Crow-AMSAA Confidence Bounds: Difference between revisions

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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.''
''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 HEADER IS LINKED FROM SEVERAL SECTIONS IN THIS PAGE. IF YOU RENAME THE SECTION, YOU MUST UPDATE THE LINK(S). -->
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.
===Beta===<!-- THIS SECTION HEADER IS LINKED FROM SEVERAL SECTIONS IN THIS PAGE. IF YOU RENAME THE SECTION, YOU MUST UPDATE THE LINK(S). -->
====Fisher Matrix Bounds====<!-- THIS SECTION HEADER IS LINKED TO: Crow-AMSAA (NHPP). IF YOU RENAME THE SECTION, YOU MUST UPDATE THE LINK. -->
====Fisher Matrix Bounds====<!-- THIS SECTION HEADER IS LINKED TO: Crow-AMSAA (NHPP). IF YOU RENAME THE SECTION, YOU MUST UPDATE THE LINK. -->
The parameter <math>\beta \,\!</math> must be positive, thus <math>\ln \beta \,\!</math> is treated as being normally distributed as well.   
The parameter <math>\beta \,\!</math> must be positive, thus <math>\ln \beta \,\!</math> is treated as being normally distributed as well.   
Line 15: Line 16:
:<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}
Line 27: Line 28:
<math>\Lambda \,\!</math> is the natural log-likelihood function:  
<math>\Lambda \,\!</math> is the natural log-likelihood function:  


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


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


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


:<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% 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}
Line 56: Line 55:
\end{align}\,\!</math>
\end{align}\,\!</math>


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


For the 2-sided <math>(1-\alpha )\,\!</math> 100% 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}
Line 72: Line 71:
\end{align}\,\!</math>
\end{align}\,\!</math>


===Bounds on Lambda===
===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]].
 
===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:  
Line 86: Line 94:
:<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#Bounds_on_Beta|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:
For the 2-sided <math>(1-\alpha )\,\!</math> 100% confidence interval, the confidence bounds on <math>\lambda \,\!</math> are:
Line 95: Line 103:
:<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{{T}^{{\hat{\beta }}}}}   
\end{align}\,\!</math>
\end{align}\,\!</math>


The fractiles can be found in the tables of the <math>{{\chi }^{2}}\,\!</math> distribution.
where:
*<math>N\,\!</math> = total number of failures.
*<math>T\,\!</math> = termination time.


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


For the 2-sided <math>(1-\alpha )\,\!</math> 100% 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:
Line 106: Line 116:
:<math>\begin{align}
:<math>\begin{align}
   {{\lambda }_{L}}= & \frac{\chi _{\tfrac{\alpha }{2},2N}^{2}}{2{{T}^{{\hat{\beta }}}}} \\  
   {{\lambda }_{L}}= & \frac{\chi _{\tfrac{\alpha }{2},2N}^{2}}{2{{T}^{{\hat{\beta }}}}} \\  
   {{\lambda }_{U}}= & \frac{\chi _{1-\tfrac{\alpha }{2},2N}^{2}}{2{{T}^{{\hat{\beta }}}}}   
   {{\lambda }_{U}}= & \frac{\chi _{1-\tfrac{\alpha }{2},2N+2}^{2}}{2{{T}^{{\hat{\beta }}}}}   
\end{align}\,\!</math>
\end{align}\,\!</math>


===Bounds on Growth Rate===
where:
Since the growth rate is equal to <math>1-\beta \,\!</math>, the confidence bounds for both the Fisher Matrix and Crow methods are:
*<math>N\,\!</math> = total number of failures.
<br>
*<math>T\,\!</math> = termination time.
:<math>G\text{row}th Rate_L=1-\beta_U\,\!</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. 
===Cumulative Number of Failures===
 
===Bounds on Cumulative MTBF===
====Fisher Matrix Bounds====
====Fisher Matrix Bounds====
The cumulative MTBF, <math>{{m}_{c}}(t)\,\!</math>, must be positive, thus <math>\ln {{m}_{c}}(t)\,\!</math> is treated as being normally distributed as well.
The cumulative 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{m}}}_{c}}(t)-\ln {{m}_{c}}(t)}{\sqrt{Var(\ln {{{\hat{m}}}_{c}}(t)})}\ \tilde{\ }\ N(0,1)\,\!</math>
:<math>\frac{\ln \hat{N}(t)-\ln N(t)}{\sqrt{Var(\ln \hat{N}(t)})}\ \tilde{\ }\ N(0,1)\,\!</math>


