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This section of the on-line reference includes the following subsections:
The parameter β
must be positive, thus lnβ
is treated as being normally distributed as well. 
The approximate confidence bounds
are given as: 
(34)
can be
obtained by 
.
All variance can be calculated using the Fisher Matrix: 
(35)
Λ is the natural log-likelihood function where
ln2T
= (lnT)2
and:
Step 1: Calculate 
.
Step 2: Calculate: 
Step 3: Calculate 
and 
. Thus
an approximate 2-sided (1 -
α) 100-percent confidence interval on
is: 
(36)
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: 
(37)where: 
The
variance calculation is the same as Eqn. (35).
For the 2-sided (1 - α) 100-percent confidence interval, the confidence bounds on λ are:
(38)
(39)
For the 2-sided (1 - α) 100-percent confidence interval, the confidence bounds on λ are:
(40)
(41)
Since the growth rate is equal to 1 - β, the confidence bounds are calculated from:
(42)
For the Fisher Matrix confidence bounds, βL and βU are obtained from Eqn. (34). For the Crow bounds, βL and βU are obtained from Eqn. (36).
The cumulative MTBF, mc(t), must be positive, thus lnmc(t) is treated as being normally distributed as
well. 
The approximate
confidence bounds on the cumulative MTBF are then estimated from:
(43) where: 
The
variance calculation is the same as Eqn. (35) and: 
Calculate the Crow cumulative failure intensity confidence bounds: 
Then:
(44)
The instantaneous MTBF, mi(t), must be positive, thus lnmi(t) is approximately treated as being normally distributed
as well. 
The approximate confidence bounds on the instantaneous MTBF are then
estimated from: 
(45) where: 
The variance calculation is the same as Eqn. (35) and:
Step 1: Calculate 
.
Step 2: Calculate: 
Step 3: Calculate 
and 
. Thus
an approximate 2-sided (1 -
α) 100-percent confidence interval on
is:
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: 
(46)
where: 
and: 
The variance calculation is the same as Eqn. (35) and: 
The Crow cumulative failure intensity confidence bounds are given as:
(47)
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: 
(48)
where λi(t) = λβtβ-1
and: 
The variance calculation is the same as Eqn. (35) and: 
The Crow instantaneous failure intensity confidence bounds are given as:
(49)
The time, T, must
be positive, thus lnT
is treated as being normally distributed. 
Confidence bounds on the time are
given by: 
(50)where: 
The
variance calculation is the same as Eqn. (35) and:
Step 1: Calculate 
.
Step 2: Use the equations presented in the Bounds on Time Given Cumulative Failure Intensity for the Crow-AMSAA Model 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: 
(51)where: 
The
variance calculation is the same as Eqn. (35) and:
Step 1: Calculate the confidence bounds on the instantaneous MTBF:
Step 2: Use equations presented in the Bounds on Instantaneous MTBF for the Crow-AMSAA Model section of this on-line reference to calculate the time given the instantaneous MTBF.
The time, T, must
be positive, thus lnT
is treated as being normally distributed. 
Confidence bounds on the time are
given by: 
where: 
The
variance calculation is the same as Eqn. (35) 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:
(52)
The time, T, must
be positive, thus lnT
is treated as being normally distributed. 
Confidence bounds on the time are
given by: 
(53)where: 
The
variance calculation is the same as Eqn. (35) and: 
Step 1: Calculate 
.
Step 2: Follow the same process presented in the Bounds on Time Given Instantaneous MTBF for the Crow-AMSAA Model 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. 
(54)
where: 
The variance calculation is the same as Eqn. (35) and: 
The Crow confidence bounds on cumulative number of failures are:
(55)
where λi(T)L and λi(T)U can be obtained from Eqn. (49).
