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Calculate the 90% 2-sided confidence bounds on the cumulative and instantaneous failure intensity for the data from Example 1 given in Table 5.1.
Fisher Matrix Bounds
Using 
and 
estimated
in Example
1, Eqns. (9),
(10)
and (11)
are: 
The Fisher Matrix
then becomes: 
For
T =
620 hr, the partial derivatives of the cumulative and instantaneous
failure intensities are: 
Therefore, the variances become: 
The cumulative and instantaneous failure intensities at T = 620
hr are: 
So, at the 90% confidence level and for T = 620 hr, the Fisher Matrix confidence bounds for the cumulative failure intensity are:
The confidence bounds for the instantaneous failure intensity are:
Figures 5.2 and 5.3 display plots of the Fisher Matrix confidence bounds for the cumulative and instantaneous failure intensity, respectively.
Figure 5.2: Cumulative failure intensity with 2-sided 90% Fisher Matrix confidence bounds |
Figure 5.3: Instantaneous failure intensity with 2-sided 90% Fisher Matrix confidence bounds |
Crow Bounds
The Crow confidence bounds for the cumulative failure intensity at the 90% confidence level and for T = 620 hr are:
The Crow confidence bounds for the instantaneous failure intensity at the 90% confidence level and for T = 620 hr are:
Figures 5.4 and 5.5 display plots of the Crow confidence bounds for the cumulative and instantaneous failure intensity, respectively.
Figure 5.4: Cumulative failure intensity with 2-sided 90% Crow confidence bounds |
Figure 5.5: Instantaneous failure intensity with 2-sided 90% Crow confidence bounds |
Calculate the confidence bounds on the cumulative and instantaneous MTBF for the data in Table 5.1.
Fisher Matrix Bounds
From the previous example: 
And for T = 620 hr, the partial derivatives of the cumulative and instantaneous MTBF are:
Therefore, the variances become: 
So, at 90% confidence level and T
= 620 hr, the Fisher Matrix confidence
bounds are: 
Figures 5.6 and 5.7 show plots of the Fisher Matrix confidence bounds for the cumulative and instantaneous MTBFs.
Figure 5.6: Cumulative MTBF with 2-sided 90% Fisher Matrix confidence bounds |
Figure 5.7: Instantaneous MTBF with 2-sided Fisher Matrix confidence bounds |
Crow Bounds
The Crow confidence bounds for the cumulative MTBF and the instantaneous MTBF at the 90% confidence level and for T = 620 hr are:
Figures 5.8 and 5.9 show plots of the Crow confidence bounds for the cumulative and instantaneous MTBF.
Figure 5.8: Cumulative MTBF with 2-sided 90% Crow confidence bounds |
Figure 5.9: Instantaneous MTBF with 2-sided 90% Crow confidence bounds |
Confidence bounds can also be obtained on the parameters 
and 
.
For Fisher Matrix confidence bounds:
and: 
For Crow confidence bounds:
and: 
