Crow-AMSAA Confidence Bounds
In this appendix, we will present the two methods used in the Weibull++ 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 
must be positive, thus
is treated as being normally distributed as well.
The approximate confidence bounds are given as:
in
is different (
,
) according to a 2-sided confidence interval or a 1-sided confidence interval, and variances can be calculated using the Fisher matrix.
![{\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]\,\!}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/d368cc8a17b5268e6b93384aeb2168f8e6ed536b)
is the natural log-likelihood function:
And:
Crow Bounds
Failure Terminated
For the 2-sided 
100% confidence interval on
, calculate:
Thus, the confidence bounds on 
are:
Time Terminated
For the 2-sided 
100% confidence interval on
, calculate:
The confidence bounds on 
are:
Growth Rate
Since the growth rate, 
, is equal to
, the confidence bounds for both the Fisher matrix and Crow methods are:
and
are obtained using the methods described above in the confidence bounds on
Beta.
Lambda
Fisher Matrix Bounds
The parameter 
must be positive; thus,
is treated as being normally distributed as well. These bounds are based on:
The approximate confidence bounds on
are given as:
where:
The variance calculation is the same as given above in the confidence bounds on Beta.
Crow Bounds
Failure Terminated
For the 2-sided 
100% confidence interval, the confidence bounds on
are:
where:
= total number of failures.
= termination time.
Time Terminated
For the 2-sided 
100% confidence interval, the confidence bounds on
are:
where:
= total number of failures.
= termination time.
Cumulative Number of Failures
Fisher Matrix Bounds
The cumulative number of failures, 
, must be positive, thus
is treated as being normally distributed.
where:
The variance calculation is the same as given above in the confidence bounds on Beta. And:
Crow Bounds
The Crow cumulative number of failure confidence bounds are:
where 
and
are calculated using the process for calculating the confidence bounds on
instantaneous failure intensity.
Cumulative Failure Intensity
Fisher Matrix Bounds
The cumulative failure intensity, 
, must be positive, thus
is treated as being normally distributed.
The approximate confidence bounds on the cumulative failure intensity are then estimated from:
where:
and:
The variance calculation is the same as given above in the confidence bounds on Beta. And:
Crow Bounds
The Crow bounds on the cumulative failure intensity
are given below. Let:
Failure Terminated
Time Terminated
Cumulative MTBF
Fisher Matrix Bounds
The cumulative MTBF, 
, must be positive, thus
is treated as being normally distributed as well.
The approximate confidence bounds on the cumulative MTBF are then estimated from:
where:
The variance calculation is the same as given above in the confidence bounds on Beta. And:
Crow Bounds
The 2-sided confidence bounds on the cumulative MTBF
are given by:
where 
and
are calculated using the process for calculating the confidence bounds on
cumulative failure intensity.
Instantaneous MTBF
Fisher Matrix Bounds
The instantaneous MTBF, 
, must be positive, thus
is treated as being normally distributed as well.
The approximate confidence bounds on the instantaneous MTBF are then estimated from:
where:
The variance calculation is the same as given above in the confidence bounds on Beta. And:
Crow Bounds
Failure Terminated
For failure terminated data and the 2-sided confidence bounds on instantaneous MTBF
, consider the following equation:
Find the values 
and
by finding the solution
and
for the lower and upper bounds, respectively.
If using the biased parameters, 
and
, then the upper and lower confidence bounds are:
where 
.
If using the unbiased parameters, 
and
, then the upper and lower confidence bounds are:
where 
.
Time Terminated
Consider the following equation where
is the modified Bessel function of order one:
Find the values 
and
by finding the solution
to
and
in the cases corresponding to the lower and upper bounds, respectively. Calculate
for each case.
If using the biased parameters, 
and
, then the upper and lower confidence bounds are:
where 
.
If using the unbiased parameters, 
and
, then the upper and lower confidence bounds are:
where 
.
Instantaneous Failure Intensity
Fisher Matrix Bounds
The instantaneous failure intensity,
, must be positive, thus
is treated as being normally distributed.
The approximate confidence bounds on the instantaneous failure intensity are then estimated from:
where
The variance calculation is the same as given above in the confidence bounds on Beta. And:
Crow Bounds
The 2-sided confidence bounds on the instantaneous failure intensity
are given by:
where 
and
are calculated using the process presented for the confidence bounds on the
instantaneous MTBF.
