Related Topics:
	Duane Model General Examples Using the Duane Model Time-to-Failure Data Discrete Data

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
• Other Bounds for the Duane Model
• Duane Model Example 5

Parameter Bounds

Apply least squares analysis on the Duane model:

The unbiased estimator of σ2 can be obtained from:

where:

Thus, the confidence bounds on α and b are:

(19)


(20)

where tn-2,α/2 denotes the percentage point of the t distribution with n - 2 degrees of freedom such that P{tn-2 ≥ tα/2,n-2} = α/2 and:

(21)

(22)

(23)

Other Bounds

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

(24)

(25)

(26)

When n is large, the approximate 100(1 - α)% 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 α 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 α 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.

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:

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

 

Go to Weibull.com

Go to ReliaSoft.com

©2009-2010. ReliaSoft Corporation. ALL RIGHTS RESERVED.