMTBF)\,\! }[/math] are given by first calculating:

[math]\displaystyle{ P\left( i \right)=\frac{{{T}_{i}}}{{{T}_{K}}};\text{ }i=1,2,...,K }[/math]

where:

  • [math]\displaystyle{ T_i\,\! }[/math] = interval end time for the [math]\displaystyle{ {{i}^{th}}\,\! }[/math] interval.
  • [math]\displaystyle{ K\,\! }[/math] = number of intervals.
  • [math]\displaystyle{ T_K\,\! }[/math] = end time for the last interval.

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)}^{\hat{\beta }}}\ln P{{(i-1)}^{{\hat{\beta }}}} \right]}^{2}}}{\left[ P{{(i)}^{{\hat{\beta }}}}-P{{(i-1)}^{{\hat{\beta }}}} \right]}\,\! }[/math]

Next:

[math]\displaystyle{ D=\sqrt{\frac{1}{A}+1} }[/math]

And:

[math]\displaystyle{ W=\frac{\left( {{z}_{1-\tfrac{\alpha }{2}}} \right)\cdot D}{\sqrt{N}} }[/math]

where:

  • [math]\displaystyle{ {{z}_{1-\tfrac{\alpha }{2}}}\,\! }[/math] = inverse standard normal.
  • [math]\displaystyle{ N\,\! }[/math] = number of failures.

The 2-sided confidence bounds on instantaneous MTBF are then [math]\displaystyle{ IMTBF\left( 1\pm W \right)\,\! }[/math].

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

[math]\displaystyle{ \begin{align} IF{{I}_{U}}= & \frac{1}{IMTB{{F}_{L}}} \\ IF{{I}_{L}}= & \frac{1}{IMTB{{F}_{U}}} \end{align}\,\! }[/math]

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

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 confidence bounds on Beta. 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

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

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

Then estimate the number of failures:

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

The confidence bounds on time given the cumulative failure intensity are then given by:

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

Time Given Cumulative MTBF (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 confidence bounds on Beta. 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

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

Time Given Instantaneous MTBF (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 confidence bounds on Beta. 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

Failure Terminated

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

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

Time Terminated

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

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

Time Given Instantaneous Failure Intensity (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 confidence bounds on Beta. 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

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