Time Given Cumulative Failure Intensity
Fisher Matrix Bounds
The time, 
, must be positive, thus
is treated as being normally distributed.
Confidence bounds on the time are given by:
- where:
The variance calculation is the same as given above in the confidence bounds on Beta. And:
Crow Bounds
The 2-sided confidence bounds on time given cumulative failure intensity
are given by:
Then estimate the number of failures,
, such that:
The lower and upper confidence bounds on time are then estimated using:
Time Given Cumulative MTBF
Fisher Matrix Bounds
The time, 
, must be positive, thus
is treated as being normally distributed.
Confidence bounds on the time are given by:
where:
The variance calculation is the same as given above in the confidence bounds on Beta. And:
Crow Bounds
The 2-sided confidence bounds on time given cumulative MTBF
are estimated using the process for calculating the confidence bounds on
time given cumulative failure intensity 
where
.
Time Given Instantaneous MTBF
Fisher Matrix Bounds
The time, 
, must be positive, thus
is treated as being normally distributed.
Confidence bounds on the time are given by:
where:
The variance calculation is the same as given above in the confidence bounds on Beta. And:
![{\displaystyle {\begin{aligned}{\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{aligned}}\,\!}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/a3ff864beb33b06c59debe4581d9a99f8ea240bf)
Crow Bounds
Failure Terminated
If the unbiased value 
is used then:
where:
= instantaneous MTBF.
= total number of failures.
Calculate the constants 
and
using procedures described for the confidence bounds on
instantaneous MTBF. The lower and upper confidence bounds on time are then given by:
Time Terminated
If the unbiased value 
is used then:
where:
= instantaneous MTBF.
= total number of failures.
Calculate the constants 
and
using procedures described for the confidence bounds on
instantaneous MTBF. The lower and upper confidence bounds on time are then given by:
Time Given Instantaneous Failure Intensity
Fisher Matrix Bounds
The time, 
, must be positive, thus
is treated as being normally distributed.
Confidence bounds on the time are given by:
where:
The variance calculation is the same as given above in the confidence bounds on Beta. And:
![{\displaystyle {\begin{aligned}{\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{aligned}}\,\!}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/18ad56d3d9be55b168b8c94916b8d8062dda504f)
Crow Bounds
The 2-sided confidence bounds on time given instantaneous failure intensity
are estimated using the process for calculating the confidence bounds on
time given instantaneous MTBF where
.
Grouped Data
This section presents the confidence bounds for the Crow-AMSAA model when using Grouped data.
Beta (Grouped)
Fisher Matrix Bounds
The parameter 
must be positive, thus
is treated as being normally distributed as well.
The approximate confidence bounds are given as:
can be obtained by
.
All variance can be calculated using the Fisher matrix:
is the natural log-likelihood function where
and:
![{\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]\,\!}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/b9ec7f1d218cbc730cf1189cce92a7e30ec0a2c4)
![{\displaystyle {\begin{aligned}{\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{aligned}}\,\!}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/33dd5daf92b59adcd1f6d7294eaeded8e1b19c85)
Crow Bounds
The 2-sided confidence bounds on 
are given by first calculating:
where:
= interval end time for the
interval.
= number of intervals.
= end time for the last interval.
Next:
![{\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 }}}]}}\,\!}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/bc0d7ceed00bf9caf744bb6080716ea04f5ec5fd)
And:
Then:
where:
= inverse standard normal.
= number of failures.
The 2-sided confidence bounds on 
are then
.
Growth Rate (Grouped)
Since the growth rate, 
, is equal to
, the confidence bounds for both the Fisher matrix and Crow methods are:
and
are obtained using the methods described above in the confidence bounds on
Beta.
Lambda (Grouped)
Fisher Matrix Bounds
The parameter 
must be positive, thus
is treated as being normally distributed as well. These bounds are based on:
The approximate confidence bounds on
are given as:
where:
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
100% confidence interval, the confidence bounds on
are:
where:
= total number of failures.
= end time of last interval.
Time Terminated
For time terminated data, the 2-sided
100% confidence interval, the confidence bounds on
are:
where:
= total number of failures.
= end time of last interval.