The approximate confidence bounds on the cumulative MTBF are then estimated from:
:<math>N(t)=\hat{N}(t){{e}^{\pm {{z}_{\alpha }}\sqrt{Var(\hat{N}(t))}/\hat{N}(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{N}(t)=\hat{\lambda }{{t}^{{\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{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 {{m}_{c}}(t)}{\partial \beta } \right)\left( \frac{\partial {{m}_{c}}(t)}{\partial \lambda } \right)cov(\hat{\beta },\hat{\lambda })\,  
   & +2\left( \frac{\partial \hat{N}(t)}{\partial \beta } \right)\left( \frac{\partial \hat{N}(t)}{\partial \lambda } \right)cov(\hat{\beta },\hat{\lambda })   
\end{align}\,\!</math>
\end{align}\,\!</math>


The variance calculation is the same as given above in the [[Crow-AMSAA_Confidence_Bounds#Bounds_on_Beta|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 \hat{N}(t)}{\partial \beta }= & \hat{\lambda }{{t}^{{\hat{\beta }}}}\ln t \\  
   \frac{\partial {{m}_{c}}(t)}{\partial \lambda }= & -\frac{1}{{{{\hat{\lambda }}}^{2}}}{{t}^{1-\hat{\beta }}}   
   \frac{\partial \hat{N}(t)}{\partial \lambda }= & {{t}^{{\hat{\beta }}}}   
\end{align}\,\!</math>
\end{align}\,\!</math>


====Crow Bounds====
====Crow Bounds====
To calculate the Crow confidence bounds on cumulative MTBF, first calculate the Crow cumulative failure intensity confidence bounds:
The Crow cumulative number of failure confidence bounds are:  
 
:<math>C{{(t)}_{L}}=\frac{\chi _{\tfrac{\alpha }{2},2N}^{2}}{2\cdot t}\,\!</math>
 
'''Time Terminated Data'''
:<math>C{{(t)}_{U}}=\frac{\chi _{1-\tfrac{\alpha }{2},2N+1}^{2}}{2\cdot t}\,\!</math>
 
'''Failure Terminated Data'''
:<math>C{{(t)}_{U}}=\frac{\chi _{1-\tfrac{\alpha }{2},2N}^{2}}{2\cdot t}\,\!</math>
 
Then:


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


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


where:  
where:  


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


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


The variance calculation is the same as given above in the [[Crow-AMSAA_Confidence_Bounds#Bounds_on_Beta|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 {{\lambda }_{c}}(t)}{\partial \beta }= & \hat{\lambda }{{t}^{\hat{\beta }-1}}\ln t \\  
   \frac{\partial {{m}_{i}}(t)}{\partial \lambda }= & -\frac{1}{{{{\hat{\lambda }}}^{2}}\hat{\beta }}{{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====
'''Failure Terminated Data'''
The Crow bounds on the cumulative failure intensity <math>(CFI)\,\!</math> are given below. Let:


Consider the following equation:  
:<math>N=\hat{\lambda }{{t}^{{\hat{\beta }}}}\,\!</math>


:<math>G(\mu |n)=\mathop{}_{0}^{\infty }\frac{{{e}^{-x}}{{x}^{n-2}}}{(n-2)!}\underset{i=0}{\overset{n-1}{\mathop \sum }}\,\frac{1}{i!}{{\left( \frac{\mu }{x} \right)}^{i}}\exp (-\frac{\mu }{x})\,dx\,\!</math>
'''Failure Terminated'''
 
:<math>\begin{align}
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:
  CFI{_{L}}= & \frac{\chi _{\tfrac{\alpha }{2},2N}^{2}}{2\cdot t} \\  
\end{align}\,\!</math>


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


where <math>MTB{{F}_{i}}=\tfrac{1}{\hat{\lambda }\hat{\beta }{{t}^{\hat{\beta }-1}}}\,\!</math>. If using the unbiased parameters, <math>\bar{\beta }\,\!</math> and <math>\bar{\lambda }\,\!</math>, then the upper and lower confidence bounds are:
'''Time Terminated'''


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


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


'''Time Terminated Data'''
:<math>\frac{\ln {{{\hat{m}}}_{c}}(t)-\ln {{m}_{c}}(t)}{\sqrt{Var(\ln {{{\hat{m}}}_{c}}(t)})}\ \tilde{\ }\ N(0,1)\,\!</math>


