Confidence Bounds for the Crow-AMSAA Model

The following sections present 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.

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:

(7)
 

in is different (, ) according to a 2-sided confidence interval or a 1-sided confidence interval, and variances can be calculated using the Fisher Matrix.

(8)

is the nature log-likelihood function:

(9)

and:

(10)

also:

(11)

Crow Bounds

Time Terminated Data

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


 

The fractiles can be found in the tables of the distribution. Thus the confidence bounds on are:

(12)

Failure Terminated Data

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


 

Thus the confidence bounds on are:

(13)

Top

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:

(14)

where:


 

The variance calculation is the same as Eqn. 8.

Crow Bounds

Time Terminated Data

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

(15)
 

The fractiles can be found in the tables of the distribution.

Failure Terminated Data

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

(16)
 

Top

Confidence Bounds on Growth Rate for the Crow-AMSAA Model

Since the growth rate is equal to , the confidence bounds based for the Fisher Matrix and Crow methods are:

(17)

For the Fisher Matrix confidence bounds, and are obtained from Eqn. 7. For the Crow bounds, and are obtained from Eqns. 12 and 13 according to time terminated data or failure terminated data.

Top

Confidence Bounds on MTBF

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:

(18)
 

where:

The variance calculation is the same as Eqn. 8 and:

Crow Bounds on Cumulative MTBF

To calculate the Crow confidence bounds on cumulative MTBF, first calculate the Crow cumulative failure intensity confidence bounds:





 

Then:

(19)

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:

(20)
 

where:



The variance calculation is the same as Eqn. 8 and:

Crow Bounds on Instantaneous MTBF

Failure Terminated Data

Consider the following equation:

Find the values and by finding the solution 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:

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: