Confidence Bounds Examples Using the Crow-AMSAA Model
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Confidence Bounds Examples Using the Crow-AMSAA Model |
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This section presents the methods used in the RGA software to estimate the confidence bounds for the Crow-AMSAA model when applied to developmental testing data. RGA provides two methods to estimate the confidence bounds. The Fisher Matrix (FM) method, which is commonly employed in the reliability field, is based on the Fisher information matrix. The Crow Bounds (Crow) method has been developed by Dr. Crow.
The parameter β
must be positive, thus lnβ
is treated as being normally distributed as well. 
The approximate confidence bounds are given as:
(7)α in za
is different (α/2, α)
according to a 2-sided confidence interval or a 1-sided confidence interval,
and variances can be calculated using the Fisher Matrix. 
(8)
Λ is the natural log-likelihood function:
For the 2-sided (1 - α) 100-percent confidence interval on β, calculate:
The fractiles can be found in the tables of the χ2 distribution. Thus the confidence bounds on β are:
(12)
For the 2-sided (1 - α) 100-percent confidence interval on β, calculate:
Thus the confidence bounds on β are:
(13)
The parameter λ
must be positive, thus lnλ
is treated as being normally distributed as well. These bounds are based
on: 
The approximate
confidence bounds on λ
are given as: 
(14)where: 
The
variance calculation is the same as Eqn. (8).
For the 2-sided (1 - α) 100-percent confidence interval, the confidence bounds on λ are:
(15)
The fractiles can be found in the tables of the χ2 distribution.
For the 2-sided (1 - α) 100-percent confidence interval, the confidence bounds on λ are:
(16)
Since the growth rate is equal to 1 - β, the confidence bounds for both the Fisher Matrix and Crow methods are:
(17)
For the Fisher Matrix confidence bounds, βL and βU are obtained from Eqn. (7). For the Crow bounds, βL and βU are obtained from Eqns. (12) and (13) depending on whether the analysis is for time terminated data or failure terminated data.
The cumulative MTBF, mc(t), must be positive, thus lnmc(t) is treated as being normally distributed as well.
(18) where: 
The
variance calculation is the same as Eqn. (8) and:
To calculate the Crow confidence bounds on cumulative MTBF, first calculate
the Crow cumulative failure intensity confidence bounds: 
Then:
(19)
The instantaneous MTBF, mi(t), must be positive, thus lnmi(t) is treated as being normally distributed as
well. 
The approximate confidence bounds on the instantaneous MTBF are then
estimated from: 
(20) where: 
The variance calculation is the same as Eqn. (8) and:
Consider the following equation: 
Find the values p1
and p2 by finding the
solution c to G(n2/c|n) = ζ for 
and 
, 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 
.
Consider the following equation where I1(.) is the modified Bessel function of order one:
Find the values Π1
and Π2 by
finding the solution x
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 
.
The cumulative failure intensity, λc(t), must be positive, thus lnλc(t) is treated as being normally distributed. 
The approximate confidence bounds on the cumulative failure intensity
are then estimated from: 
(21)
where: 
and: 
The variance calculation is the same as Eqn. (8) and: 
The Crow cumulative failure intensity confidence bounds are given as:
(22)
The instantaneous failure intensity, λi(t), must be positive, thus lnλi(t) is treated as being normally distributed. 
The
approximate confidence bounds on the instantaneous failure intensity are
then estimated from: 
(23)
where
The variance calculation is the same as Eqn. (8) and: 
The Crow instantaneous failure intensity confidence bounds are given as:
(24)
The time, T, must
be positive, thus lnT
is treated as being normally distributed. 
Confidence bounds on the time are
given by: 
(25)where: 
The
variance calculation is the same as Eqn. (8) and: 
Step 1: Calculate:
Step 2: Estimate the number of failures:
Step 3: Obtain the confidence bounds on time given the cumulative failure
intensity by solving for t1
and tu
in the following equations: 
(26)
The time, T, must
be positive, thus lnT
is treated as being normally distributed. 
Confidence bounds on the time are
given by: 
(27)where: 
The
variance calculation is the same as Eqn. (8) and: 
Step 1: Calculate 
.
Step 2: Use the equations presented in the Bounds on Time Given Cumulative Failure Intensity section of this on-line reference to calculate the bounds on time given the cumulative failure intensity.
The time, T, must
be positive, thus lnT
is treated as being normally distributed. 
Confidence bounds on the time are
given by: 
(28)where: 
The
variance calculation is the same as Eqn. (8) and:
Step 1: Calculate the confidence bounds on the instantaneous MTBF as presented in the Confidence Bounds on MTBF for Repairable Systems subsection of the Repairable Systems section of this on-line reference.
Step 2: Calculate the bounds on time as follows.
So the lower an upper bounds on time are:
So the lower and upper bounds on time are:
The time, T, must be positive, thus lnT is treated as being normally distributed.
Confidence bounds on the time are given by:
(29)
where:
The variance calculation is the same as Eqn. (8) and:
Step 1: Calculate 
.
Step 2: Use the equations in the Bounds on Time Given Instantaneous MTBF section of this on-line reference to calculate the bounds on time given the instantaneous failure intensity.
The cumulative number of failures, N(t), must be positive, thus lnN(t)
is treated as being normally distributed. 
(30)
where: 
The variance calculation is the same as Eqn. (8) and: 
The Crow cumulative number of failure confidence bounds are:
(31)
where λi(T)L and λi(T)U can be obtained from Eqn. (24).
