Confidence Bounds for Repairable Systems Analysis

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

Beta

Fisher Matrix Bounds

The parameter \beta \,\! must be positive, thus \ln \beta \,\! is approximately treated as being normally distributed.

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

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

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

Crow Bounds

Calculate the conditional maximum likelihood estimate of \tilde{\beta \,\!}\,\! :

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

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

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

Lambda

Fisher Matrix Bounds

The parameter \lambda \,\! must be positive, thus \ln \lambda \,\! is approximately treated as being normally distributed. These bounds are based on:

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

The approximate confidence bounds on \lambda \,\! are given as:

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

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

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

Crow Bounds

Failure Terminated

The confidence bounds on \lambda \,\! for failure terminated data are calculated using:

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

where:

  • N\,\! = total number of failures.
  • K\,\! = number of systems.
  • {{T}_{q}}\,\! = end time for the qth system.

Time Terminated

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

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

where:

  • N\,\! = total number of failures.
  • K\,\! = number of systems.
  • {{T}_{q}}\,\! = end time for the qth system.

Cumulative Number of Failures

Fisher Matrix Bounds

The cumulative number of failures, N(t)\,\!. must be positive, thus \ln \left( N(t) \right)\,\! is approximately treated as being normally distributed.

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

where:

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

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

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

Crow Bounds

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

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

where:

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

Cumulative Failure Intensity

Fisher Matrix Bounds

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

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

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

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

where:

{{\hat{\lambda }}_{c}}(t)=\hat{\lambda }{{t}^{\hat{\beta }-1}}\,\!

and:

\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}\,\!

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

\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}\,\!

Crow Bounds

The 2-sided confidence bounds on the cumulative failure intensity are given by:

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

where:

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

Cumulative MTBF

Fisher Matrix Bounds

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

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

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

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

where:

{{\hat{m}}_{c}}(t)=\frac{1}{\hat{\lambda }}{{t}^{1-\hat{\beta }}}\,\!
\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}\,\!

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

\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}\,\!

Crow Bounds

The 2-sided confidence bounds on the cumulative MTBF (CMTBF)\,\! are given by:

\begin{align}
 & CMTBF_{L}=\frac{1}{CFI_{U}} \\ 
 & CMTBF_{U}=\frac{1}{CFI_{L}}  
\end{align}\,\!

where CFI_L\,\! and CFI_U\,\! are calculated using the process for the confidence bounds on cumulative failure intensity.

Instantaneous MTBF

Fisher Matrix Bounds

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

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

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

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

where:

{{\hat{m}}_{i}}(t)=\frac{1}{\lambda \beta {{t}^{\beta -1}}}\,\!
\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}\,\!

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

\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}\,\!

Crow Bounds

Failure Terminated

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

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\,\!

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

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

\begin{align}
  {{IMTBF}_{L}}= & IMTBF\cdot {{p}_{1}} \\ 
  {{IMTBF}_{U}}= & IMTBF\cdot {{p}_{2}}  
\end{align}\,\!

where IMTBF=\tfrac{1}{\hat{\lambda }\hat{\beta }{{t}^{\hat{\beta }-1}}}\,\!.

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

\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}\,\!

where IMTBF=\tfrac{1}{\bar{\lambda }\bar{\beta }{{t}^{\bar{\beta }-1}}}\,\!.

Time Terminated

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

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

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

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

\begin{align}
   {{IMTBF}_{L}}= & IMTBF\cdot {{\Pi }_{1}} \\ 
  {{IMTBF}_{U}}= & IMTBF\cdot {{\Pi }_{2}}  
\end{align}\,\!

where IMTBF=\tfrac{1}{\hat{\lambda }\hat{\beta }{{t}^{\hat{\beta }-1}}}\,\!.

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

\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}\,\!

where IMTBF=\tfrac{1}{\bar{\lambda }\bar{\beta }{{t}^{\bar{\beta }-1}}}\,\!.