Consider the following equation where <math>{{I}_{1}}(.)\,\!</math> is the modified Bessel function of order one:
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>
 
where:
 
:<math>{{\hat{m}}_{c}}(t)=\frac{1}{{\hat{\lambda }}}{{t}^{1-\hat{\beta }}}\,\!</math>


:<math>H(x|k)=\underset{j=1}{\overset{k}{\mathop \sum }}\,\frac{{{x}^{2j-1}}}{{{2}^{2j-1}}(j-1)!j!{{I}_{1}}(x)}\,\!</math>
:<math>\begin{align}
  Var({{{\hat{m}}}_{c}}(t))= & {{\left( \frac{\partial {{m}_{c}}(t)}{\partial \beta } \right)}^{2}}Var(\hat{\beta })+{{\left( \frac{\partial {{m}_{c}}(t)}{\partial \lambda } \right)}^{2}}Var(\hat{\lambda }) \\
  & +2\left( \frac{\partial {{m}_{c}}(t)}{\partial \beta } \right)\left( \frac{\partial {{m}_{c}}(t)}{\partial \lambda } \right)cov(\hat{\beta },\hat{\lambda })\, 
\end{align}\,\!</math>


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


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


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


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


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


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


where:  
where:  


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


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


The variance calculation is the same as given above in the [[Crow-AMSAA_Confidence_Bounds#Bounds_on_Beta|Bounds on Beta]] section and:  
====Crow Bounds====
'''Failure Terminated'''
 
For failure terminated data and the 2-sided confidence bounds on instantaneous MTBF <math>(IMTBF)\,\!</math>, consider the following equation:
 
:<math>G(\mu |n)=\mathop{}_{0}^{\infty }\frac{{{e}^{-x}}{{x}^{n-2}}}{(n-2)!}\underset{i=0}{\overset{n-1}{\mathop \sum }}\,\frac{1}{i!}{{\left( \frac{\mu }{x} \right)}^{i}}\exp (-\frac{\mu }{x})\,dx\,\!</math>
 
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.
 
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}
  \frac{\partial {{\lambda }_{c}}(t)}{\partial \beta }= & \hat{\lambda }{{t}^{\hat{\beta }-1}}\ln t \\  
  {{IMTBF}_{L}}= & IMTBF\cdot {{p}_{1}} \\  
   \frac{\partial {{\lambda }_{c}}(t)}{\partial \lambda }= & {{t}^{\hat{\beta }-1}}   
   {{IMTBF}_{U}}= & IMTBF\cdot {{p}_{2}}   
\end{align}\,\!</math>
\end{align}\,\!</math>


====Crow Bounds====
where <math>IMTBF=\tfrac{1}{\hat{\lambda }\hat{\beta }{{t}^{\hat{\beta }-1}}}\,\!</math>.
The Crow lower cumulative failure intensity confidence bound is given by:
 
If using the unbiased parameters, <math>\bar{\beta }\,\!</math> and <math>\bar{\lambda }\,\!</math>, then the upper and lower confidence bounds are:


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


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


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


'''Failure Terminated'''
where <math>IMTBF=\tfrac{1}{\hat{\lambda }\hat{\beta }{{t}^{\hat{\beta }-1}}}\,\!</math>.
 
If using the unbiased parameters, <math>\bar{\beta }\,\!</math> and <math>\bar{\lambda }\,\!</math>, then the upper and lower confidence bounds are:


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


===Bounds on Instantaneous Failure Intensity===
where <math>IMTBF=\tfrac{1}{\bar{\lambda }\bar{\beta }{{t}^{\bar{\beta }-1}}}\,\!</math>.
 
===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.  
Line 299: Line 341:
\end{align}\,\!</math>
\end{align}\,\!</math>


The variance calculation is the same as given above in the [[Crow-AMSAA_Confidence_Bounds#Bounds_on_Beta|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}
Line 307: Line 349:


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


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


===Bounds on Time Given Cumulative Failure Intensity===
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]].
 
