Crow-AMSAA Confidence Bounds: Difference between revisions

From ReliaWiki
Jump to navigation Jump to search
No edit summary
 
(26 intermediate revisions by 2 users not shown)
Line 1: Line 1:
{{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==
Line 16: 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 39: Line 39:


====Crow Bounds====
====Crow Bounds====
'''Failure 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:
Line 55: Line 55:
\end{align}\,\!</math>
\end{align}\,\!</math>


'''Time 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:
Line 72: Line 72:


===Growth Rate===
===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:
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>
<br>


Line 78: Line 78:
:<math>\alpha_U=1-\beta_L\,\!</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]] section.
<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===
===Lambda===
Line 94: 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 confidence bounds on [[Crow-AMSAA_Confidence_Bounds#Beta|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====
'''Failure 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 110: Line 110:
*<math>T\,\!</math> = termination time.
*<math>T\,\!</math> = termination time.


'''Time 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 140: Line 140:
\end{align}\,\!</math>
\end{align}\,\!</math>


The variance calculation is the same as given above in the confidence bounds on [[Crow-AMSAA_Confidence_Bounds#Beta|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 155: Line 155:
\end{align}\,\!</math>
\end{align}\,\!</math>


where <math>IFI{{(t)}_{L}}\,\!</math> and <math>IFI{{(t)}_{U}}\,\!</math> are calculated using the process for the confidence bounds on [[Crow-AMSAA_Confidence_Bounds#Instantaneous_Failure_Intensity|instantaneous failure intensity]].
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===
===Cumulative Failure Intensity===
Line 178: Line 178:
\end{align}\,\!</math>
\end{align}\,\!</math>


The variance calculation is the same as given above in the confidence bounds on [[Crow-AMSAA_Confidence_Bounds#Beta|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 225: Line 225:
\end{align}\,\!</math>
\end{align}\,\!</math>


The variance calculation is the same as given above in the confidence bounds on [[Crow-AMSAA_Confidence_Bounds#Beta|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 240: Line 240:
\end{align}\,\!</math>
\end{align}\,\!</math>


where <math>CFI_L\,\!</math> and <math>CFI_U\,\!</math> are calculated using the process for the confidence bounds on [[Crow-AMSAA_Confidence_Bounds#Cumulative_Failure_Intensity|cumulative failure intensity]].
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]].


===Instantaneous MTBF===
===Instantaneous MTBF===
Line 261: Line 261:
\end{align}\,\!</math>
\end{align}\,\!</math>


The variance calculation is the same as given above in the confidence bounds on [[Crow-AMSAA_Confidence_Bounds#Beta|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 269: Line 269:


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


For failure terminated data and the 2-sided confidence bounds on instantaneous MTBF <math>(IMTBF)\,\!</math>, consider the following equation:  
For failure terminated data and the 2-sided confidence bounds on instantaneous MTBF <math>(IMTBF)\,\!</math>, consider the following equation:  
Line 290: Line 290:


:<math>\begin{align}
:<math>\begin{align}
   {{IMTBF}_{L}}= & MTB{{F}_{i}}\cdot \left( \frac{N-2}{N} \right)\cdot {{p}_{1}} \\  
   {{IMTBF}_{L}}= & IMTBF\cdot \left( \frac{N-2}{N} \right)\cdot {{p}_{1}} \\  
   {{IMTBF}_{U}}= & MTB{{F}_{i}}\cdot \left( \frac{N-2}{N} \right)\cdot {{p}_{2}}   
   {{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>.
where <math>IMTBF=\tfrac{1}{\bar{\lambda }\bar{\beta }{{t}^{\bar{\beta }-1}}}\,\!</math>.


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


Consider the following equation where <math>{{I}_{1}}(.)\,\!</math> is the modified Bessel function of order one:  
Consider the following equation where <math>{{I}_{1}}(.)\,\!</math> is the modified Bessel function of order one:  
Line 306: Line 306:
If using the biased parameters, <math>\hat{\beta }\,\!</math> and <math>\hat{\lambda }\,\!</math>, then the upper and lower confidence bounds are:
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}
   {{IMTBF}_{L}}= & IMTBF\cdot {{\Pi }_{1}} \\  
   {{IMTBF}_{L}}= & IMTBF\cdot {{\Pi }_{1}} \\  
   {{IMTBF}_{U}}= & IMTBF\cdot {{\Pi }_{2}}   
   {{IMTBF}_{U}}= & IMTBF\cdot {{\Pi }_{2}}   
Line 341: Line 341:
\end{align}\,\!</math>
\end{align}\,\!</math>


The variance calculation is the same as given above in the confidence bounds on [[Crow-AMSAA_Confidence_Bounds#Beta|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 356: Line 356:
\end{align}\,\!</math>
\end{align}\,\!</math>


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


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


The variance calculation is the same as given above in the confidence bounds on [[Crow-AMSAA_Confidence_Bounds#Beta|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 387: Line 387:
:<math>\hat{t}={{\left( \frac{CFI}{{\hat{\lambda }}} \right)}^{\tfrac{1}{\hat{\beta }-1}}}\,\!</math>
:<math>\hat{t}={{\left( \frac{CFI}{{\hat{\lambda }}} \right)}^{\tfrac{1}{\hat{\beta }-1}}}\,\!</math>


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


:<math>N=\hat{\lambda }{{\hat{t}}^{{\hat{\beta }}}}\,\!</math>
:<math>N=\hat{\lambda }{{\hat{t}}^{{\hat{\beta }}}}\,\!</math>
Line 415: Line 415:
\end{align}\,\!</math>
\end{align}\,\!</math>


The variance calculation is the same as given above in the confidence bounds on [[Crow-AMSAA_Confidence_Bounds#Beta|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 425: Line 425:


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


===Time Given Instantaneous MTBF===
===Time Given Instantaneous MTBF===
Line 444: Line 444:
\end{align}\,\!</math>
\end{align}\,\!</math>


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


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


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


:<math>IMTBF=IMTBF\cdot \frac{N-1}{N}\,\!</math>
:<math>IMTBF=IMTBF\cdot \frac{N-2}{N}\,\!</math>


where:
where:
Line 464: Line 464:
*<math>N\,\!</math> = total number of failures.
*<math>N\,\!</math> = total number of failures.


Calculate the constants <math>p_1\,\!</math> and <math>p_2\,\!</math> using procedures described for the confidence bounds on [[Crow-AMSAA_Confidence_Bounds#Instantaneous_MTBF|instantaneous MTBF]]. The lower and upper confidence bounds on time are then given by:
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}}_{L}}={{\left( \frac{\lambda \beta \cdot IMTBF}{{{p}_{1}}} \right)}^{\tfrac{1}{1-\beta }}}</math>
:<math>{{\hat{t}}_{L}}={{\left( \frac{\lambda \beta \cdot IMTBF}{{{p}_{1}}} \right)}^{\tfrac{1}{1-\beta }}}</math>
Line 470: Line 470:
:<math>{{\hat{t}}_{U}}={{\left( \frac{\lambda \beta \cdot IMTBF}{{{p}_{2}}} \right)}^{\tfrac{1}{1-\beta }}}</math>
:<math>{{\hat{t}}_{U}}={{\left( \frac{\lambda \beta \cdot IMTBF}{{{p}_{2}}} \right)}^{\tfrac{1}{1-\beta }}}</math>


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


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


:<math>IMTBF=IMTBF\cdot \frac{N-2}{N}\,\!</math>
:<math>IMTBF=IMTBF\cdot \frac{N-1}{N}\,\!</math>


where:
where:
Line 480: Line 480:
*<math>N\,\!</math> = total number of failures.
*<math>N\,\!</math> = total number of failures.


Calculate the constants <math>{{\Pi }_{1}}\,\!</math> and <math>{{\Pi }_{2}}\,\!</math> using procedures described for the confidence bounds on [[Crow-AMSAA_Confidence_Bounds#Instantaneous_MTBF|instantaneous MTBF]]. The lower and upper confidence bounds on time are then given by:
Calculate the constants <math>{{\Pi }_{1}}\,\!</math> and <math>{{\Pi }_{2}}\,\!</math> using procedures described for the confidence bounds on [[Crow-AMSAA_Confidence_Bounds#Crow_Bounds_6|instantaneous MTBF]]. The lower and upper confidence bounds on time are then given by:


:<math>{{\hat{t}}_{L}}={{\left( \frac{\lambda \beta \cdot IMTBF}{{{\Pi }_{1}}} \right)}^{\tfrac{1}{1-\beta }}}\,\!</math>
:<math>{{\hat{t}}_{L}}={{\left( \frac{\lambda \beta \cdot IMTBF}{{{\Pi }_{1}}} \right)}^{\tfrac{1}{1-\beta }}}\,\!</math>
Line 503: Line 503:
\end{align}\,\!</math>
\end{align}\,\!</math>


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


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


==Grouped Data==
==Grouped Data==
Line 529: 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 539: 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 583: Line 583:


===Growth Rate (Grouped)===
===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:
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>
<br>


Line 589: Line 589:
:<math>\alpha_U=1-\beta_L\,\!</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_.28Grouped.29|Beta]] section.
<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)===
===Lambda (Grouped)===
====Fisher Matrix Bounds====
====Fisher Matrix Bounds====
Line 604: 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 confidence bounds on [[Crow-AMSAA_Confidence_Bounds#Beta_.28Grouped.29|Beta]] section.
The variance calculation is the same as given above in the confidence bounds on [[Crow-AMSAA_Confidence_Bounds#Beta_.28Grouped.29|Beta]].


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


For failure terminated data, 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:
Line 620: Line 620:
*<math>T_K\,\!</math> = end time of last interval.
*<math>T_K\,\!</math> = end time of last interval.


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


For time terminated data, the 2-sided <math>(1-\alpha )\,\!</math> 100% confidence interval, the confidence bounds on <math>\lambda \,\!</math> are:
For time terminated data, the 2-sided <math>(1-\alpha )\,\!</math> 100% confidence interval, the confidence bounds on <math>\lambda \,\!</math> are:
Line 650: Line 650:
\end{align}\,\!</math>
\end{align}\,\!</math>


The variance calculation is the same as given above in the confidence bounds on [[Crow-AMSAA_Confidence_Bounds#Beta_.28Grouped.29|Beta]] 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 664: Line 664:
:<math>N{{(t)}_{U}}=\frac{t}{{\hat{\beta }}}IF{{I}_{U}}\,\!</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#Instantaneous_Failure_Intensity_.28Grouped.29|instantaneous failure intensity]].
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)===
===Cumulative Failure Intensity (Grouped)===
Line 687: Line 687:
\end{align}\,\!</math>
\end{align}\,\!</math>


The variance calculation is the same as given above in the confidence bounds on [[Crow-AMSAA_Confidence_Bounds#Beta_.28Grouped.29|Beta]] 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 725: Line 725:
\end{align}\,\!</math>
\end{align}\,\!</math>


The variance calculation is the same as given above in the confidence bounds on [[Crow-AMSAA_Confidence_Bounds#Beta_.28Grouped.29|Beta]] 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 739: Line 739:
:<math>CMTB{{F}_{U}}=\frac{1}{CF{{I}_{L}}}\,\!</math>
:<math>CMTB{{F}_{U}}=\frac{1}{CF{{I}_{L}}}\,\!</math>


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


===Instantaneous MTBF (Grouped)===
===Instantaneous MTBF (Grouped)===
Line 760: Line 760:
\end{align}\,\!</math>
\end{align}\,\!</math>


The variance calculation is the same as given above in the confidence bounds on [[Crow-AMSAA_Confidence_Bounds#Beta_.28Grouped.29|Beta]] 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 814: Line 814:
\end{align}\,\!</math>
\end{align}\,\!</math>


The variance calculation is the same as given above in the confidence bounds on [[Crow-AMSAA_Confidence_Bounds#Beta_.28Grouped.29|Beta]] 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 829: Line 829:
\end{align}\,\!</math>
\end{align}\,\!</math>
   
   
where <math>IMTB{{F}_{L}}\,\!</math>and <math>IMTB{{F}_{U}}\,\!</math> are calculated using the process for the confidence bounds on the [[Crow-AMSAA_Confidence_Bounds#Instantaneous_MTBF_.28Grouped.29|instantaneous MTBF]].
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]].


