This section of the on-line reference includes the following subsections:
Grouped Data Confidence Bounds on Beta for the Crow-AMSAA Model
Grouped Data Confidence Bounds on Lambda for the Crow-AMSAA Model
Grouped Data Confidence Bounds on Growth Rate for the Crow-AMSAA Model
Grouped Data Confidence Bounds on MTBF for the Crow-AMSAA Model
Grouped Data Confidence Bounds on Failure Intensity for the Crow-AMSAA Model
Grouped Data Confidence Bounds on Time Given MTBF for the Crow-AMSAA Model
Grouped Data Confidence Bounds on Time Given Failure Intensity for the Crow-AMSAA Model
Grouped Data Confidence Bounds on Cumulative Number of Failures for the Crow-AMSAA Model
Grouped Data Confidence Bounds Examples for the Crow-AMSAA Model
The parameter must be positive, thus is approximately 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)
is the natural log-likelihood function where ln and:

Step 1: Calculate .
Step 2: Calculate:
Step 3: Calculate and
. Thus an approximate 2-sided 100-percent confidence interval on is:
(36)
The parameter must be positive, thus is approximately treated as being normally distributed as well. These bounds are based on:
The approximate confidence bounds on are given as:
(37)
where:
The variance calculation is the same as Eqn. 35.
For the 2-sided 100-percent confidence interval on , the confidence bounds on are:
(38)
(39)
For the 2-sided 100-percent confidence interval on , the confidence bounds on are:
(40)
(41)
Since the growth rate is equal to , the confidence bounds are calculated from:
(42)
For the Fisher Matrix confidence bounds, and are obtained from Eqn. 34. For the Crow bounds, and are obtained from Eqn. 36.
This section presents the confidence bounds on both the cumulative and instantaneous MTBF.
The cumulative MTBF, mc(t), must be positive, thus ln mc(t) is approximately 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 ln mi(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 100-percent confidence interval on is:
This section presents the confidence bounds on both the cumulative and instantaneous failure intensity.
The cumulative failure intensity, , must be positive, thus is approximately 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, , must be positive, thus is approximately treated as being normally distributed.
The approximate confidence bounds on the instantaneous failure intensity are then estimated from:
(48)
where and:
The variance calculation is the same as Eqn. 35 and:
The Crow instantaneous failure intensity confidence bounds are given as:
(49)
This section presents the confidence bounds on both time given cumulative MTBF and time given instantaneous MTBF.
The time, T, must be positive, thus ln T is approximately 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: Calculate the bounds on time given the cumulative failure intensity.
The time, T, must be positive, thus ln T is approximately 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: Calculate the time given the instantaneous MTBF.
This section presents the confidence bounds on both time given cumulative failure intensity and time given instantaneous failure intensity.
The time, T, must be positive, thus ln