===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:  
Line 331: Line 375:
\end{align}\,\!</math>
\end{align}\,\!</math>


The variance calculation is the same as given above in the [[Crow-AMSAA_Confidence_Bounds#Bounds_on_Beta|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}
Line 339: Line 383:


====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.  
Line 371: Line 415:
\end{align}\,\!</math>
\end{align}\,\!</math>


The variance calculation is the same as given above in the [[Crow-AMSAA_Confidence_Bounds#Bounds_on_Beta|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>
Line 381: Line 425:


====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 the [[Crow-AMSAA_Confidence_Bounds#Bounds_on_Time_Given_Instantaneous_Failure_Intensity|Bounds on Time Given Instantaneous Failure Intensity]] section to calculate the bounds.


===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.  
Line 402: Line 444:
\end{align}\,\!</math>
\end{align}\,\!</math>


The variance calculation is the same as given above in the [[Crow-AMSAA_Confidence_Bounds#Bounds_on_Beta|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>
Line 412: Line 454:


====Crow Bounds====
====Crow Bounds====
Step 1: Calculate the confidence bounds on the instantaneous MTBF as presented in the [[Crow-AMSAA_Confidence_Bounds#Bounds_on_Instantaneous_MTBF|Bounds on Instantaneous MTBF]] section.
'''Failure Terminated'''
Step 2: Calculate the bounds on time as follows.
 
If the unbiased value <math>\bar{\beta }\,\!</math> is used then:
 
:<math>IMTBF=IMTBF\cdot \frac{N-2}{N}\,\!</math>
 
where:
*<math>IMTBF\,\!</math> = instantaneous MTBF.
*<math>N\,\!</math> = total number of failures.


'''Failure Terminated Data'''
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}={{(\frac{\lambda \beta \cdot MTB{{F}_{i}}}{c})}^{1/(1-\beta )}}\,\!</math>
:<math>{{\hat{t}}_{L}}={{\left( \frac{\lambda \beta \cdot IMTBF}{{{p}_{1}}} \right)}^{\tfrac{1}{1-\beta }}}</math>


So the lower an upper bounds on time are:
:<math>{{\hat{t}}_{U}}={{\left( \frac{\lambda \beta \cdot IMTBF}{{{p}_{2}}} \right)}^{\tfrac{1}{1-\beta }}}</math>


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


:<math>{{\hat{T}}_{U}}={{(\frac{\lambda \beta \cdot MTB{{F}_{i}}}{{{c}_{2}}})}^{1/(1-\beta )}}\,\!</math>
If the unbiased value <math>\bar{\beta }\,\!</math> is used then:


'''Time Terminated Data'''
:<math>IMTBF=IMTBF\cdot \frac{N-1}{N}\,\!</math>


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


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


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


===Bounds on 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). -->
===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.  
Line 452: Line 503:
\end{align}\,\!</math>
\end{align}\,\!</math>


The variance calculation is the same as given above in the [[Crow-AMSAA_Confidence_Bounds#Bounds_on_Beta|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>
Line 462: Line 513:


====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 the [[Crow-AMSAA_Confidence_Bounds#Bounds_on_Time_Given_Instantaneous_MTBF|Bounds on Time Given Instantaneous MTBF]] section to calculate the bounds on time given the instantaneous failure intensity.
 
===Bounds on Cumulative Number of Failures===
====Fisher Matrix Bounds====
The cumulative number of failures, <math>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|Bounds on Beta]] 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 cumulative number of failure confidence bounds are:
 
:<math>\begin{align}
  {{N}_{L}}(T)= & \frac{T}{{\hat{\beta }}}{{\lambda }_{i}}{{(T)}_{L}} \\
  {{N}_{U}}(T)= & \frac{T}{{\hat{\beta }}}{{\lambda }_{i}}{{(T)}_{U}} 
\end{align}\,\!</math>
 
where <math>{{\lambda }_{i}}{{(T)}_{L}}\,\!</math> and <math>{{\lambda }_{i}}{{(T)}_{U}}\,\!</math> can be obtained from the [[Crow-AMSAA_Confidence_Bounds#Bounds_on_Instantaneous_Failure_Intensity|Crow instantaneous failure intensity confidence bounds equations]] given above.


==Grouped Data==
==Grouped Data==
====Bounds on Beta (Grouped)====<!-- THIS SECTION HEADER IS LINKED TO: Crow-AMSAA (NHPP) and to several sections in this page. IF YOU RENAME THE SECTION, YOU MUST UPDATE THE LINK. -->
This section presents the confidence bounds for the Crow-AMSAA model when using Grouped data.
====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=====
=====Fisher Matrix Bounds=====
The parameter <math>\beta \,\!</math> must be positive, thus <math>\ln \beta \,\!</math> is treated as being normally distributed as well.   
The parameter <math>\beta \,\!</math> must be positive, thus <math>\ln \beta \,\!</math> is treated as being normally distributed as well.   
Line 512: Line 529:
:<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>\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>.


