Crow Extended Confidence Bounds
In this appendix, we will present the two methods used in the Weibull++ software to estimate the confidence bounds for the Crow extended 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.
Bounds on Demonstrated Failure Intensity
Fisher Matrix Bounds
If there are no BC failure modes, the demonstrated failure intensity is
.
Thus:
and:
where 
.
If there are BC failure modes, the demonstrated failure intensity,
, is actually the instantaneous failure intensity based on all of the data.
must be positive; thus,
is approximately 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 described in the Crow-AMSAA Confidence Bounds appendix.
Crow Bounds
If there are no BC failure modes then:
![{\displaystyle {\begin{aligned}{{[{{\lambda }_{D}}(T)]}_{l}}=&{{\widehat {\lambda }}_{D}}(T){\frac {\chi _{(2N,1-\alpha /2)}^{2}}{2N}}\\{{[{{\lambda }_{D}}(T)]}_{u}}=&{{\widehat {\lambda }}_{D}}(T){\frac {\chi _{(2N,\alpha /2)}^{2}}{2N}}\end{aligned}}\,\!}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/8473eb8019cd26d83f2216caa23442cfe4a71cba)
where 
.
If there are BC modes then the confidence bounds on the demonstrated failure intensity are calculated as presented in the Crow-AMSAA Confidence Bounds appendix.
Bounds on Demonstrated MTBF
Fisher Matrix Bounds
![{\displaystyle {\begin{aligned}&MTB{{F}_{{D}_{L}}}=&{\frac {1}{{[{{\lambda }_{D}}(T)]}_{U}}}\\&MTB{{F}_{{D}_{U}}}=&{\frac {1}{{[{{\lambda }_{D}}(T)]}_{L}}}\end{aligned}}\,\!}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/6864478e16545f208cae71afea4b0fb3737ffd1a)
where 
and
can be obtained from the equation given above for
Bounds on Demonstrated Failure Intensity.
![{\displaystyle {{[{{\lambda }_{D}}(T)]}_{L}}\,\!}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/24720dd2de266673a7c862fe4139b6df9922933b)
![{\displaystyle {{[{{\lambda }_{D}}(T)]}_{U}}\,\!}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/5b4ffe125a0840f40bfc330078947ebb147448ac)
Crow Bounds
![{\displaystyle {\begin{aligned}MTB{{F}_{{D}_{L}}}=&{\frac {1}{{[{{\lambda }_{D}}(T)]}_{U}}}\\MTB{{F}_{{D}_{U}}}=&{\frac {1}{{[{{\lambda }_{D}}(T)]}_{L}}}\end{aligned}}\,\!}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/09fc3ff3e244a66b9a042335c2d4f2ea59454199)
where 
and
can be obtained from the equation given above for
Bounds on Demonstrated Failure Intensity.
Bounds on Projected Failure Intensity
Fisher Matrix Bounds
The projected failure intensity 
must be positive; thus,
is approximately treated as being normally distributed as well:
where:
-
when there are no BC modes.
when there are BC modes.
is the total failure number of the
distinct BD mode.
You can then get:
where:
The 
can be obtained from Fisher Matrix based on
distinct BD modes.
Crow Bounds
![{\displaystyle {\begin{aligned}{{[{{\lambda }_{P}}(T)]}_{L}}=&{{\hat {\lambda }}_{P}}(T)+{\frac {{C}^{2}}{2}}-{\sqrt {{{\hat {\lambda }}_{P}}(T)\cdot {{C}^{2}}+{\frac {{C}^{4}}{4}}}}\\{{[{{\lambda }_{P}}(T)]}_{U}}=&{{\hat {\lambda }}_{P}}(T)+{\frac {{C}^{2}}{2}}+{\sqrt {{{\hat {\lambda }}_{P}}(T)\cdot \ \,{{C}^{2}}+{\frac {{C}^{4}}{4}}}}\end{aligned}}\,\!}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/3d3f86487ecc9e1d7635c55019bf28568bcd665b)
where 
.
Bounds on Projected MTBF
Fisher Matrix Bounds
![{\displaystyle {\begin{aligned}MTB{{F}_{{P}_{L}}}=&{\frac {1}{{[{{\lambda }_{P}}(T)]}_{U}}}\\MTB{{F}_{{P}_{U}}}=&{\frac {1}{{[{{\lambda }_{P}}(T)]}_{L}}}\end{aligned}}\,\!}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/d6f3efba549548223a4b7b2ee6409c65f0a521f4)
and
can be obtained from the equation given above for
Bounds on Projected Failure Intensity.
![{\displaystyle {{[{{\lambda }_{P}}(T)]}_{U}}\,\!}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/697fc78f624b3dfb4fd71bef1ca6d18708926708)
![{\displaystyle {{[{{\lambda }_{P}}(T)]}_{L}}\,\!}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/e9bde2f6cb291dfb947dfab03836bdced15eadc4)
Crow Bounds
and
can be obtained from the equation given above for
Bounds on Projected Failure Intensity.
Bounds on Growth Potential Failure Intensity
Fisher Matrix Bounds
If there are no BC failure modes, the growth potential failure intensity is
.
Then:
![{\displaystyle {\begin{aligned}Var({{\widehat {r}}_{GP}})=&{\frac {1}{T}}\left[{\frac {{N}_{A}}{T}}+{\underset {i=1}{\overset {M}{\mathop {\sum } }}}\,{{(1-{{d}_{i}})}^{2}}{\frac {{N}_{i}}{T}}\right]\\\leq &{\frac {1}{T}}\left[{\frac {{N}_{A}}{T}}+{\underset {i=1}{\overset {M}{\mathop {\sum } }}}\,(1-{{d}_{i}}){\frac {{N}_{i}}{T}}\right]\\=&{\frac {{r}_{GP}}{T}}\end{aligned}}\,\!}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/783e9e6f7f3ade5b463455104b8f881fb747fef7)
If there are BC failure modes, the growth potential failure intensity is
.
Therefore:
The confidence bounds on the growth potential failure intensity are as follows:
where 
.
Crow Bounds
The Crow bounds for the growth potential failure intensity are the same as the Fisher Matrix bounds.
Bounds on Growth Potential MTBF
Fisher Matrix Bounds
where 
and
can be obtained from the equation given above for
Bounds on Growth Potential Failure Intensity.
Crow Bounds
The Crow bounds for the growth potential MTBF are the same as the Fisher Matrix bounds.