Instantaneous Failure Intensity

Fisher Matrix Bounds

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

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

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

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

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

\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}\,\!

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

\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}\,\!

Crow Bounds

The 2-sided confidence bounds on the instantaneous failure intensity (IFI)\,\! are given by:

\begin{align}
   {IFI_{L}}= & \frac{1}{{IMTBF}_{U}} \\ 
  {IFI_{U}}= & \frac{1}{{IMTBF}_{L}}  
\end{align}\,\!

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

Time Given Cumulative Failure Intensity

Fisher Matrix Bounds

The time, T\,\!. must be positive, thus \ln T\,\! is approximately treated as being normally distributed.

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

The confidence bounds on the time are given by:

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

where:

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 })\,\!

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

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

Crow Bounds

The 2-sided confidence bounds on time given cumulative failure intensity (CFI)\,\! are given by:

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

Then estimate, the number of failures, N\,\!, such that:

N=\hat{\lambda }{{\hat{t}}^{{\hat{\beta }}}}\,\!

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

\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}\,\!

Time Given Cumulative MTBF

Fisher Matrix Bounds

The time, T\,\!. must be positive, thus \ln T\,\! is approximately treated as being normally distributed.

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

The confidence bounds on the time are given by:

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

where:

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 })\,\!

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

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

Crow Bounds

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

Time Given Instantaneous MTBF

Fisher Matrix Bounds

The time, T\,\!. must be positive, thus \ln T\,\! is approximately treated as being normally distributed.

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

The confidence bounds on the time are given by:

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

where:

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 })\,\!

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

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

Crow Bounds

Failure Terminated

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

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

Time Terminated

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

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

Time Given Instantaneous Failure Intensity

Fisher Matrix Bounds

These bounds are based on:

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

The confidence bounds on the time are given by:

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

where:

\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}\,\!

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

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

Crow Bounds

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

Reliability

Fisher Matrix Bounds

These bounds are based on:

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

The confidence bounds on reliability are given by:

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

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

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

Crow Bounds

Failure Terminated

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

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

where:

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

Time Terminated

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

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

where:

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

Time Given Reliability and Mission Time

Fisher Matrix Bounds

The time, t\,\!. must be positive, thus \ln t\,\! is approximately treated as being normally distributed.

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

The confidence bounds on time are calculated by using:

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

where:

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 })\,\!

\hat{t}\,\! is calculated numerically from:

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

The variance calculations are done by:

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

Crow Bounds

Failure Terminated

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

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

where p_1\,\! and p_2\,\! are obtained from the confidence bounds on instantaneous MTBF for failure terminated data.

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

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

If {{t}_{1}}<{{t}_{2}}\,\! then {{t}_{L}}={{t}_{1}}\,\! and {{t}_{U}}={{t}_{2}}\,\!. If {{t}_{1}}>{{t}_{2}}\,\! then {{t}_{L}}={{t}_{2}}\,\! and {{t}_{U}}={{t}_{1}}\,\!.

Time Terminated

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

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

where \Pi_1\,\! and \Pi_2\,\! are obtained from the confidence bounds on instantaneous MTBF for time terminated data.

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

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

If {{t}_{1}}<{{t}_{2}}\,\!. then {{t}_{L}}={{t}_{1}}\,\! and {{t}_{U}}={{t}_{2}}\,\!. If {{t}_{1}}>{{t}_{2}}\,\!. then {{t}_{L}}={{t}_{2}}\,\! and {{t}_{U}}={{t}_{1}}\,\!.

Mission Time Given Reliability and Time

Fisher Matrix Bounds

The mission time, d\,\!. must be positive, thus \ln \left( d \right)\,\! is approximately treated as being normally distributed.

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

The confidence bounds on mission time are given by using:

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

where:

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

Calculate \hat{d}\,\! from:

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

The variance calculations are done by:

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

Crow Bounds

Failure Terminated

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

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

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

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

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

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

Time Terminated

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

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

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

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

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