All variance can be calculated using the Fisher Matrix:  
All variance can be calculated using the Fisher matrix:  


:<math>\left[ \begin{matrix}
:<math>\left[ \begin{matrix}
Line 522: Line 539:
\end{matrix} \right]\,\!</math>
\end{matrix} \right]\,\!</math>


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


:<math>\Lambda =\underset{i=1}{\overset{k}{\mathop \sum }}\,\left[ {{n}_{i}}\ln (\lambda T_{i}^{\beta }-\lambda T_{i-1}^{\beta })-(\lambda T_{i}^{\beta }-\lambda T_{i-1}^{\beta })-\ln {{n}_{i}}! \right]\,\!</math>
:<math>\Lambda =\underset{i=1}{\overset{k}{\mathop \sum }}\,\left[ {{n}_{i}}\ln (\lambda T_{i}^{\beta }-\lambda T_{i-1}^{\beta })-(\lambda T_{i}^{\beta }-\lambda T_{i-1}^{\beta })-\ln {{n}_{i}}! \right]\,\!</math>
Line 536: Line 553:


=====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 <math>\hat{\beta }\,\!</math> are given by first calculating:
Step 2: Calculate:  
 
:<math>P\left( i \right)=\frac{{{T}_{i}}}{{{T}_{K}}};\text{ }i=1,2,...,K</math>
 
where:
 
*<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.
 
Next:  


:<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>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>


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% confidence interval on <math>\hat{\beta }\,\!</math> is:
And:
 
:<math>c=\frac{1}{\sqrt{A}}</math>  
 
Then:
 
:<math>S=\frac{\left( {{z}_{1-\tfrac{\alpha }{2}}} \right)\cdot c}{\sqrt{N}}</math>
 
where:
 
*<math>{{z}_{1-\tfrac{\alpha }{2}}}\,\!</math> = inverse standard normal.
*<math>N\,\!</math> = number of failures.
 
The 2-sided confidence bounds on <math>\beta\,\!</math> are then <math>\hat{\beta }\left( 1\pm S \right)\,\!</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>
 
:<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:  
Line 557: Line 604:
:<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====
'''Time Terminated Data'''
'''Failure Terminated'''
For the 2-sided <math>(1-\alpha )\,\!</math> 100% confidence interval, the confidence bounds on <math>\lambda \,\!</math> are:
 
For failure 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 }_{U}}= & \frac{\chi _{1-\tfrac{\alpha }{2},2N}^{2}}{2\cdot T_{k}^{\beta }}
\end{align}\,\!</math>
 
where:
*<math>N\,\!</math> = total number of failures.
*<math>T_K\,\!</math> = end time of last interval.
 
'''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}
:<math>\begin{align}
Line 568: Line 629:
\end{align}\,\!</math>
\end{align}\,\!</math>


'''Failure Terminated Data'''
where:
For the 2-sided <math>(1-\alpha )\,\!</math> 100% confidence interval, the confidence bounds on <math>\lambda \,\!</math> are:
*<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>
 
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}
   {{\lambda }_{L}}= & \frac{\chi _{\tfrac{\alpha }{2},2N}^{2}}{2\cdot T_{k}^{\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\cdot T_{k}^{\beta }}   
   \frac{\partial \hat{N}(t)}{\partial \lambda }= & {{t}^{{\hat{\beta }}}}   
\end{align}\,\!</math>
\end{align}\,\!</math>


===Bounds on Growth Rate (Grouped)===
====Crow Bounds====
The 2-sided confidence bounds on the cumulative number of failures are given by:
 
:<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]].
 
