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In this section, we will present the methods used in RGA to estimate the confidence bounds for the Crow Extended model when applied to developmental testing data. RGA provides two methods to estimate the confidence bounds on demonstrated MTBF and failure intensity, projected MTBF and failure intensity and growth potential MTBF and failure intensity. The two methods are the Fisher Matrix (FM) method and Crow bounds. 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.
This section of the on-line reference includes the following subsections:
If there are no BC failure modes, the demonstrated failure intensity
is 
. Thus:
and: 
(12)
where 
.
If there are BC failure modes, the demonstrated
failure intensity, 
,
is actually the instantaneous failure intensity based on all of the data.
λCA(T) must be positive, thus lnλCA(T) is approximately treated as being normally
distributed. 
The
approximate confidence bounds on the instantaneous failure intensity are
then estimated from: 
(13)
where λCA(t) = λβTβ-1.
The variance calculation is the same as described in the Crow-AMSAA (NHPP) chapter.
If there are no BC failure modes then:
(14)
(15)
where 
.
If there are BC modes then the confidence bounds on the demonstrated failure intensity are calculated as presented in the Crow-AMSAA (NHPP) chapter.
(16)
where [λD(T)]L
and [λD(T)]U
can be obtained from Eqn. (12).
(17)
where [λD(T)]L
and [λD(T)]U
can be obtained from Eqn. (14).
The projected failure intensity λP(T) must be positive, thus lnλP(T) is approximately treated as being normally distributed
as well: 
(18) where:
when there are no BC modes.
when there are BC modes.You can then get: 
where: 
The
can be obtained
from Fisher Matrix based on M
distinct BD modes.
(19)
where 
.
(20)
[λP(T)]U and [λP(T)]L can be obtained from Eqn. (18).
(21)
[λP(T)]U and [λP(T)]L can be obtained from Eqn. (19).
If there are no BC failure modes, the growth potential failure intensity
is 
. Then:
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:
(22)
where 
.
The Crow bounds for the growth potential failure intensity are the same as the Fisher Matrix bounds.
(23)
where rU and rL can be obtained from Eqn. (22).
The Crow bounds for the growth potential MTBF are the same as the Fisher Matrix bounds.
