Grouped Data Confidence Bounds Examples for the Crow-AMSAA Model

Crow-AMSAA Example 5

A new helicopter system is under development. System failure data has been collected on five helicopters for the final test phase. The actual failure times cannot be determined since the failures are not discovered until after the helicopters are brought into the maintenance area. Total flying hours are known when the helicopters are brought in for service. Every two weeks, each helicopter undergoes a thorough inspection to uncover any failures that may have occurred, but were not discovered, since the last inspection. Therefore, the cumulative total number of flight hours and the cumulative total number of failures for the five helicopters are known for each two-week period. The total number of flight hours is 500, which was accrued over a period of 12 weeks. The 500 flight hours are partitioned into 6 intervals, each covering a two-week period. For each period, the total number of flight hours and total number of assigned failures for the five helicopters are recorded. The group data set is collected and displayed in Table 5.3.

Table 5.3 - Grouped data for a new helicopter system

Interval	Interval Length	Failures In Interval
1	0 - 62	12
2	62 -100	6
3	100 - 187	15
4	187 - 210	3
5	210 - 350	18
6	350 - 500	16

Determine the maximum likelihood estimators.
Calculate the confidence bounds on the cumulative and instantaneous MTBF using the Fisher Matrix and Crow methods.

Solution to Crow-AMSAA Example 5

Obtain the estimator of using Eqn. 33. Using RGA, the value of is 0.81361. Now plug this value into Eqn. 32 and is:

Fisher Matrix confidence bounds can be obtained on the parameters and at the 90% confidence level by:

and:

Crow confidence bounds can also be obtained on the parameters and at the 90% confidence level, as:

and:

The Fisher Matrix confidence bounds for the cumulative MTBF and the instantaneous MTBF at the 90% 2-sided confidence level and for T = 500 hr are:

and:

Figures 5.10 and 5.11 show plots of the Fisher Matrix confidence bounds for the cumulative and instantaneous MTBF, respectively.

Figure 5.10: Cumulative MTBF with 2-sided 90% Fisher Matrix confidence bounds

Figure 5.11: Instantaneous MTBF with 2-sided 90% Fisher Matrix confidence bounds

The Crow confidence bounds for the cumulative and instantaneous MTBF at the 90% 2-sided confidence level and for T = 500 hours are:

and:

Figures 5.12 and 5.13 show plots of the Crow confidence bounds for the cumulative and instantaneous MTBF.

Figure 5.12: Cumulative MTBF with 2-sided 90% Crow confidence bounds

Figure 5.13: Instantaneous MTBF with 2-sided 90% Crow confidence bounds