Confidence Bounds Examples for the Crow-AMSAA Model

Crow-AMSAA Example 2

Calculate the confidence bounds on the cumulative and instantaneous failure intensity for the data in Table 5.1.

Using and estimated in Example 1, Eqns. 9, 10, 11 are:

The Fisher Matrix then becomes:

And 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

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 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.

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 MTBF.

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

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: