Confidence Bounds for Repairable Systems

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

Confidence Bounds on β for Repairable Systems

Fisher Matrix Bounds

The parameter must be positive, thus is approximately treated as being normally distributed.

(8)

All variance can be calculated using the Fisher Information Matrix.

is the natural log-likelihood function.

(9)

(10)

(11)

Crow Bounds

Calculate conditional maximum likelihood estimate :


 

The Crow 2-sided (1 - a) 100-percent confidence bounds on are:

(12)
 

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Confidence Bounds on λ for Repairable Systems

Fisher Matrix Bounds

The parameter must be positive, thus is approximately treated as being normally distributed. These bounds are based on

The approximate confidence bounds on are given as:


 

where .

The variance calculation is the same as Eqns. 9, 10 and 11.

Crow Bounds

Time Terminated

The confidence bounds on are calculated using:


 

Failure Terminated

The confidence bounds on are calculated using:

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Confidence Bounds on Growth Rate for Repairable Systems

Since the growth rate is equal to , the confidence bounds are:

(13)

If Fisher Matrix confidence bounds are used then and are obtained from Eqn. 8. If Crow bounds are used then and are obtained from Eqn. 12.

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Confidence Bounds on MTBF for Repairable Systems

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.

The approximate confidence bounds on the cumulative MTBF are then estimated from:


 

where:

The variance calculation is the same as Eqns. 9, 10 and 11.

Crow Bounds on Cumulative MTBF


 


 

Then

(14)

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.

The approximate confidence bounds on the instantaneous MTBF are then estimated from:


 

where:

The variance calculation is the same as Eqns. 9, 10 and 11.

Crow Bounds on Instantaneous MTBF

Failure Terminated Data

Consider the following equation:

(15)

Find the values p1 and p2 by finding the solution c to 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 .

Time Terminated Data

Consider the following equation where I1(.) is the modified Bessel function of order one:

(16)

Find the values and 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 .

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Confidence Bounds on Failure Intensity for Repairable Systems

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 using:


 

where:

and:

The variance calculation is the same as Eqns. 9, 10 and 11:

Crow Bounds on Cumulative Failure Intensity

The Crow cumulative failure intensity confidence bounds are given by:


 


 

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:


 

where