Confidence Bounds for the Duane Model

Least squares confidence bounds can be computed for both the model parameters and metrics of interest for the Duane model.

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

Parameter Bounds for the Duane Model

Apply least squares analysis on the Duane model:

The unbiased estimator of can be obtained from:

where:

Thus, the confidence bounds on and b are:

(19)

 
(20)
 

where denotes the percentage point of the t distribution with n - 2 degrees of freedom such that and:

(21)
(22)

(23)

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Other Bounds for the Duane Model

Confidence bounds can also be obtained on the cumulative MTBF and the cumulative failure intensity:

(24)

(25)

(26)

When n is large, the approximate confidence bounds for instantaneous MTBF are given by:

(27)

(28)

and ; therefore, the confidence bounds on the instantaneous failure intensity are:

(29)

(30)

Duane Example 5

For the data given in Table 4.3, calculate the 90% confidence bounds for:

  1. The parameters and b.

  2. The cumulative and instantaneous failure intensity.

  3. The cumulative and instantaneous MTBF.

Solution to Duane Example 5

  1. Using the values of and estimated from the least squares analysis in Example 3:

Eqn. 23 is:

Eqn. 21 is:

Eqn. 22 is:

Thus, 90% confidence bounds on parameter using Eqn. 19 are:

And 90% confidence bounds on parameter b using Eqn. 20 are:

2. The cumulative failure intensity is:

And the instantaneous failure intensity is equal to:

So, at the 90% confidence level and for T = 22,000 hr, the confidence bounds on cumulative failure intensity are:

For the instantaneous failure intensity:

Figures 4.7 and 4.8 show the graphs of the cumulative and instantaneous failure intensity. Both are plotted with confidence bounds.

Figure 4.7: Cumulative Failure Intensity plot with 2-sided 90% confidence bounds

Figure 4.8: Instantaneous Failure Intensity plot with 2-sided 90% confidence bounds

3. The cumulative MTBF is:

And the instantaneous MTBF is:

So, at 90% confidence level and for T = 22,000 hr, the confidence bounds on the cumulative MTBF are:

The confidence bounds for the instantaneous MTBF are:

Figure 4.9 displays the cumulative MTBF while Figure 4.10 displays the instantaneous MTBF. Both are plotted with confidence bounds.

Figure 4.9: Cumulative MTBF plot with 2-sided 90% confidence bounds

Figure 4.10: Instantaneous MTBF plot with 2-sided 90% confidence bounds

 
See Also:
Duane Model History and Development

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