Grouped Data Confidence Bounds for the Crow-AMSAA Model

This section of the on-line reference includes the following subsections:

Grouped Data Confidence Bounds on β for the Crow-AMSAA Model

Fisher Matrix Bounds

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:




 

Crow Bounds

Step 1: Calculate .

Step 2: Calculate:

 

Step 3: Calculate and . Thus an approximate 2-sided 100-percent confidence interval on is:

(36)

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Grouped Data Confidence Bounds on λ for the Crow-AMSAA Model

Fisher Matrix Bounds

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.

Crow Bounds

Time Terminated Data

For the 2-sided 100-percent confidence interval on , the confidence bounds on are:

(38)
 

(39)
 

Failure Terminated Data

For the 2-sided 100-percent confidence interval on , the confidence bounds on are:

(40)
 

(41)
 

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Grouped Data Confidence Bounds on Growth Rate for the Crow-AMSAA Model

Fisher Matrix Bounds

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.

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Grouped Data Confidence Bounds on MTBF for the Crow-AMSAA Model

This section presents the confidence bounds on both the cumulative and instantaneous MTBF.

Bounds on Cumulative MTBF

Fisher Matrix Bounds on Cumulative 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:

Crow Bounds on Cumulative MTBF

Calculate the Crow cumulative failure intensity confidence bounds:





 

Then:

(44)

Bounds on Instantaneous MTBF

Fisher Matrix Bounds on Instantaneous MTBF

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:

Crow Bounds on Instantaneous MTBF

Step 1: Calculate .

Step 2: Calculate:

Step 3: Calculate and . Thus an approximate 2-sided 100-percent confidence interval on is:

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Grouped Data Confidence Bounds on Failure Intensity for the Crow-AMSAA Model

This section presents the confidence bounds on both the cumulative and instantaneous failure intensity.

Bounds on Cumulative Failure Intensity

Fisher Matrix Bounds on Cumulative 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:

Crow Bounds on Cumulative Failure Intensity

The Crow cumulative failure intensity confidence bounds are given as:

(47)
 

Bounds on Instantaneous Failure Intensity

Fisher Matrix Bounds on Instantaneous Failure Intensity

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:

Crow Bounds on Instantaneous Failure Intensity

The Crow instantaneous failure intensity confidence bounds are given as:

(49)

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Grouped Data Confidence Bounds on Time Given MTBF for the Crow-AMSAA Model

This section presents the confidence bounds on both time given cumulative MTBF and time given instantaneous MTBF.

Bounds on Time Given Cumulative MTBF

Fisher Matrix Bounds on Time Given Cumulative 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:



Crow Bounds on Time Given Cumulative MTBF

Step 1: Calculate .

Step 2: Calculate the bounds on time given the cumulative failure intensity.

Bounds on Time Given Instantaneous MTBF

Fisher Matrix Bounds on 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:

(51)
 

where:

The variance calculation is the same as Eqn. 35 and:



Crow Bounds on Time Given Instantaneous MTBF

Step 1: Calculate the confidence bounds on the instantaneous MTBF:

Step 2: Calculate the time given the instantaneous MTBF.

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Grouped Data Confidence Bounds on Time Given Failure Intensity for the Crow-AMSAA Model

This section presents the confidence bounds on both time given cumulative failure intensity and time given instantaneous failure intensity.

Bounds on Time Given Cumulative Failure Intensity

Fisher Matrix Bounds on Time Given Cumulative Failure Intensity

The time, T, must be positive, thus ln