Cumulative Number of Failures (Grouped)
Fisher Matrix Bounds
The cumulative number of failures, 
, must be positive, thus
is treated as being normally distributed.
where:
The variance calculation is the same as given above in the confidence bounds on Beta. And:
Crow Bounds
The 2-sided confidence bounds on the cumulative number of failures are given by:
where 
and
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, 
, must be positive, thus
is treated as being normally distributed.
The approximate confidence bounds on the cumulative failure intensity are then estimated from:
where:
and:
The variance calculation is the same as given above in the confidence bounds on Beta. And:
Crow Bounds
The 2-sided confidence bounds on the cumulative failure intensity
are given below. Let:
Then:
Cumulative MTBF (Grouped)
Fisher Matrix Bounds
The cumulative MTBF, 
, must be positive, thus
is treated as being normally distributed as well.
The approximate confidence bounds on the cumulative MTBF are then estimated from:
where:
The variance calculation is the same as given above in the confidence bounds on Beta. And:
Crow Bounds
The 2-sided confidence bounds on cumulative MTBF
are given by:
where 
and
are calculating using the process for calculating the confidence bounds on the
cumulative failure intensity.
Instantaneous MTBF (Grouped)
Fisher Matrix Bounds
The instantaneous MTBF, 
, must be positive, thus
is approximately treated as being normally distributed as well.
The approximate confidence bounds on the instantaneous MTBF are then estimated from:
where:
The variance calculation is the same as given above in the confidence bounds on Beta. And:
Crow Bounds
The 2-sided confidence bounds on instantaneous MTBF
are given by first calculating:
where:
= interval end time for the
interval.
= number of intervals.
= end time for the last interval.
Calculate:
![{\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]}}\,\!}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/44d08cc7b0f5e94e681283356e19811df8cf596a)
Next:
And:
where:
= inverse standard normal.
= number of failures.
The 2-sided confidence bounds on instantaneous MTBF are then
.
Instantaneous Failure Intensity (Grouped)
Fisher Matrix Bounds
The instantaneous failure intensity,
, must be positive, thus
is treated as being normally distributed.
The approximate confidence bounds on the instantaneous failure intensity are then estimated from:
where 
and:
The variance calculation is the same as given above in the confidence bounds on Beta. And:
Crow Bounds
The 2-sided confidence bounds on the instantaneous failure intensity
are given by:
where 
and
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, 
, must be positive, thus
is treated as being normally distributed.
Confidence bounds on the time are given by:
where:
The variance calculation is the same as given above in the confidence bounds on Beta. And:
Crow Bounds
The 2-sided confidence bounds on time given cumulative failure intensity
are presented below. Let:
Then estimate the number of failures:
The confidence bounds on time given the cumulative failure intensity are then given by:
Time Given Cumulative MTBF (Grouped)
Fisher Matrix Bounds
The time, 
, must be positive, thus
is treated as being normally distributed.
Confidence bounds on the time are given by:
where:
The variance calculation is the same as given above in the confidence bounds on Beta. And:
Crow Bounds
The 2-sided confidence bounds on time given cumulative MTBF
are estimated using the process for calculating the confidence bounds on
time given cumulative failure intensity 
where
.
Time Given Instantaneous MTBF (Grouped)
Fisher Matrix Bounds
The time, 
, must be positive, thus
is treated as being normally distributed.
Confidence bounds on the time are given by:
where:
The variance calculation is the same as given above in the confidence bounds on Beta. And:
![{\displaystyle {\begin{aligned}{\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{aligned}}\,\!}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/8e610a4fc8b49caa9a68aac9cffdde11042744fa)
Crow Bounds
Failure Terminated
Calculate the constants 
and
using procedures described for the confidence bounds on
instantaneous MTBF. The lower and upper confidence bounds on time are then given by:
Time Terminated
Calculate the constants 
and
using procedures described for the confidence bounds on
instantaneous MTBF. The lower and upper confidence bounds on time are then given by:
Time Given Instantaneous Failure Intensity (Grouped)
Fisher Matrix Bounds
The time, 
, must be positive, thus
is treated as being normally distributed.
Confidence bounds on the time are given by:
where:
The variance calculation is the same as given above in the confidence bounds on Beta. And:
Crow Bounds
The 2-sided confidence bounds on time given instantaneous failure intensity
are estimated using the process for calculating the confidence bounds on
time given instantaneous MTBF where
.