===Time Given Cumulative Failure Intensity (Grouped)===
===Time Given Cumulative Failure Intensity (Grouped)===
Line 848: Line 848:
\end{align}\,\!</math>
\end{align}\,\!</math>


The variance calculation is the same as given above in the confidence bounds on [[Crow-AMSAA_Confidence_Bounds#Beta_.28Grouped.29|Beta]] 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 888: Line 888:
\end{align}\,\!</math>
\end{align}\,\!</math>


The variance calculation is the same as given above in the confidence bounds on [[Crow-AMSAA_Confidence_Bounds#Beta_.28Grouped.29|Beta]] 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 \cdot {{m}_{c}})}^{1/(1-\beta )}}\,\!</math>
:<math>\hat{T}={{(\lambda \cdot {{m}_{c}})}^{1/(1-\beta )}}\,\!</math>
Line 898: Line 898:


====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_19|time given cumulative failure intensity]] <math>(CFI)\,\!</math> where <math>CFI=\frac{1}{CMTBF}\,\!</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.


===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). -->
===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). -->
Line 918: Line 917:
\end{align}\,\!</math>
\end{align}\,\!</math>


The variance calculation is the same as given above in the confidence bounds on [[Crow-AMSAA_Confidence_Bounds#Beta_.28Grouped.29|Beta]] 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 \beta \cdot {{m}_{i}}(T))}^{1/(1-\beta )}}\,\!</math>
Line 928: Line 927:


====Crow Bounds====
====Crow Bounds====
Step 1: Calculate the confidence bounds on the instantaneous MTBF:
'''Failure Terminated'''


:<math>MTB{{F}_{i}}={{\hat{m}}_{i}}(1\pm W)\,\!</math>
Calculate the constants <math>p_1\,\!</math> and <math>p_2\,\!</math> using procedures described for the confidence bounds on [[Crow-AMSAA_Confidence_Bounds#Crow_Bounds_17|instantaneous MTBF]]. The lower and upper confidence bounds on time are then given by:


Step 2: Use equations in the confidence bounds on [[Crow-AMSAA_Confidence_Bounds#Time_Given_Instantaneous_MTBF_.28Grouped.29|Time Given Instantaneous MTBF]] section to calculate the time given the instantaneous MTBF.
:<math>{{\hat{t}}_{L}}={{\left( \frac{\lambda \beta \cdot IMTBF}{{{p}_{1}}} \right)}^{\tfrac{1}{1-\beta }}}</math>
 
:<math>{{\hat{t}}_{U}}={{\left( \frac{\lambda \beta \cdot IMTBF}{{{p}_{2}}} \right)}^{\tfrac{1}{1-\beta }}}</math>
 
'''Time Terminated'''
 
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:
 
:<math>{{\hat{t}}_{L}}={{\left( \frac{\lambda \beta \cdot IMTBF}{{{\Pi }_{1}}} \right)}^{\tfrac{1}{1-\beta }}}\,\!</math>
 
:<math>{{\hat{t}}_{U}}={{\left( \frac{\lambda \beta \cdot IMTBF}{{{\Pi }_{2}}} \right)}^{\tfrac{1}{1-\beta }}}\,\!</math>


===Time Given Instantaneous Failure Intensity (Grouped)===
===Time Given Instantaneous Failure Intensity (Grouped)===
Line 951: Line 960:
\end{align}\,\!</math>
\end{align}\,\!</math>


The variance calculation is the same as given above in the confidence bounds on [[Crow-AMSAA_Confidence_Bounds#Beta_.28Grouped.29|Beta]] 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 961: Line 970:


====Crow Bounds====
====Crow Bounds====
The 2-sided confidence bounds on time given instantaneous failure intensity <math>(IFI)\,\!</math> are estimated using the process for the confidence bounds on [[Crow-AMSAA_Confidence_Bounds#Time_Given_Instantaneous_MTBF_.28Grouped.29|time given instantaneous MTBF]] where <math>IMTBF=\frac{1}{IFI}\,\!</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>.

Latest revision as of 20:42, 18 September 2023

New format available! This reference is now available in a new format that offers faster page load, improved display for calculations and images, more targeted search and the latest content available as a PDF. As of September 2023, this Reliawiki page will not continue to be updated. Please update all links and bookmarks to the latest reference at help.reliasoft.com/reference/reliability_growth_and_repairable_system_analysis

Chapter Appendix C: Crow-AMSAA Confidence Bounds


RGAbox.png

Chapter Appendix C  
Crow-AMSAA Confidence Bounds  

Synthesis-icon.png

Available Software:
RGA

Examples icon.png

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].