Confidence Bounds for Repairable Systems Analysis

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.

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


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


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


 * $$\hat{\beta }=\frac{\underset{q=1}{\overset{K}{\mathop{\sum }}}\,{{N}_{q}}}{\hat{\lambda }\underset{q=1}{\overset{K}{\mathop{\sum }}}\,\left[ (T_{q}^{\hat{\beta }}\ln ({{T}_{q}})-S_{q}^{\hat{\beta }}\ln ({{S}_{q}}) \right]-\underset{q=1}{\overset{K}{\mathop{\sum }}}\,\underset{i=1}{\overset{\mathop{\sum }}}\,\ln ({{X}_{i}}{{}_{q}})}\,\!$$

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


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


 * $$\frac{{{\partial }^{2}}\Lambda }{\partial {{\lambda }^{2}}}=-\frac{\underset{q=1}{\overset{K}{\mathop{\sum }}}\,{{N}_{q}}}\,\!$$


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


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

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


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

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}\,\!$$

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

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 $${{q}^{th}}$$ 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 $${{q}^{th}}$$ system.

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 $$t\,\!$$.
 * $$S=\frac{\left( \frac{N} \right)}\,\!$$
 * $$t\,\!$$ = time at which calculations are being conducted.

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 $$t\,\!$$.
 * $$S=\frac{\left( \frac{N} \right)}\,\!$$
 * $$t\,\!$$ = time at which calculations are being conducted.

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}{{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.

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:


 * $$G(\mu |n)=\mathop{}_{0}^{\infty }\frac{(n-2)!}\underset{i=0}{\overset{n-1}{\mathop \sum }}\,\frac{1}{i!}{{\left( \frac{\mu }{x} \right)}^{i}}\exp (-\frac{\mu }{x})\,dx\,\!$$

Find the values $${{p}_{1}}\,\!$$ and $${{p}_{2}}\,\!$$ by finding the solution $$G\left( \left. \frac{c} \right|n \right)=\frac{\alpha }{2}$$ and $$G\left( \left. \frac{c} \right|n \right)=1-\frac{\alpha }{2}$$ 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:


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

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 $$\Pi =\tfrac{4{{n}^{2}}}\,\!$$ 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 $$IMTBF=\tfrac{1}{\bar{\lambda }\bar{\beta }{{t}^{\bar{\beta }-1}}}\,\!$$.

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.

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)} \\ \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:


 * $$N=\hat{\lambda }{{\hat{t}}^}\,\!$$

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}\,\!$$

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}})} \\ \frac{\partial T}{\partial \lambda }= & \frac{\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 $$CFI=\frac{1}{CMTBF}\,\!$$.

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}\ln (\lambda \beta \cdot MTB{{F}_{i}})+\frac{1}{\beta (1-\beta )}] \\ \frac{\partial T}{\partial \lambda }= & \frac{\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:


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


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

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 })}+\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 $$IMTBF=\frac{1}{IFI}\,\!$$.

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:


 * $$\left( {{\left[ \hat{R}\left( d \right) \right]}^{\tfrac{1}}},{{\left[ \hat{R}\left( d \right) \right]}^{\tfrac{1}}} \right)\,\!$$

where:


 * $$\hat{R}\left( d \right)={{e}^{-\left[ \hat{\lambda }{{\left( t+d \right)}^}-\hat{\lambda }{{t}^} \right]}}\,\!$$
 * $$p_1\,\!$$ 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:


 * $$\left( {{\left[ \hat{R}\left( d \right) \right]}^{\tfrac{1}}},{{\left[ \hat{R}\left( d \right) \right]}^{\tfrac{1}}} \right)\,\!$$

where:


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

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}}}^}\ln (\hat{t})-{{(\hat{t}+d)}^}\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{t}+d)}^}}{\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:


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

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:


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

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 $${{t}_{U}}={{t}_{1}}\,\!$$.

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:


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

The variance calculations are done by:


 * $$\begin{align}

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

Crow Bounds
Failure Terminated

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

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


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

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


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

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 $$({{\hat{R}}_{lower}},{{\hat{R}}_{upper}})=({{R}^{\tfrac{1}}},{{R}^{\tfrac{1}}})\,\!$$.

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 $${{d}_{upper}}={{d}_{1}}\,\!$$.