===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.
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.  
Line 606: Line 725:
\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}
Line 614: Line 733:


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


Then:
:<math>CMTB{{F}_{U}}=\frac{1}{CF{{I}_{L}}}\,\!</math>


:<math>\begin{align}
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]].
  {{[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)===
===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.  
Line 646: Line 760:
\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}
Line 654: Line 768:


====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)}^{\hat{\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% confidence interval on <math>{{\hat{m}}_{i}}(t)\,\!</math> is:


:<math>MTB{{F}_{i}}={{\hat{m}}_{i}}(1\pm W)\,\!</math>
:<math>P\left( i \right)=\frac{{{T}_{i}}}{{{T}_{K}}};\text{ }i=1,2,...,K</math>


===Bounds on Cumulative Failure Intensity (Grouped)===
where:
====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>
*<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 on the cumulative failure intensity are then estimated from:  
Calculate:  


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


where:  
Next:


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


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


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.  
Line 716: Line 814:
\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}
Line 724: Line 822:


====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.  
Line 748: Line 848:
\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}
  \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>\hat{T}={{(\lambda \cdot {{m}_{c}})}^{1/(1-\beta )}}\,\!</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>
 
Then estimate the number of failures:
 
:<math>N=\hat{\lambda }{{\hat{T}}^{{\hat{\beta }}}}\,\!</math>
 
The confidence bounds on time given the cumulative failure intensity are then given by:


:<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}}} \\  
   {{t}_{L}}= & \frac{\chi _{\tfrac{\alpha }{2},2N}^{2}}{2\cdot {CFI}} \\  
   \frac{\partial T}{\partial \lambda }= & \frac{{{(\lambda \cdot {{m}_{c}})}^{1/(1-\beta )}}}{\lambda (1-\beta )}   
   {{t}_{U}}= & \frac{\chi _{1-\tfrac{\alpha }{2},2N+2}^{2}}{2\cdot {CFI}}   
\end{align}\,\!</math>
\end{align}\,\!</math>


====Crow Bounds====
===Time Given Cumulative MTBF (Grouped)===
Step 1: Calculate <math>{{\lambda }_{c}}(T)=\tfrac{1}{MTB{{F}_{c}}}\,\!</math>.
Step 2: Use equations the [[Crow-AMSAA_Confidence_Bounds#Bounds_on_Time_Given_Cumulative_Failure_Intensity_.28Grouped.29|Bounds on Time Given Cumulative Failure Intensity]] section to calculate the bounds.
 
===Bounds on 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). -->
====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.  
Line 778: Line 888:
\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}}={{\hat{m}}_{i}}(1\pm W)\,\!</math>
===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). -->
 
Step 2: Use equations in the [[Crow-AMSAA_Confidence_Bounds#Bounds_on_Time_Given_Instantaneous_MTBF_.28Grouped.29|Bounds on Time Given Instantaneous MTBF]] section to calculate the time given the instantaneous MTBF.
 
===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.  
Line 811: Line 917:
\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>\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 }= & {{\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 }= & {{\left( \frac{{{\lambda }_{c}}(T)}{\lambda } \right)}^{1/(\beta -1)}}\frac{1}{\lambda (1-\beta )}   
   \frac{\partial T}{\partial \lambda }= & \frac{{{(\lambda \beta \cdot \text{ }{{m}_{i}}(T))}^{1/(1-\beta )}}}{\lambda (1-\beta )}   
\end{align}\,\!</math>
\end{align}\,\!</math>


====Crow Bounds====
====Crow Bounds====
Step 1: Calculate:
'''Failure Terminated'''
 
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:
 
:<math>{{\hat{t}}_{L}}={{\left( \frac{\lambda \beta \cdot IMTBF}{{{p}_{1}}} \right)}^{\tfrac{1}{1-\beta }}}</math>


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


Step 2: Estimate the number of failures:
'''Time Terminated'''


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


Step 3: Obtain the confidence bounds on time given the cumulative failure intensity by solving for <math>{{t}_{l}}\,\!</math> and <math>{{t}_{u}}\,\!</math> in the following equations:
:<math>{{\hat{t}}_{L}}={{\left( \frac{\lambda \beta \cdot IMTBF}{{{\Pi }_{1}}} \right)}^{\tfrac{1}{1-\beta }}}\,\!</math>


:<math>\begin{align}
:<math>{{\hat{t}}_{U}}={{\left( \frac{\lambda \beta \cdot IMTBF}{{{\Pi }_{2}}} \right)}^{\tfrac{1}{1-\beta }}}\,\!</math>
  {{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)===
===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.  
Line 851: Line 960:
\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>
Line 861: Line 970:


====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 the [[Crow-AMSAA_Confidence_Bounds#Bounds_on_Time_Given_Instantaneous_MTBF_.28Grouped.29|Bounds on Time Given Instantaneous MTBF]] section 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].