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The probability density function (pdf) of the ith event given that the (i - 1)th event occurred at Ti-1 is:
The likelihood function is:
where 
is the termination time and is given by:
Taking the natural log on both sides:
(4)
And differentiating with respect to λ yields:
Set equal to zero and solve for λ:
(5)
Now differentiate Eqn. (4) with respect to β:
Set equal to zero and solve for β:
Eqn. (6) returns the biased estimate of β. The unbiased estimate of β can be calculated by using the following relationships. For time terminated data (meaning that the test ends after a specified number of failures):
For failure terminated data (meaning that the test ends after a specified test time):
Two prototypes of a system were tested simultaneously with design changes incorporated during the test. Table 5.1 presents the data collected over the entire test. Find the Crow-AMSAA parameters and the intensity function using maximum likelihood estimators.
Table 5.1 - Developmental test data for two identical systems
Failure |
Failed |
Test Time |
Test Time |
Total Test |
ln(T) |
1 |
1 |
1.0 |
1.7 |
2.7 |
0.99325 |
2 |
1 |
7.3 |
3.0 |
10.3 |
2.33214 |
3 |
2 |
8.7 |
3.8 |
12.5 |
2.52573 |
4 |
2 |
23.3 |
7.3 |
30.6 |
3.42100 |
5 |
2 |
46.4 |
10.6 |
57.0 |
4.04305 |
6 |
1 |
50.1 |
11.2 |
61.3 |
4.11578 |
7 |
1 |
57.8 |
22.2 |
80.0 |
4.38203 |
8 |
2 |
82.1 |
27.4 |
109.5 |
4.69592 |
9 |
2 |
86.6 |
38.4 |
125.0 |
4.82831 |
10 |
1 |
87.0 |
41.6 |
128.6 |
4.85671 |
11 |
2 |
98.7 |
45.1 |
143.8 |
4.96842 |
12 |
1 |
102.2 |
65.7 |
167.9 |
5.12337 |
13 |
1 |
139.2 |
90.0 |
229.2 |
5.43459 |
14 |
1 |
166.6 |
130.1 |
296.7 |
5.69272 |
15 |
2 |
180.8 |
139.8 |
320.6 |
5.77019 |
16 |
1 |
181.3 |
146.9 |
328.2 |
5.79362 |
17 |
2 |
207.9 |
158.3 |
366.2 |
5.90318 |
18 |
2 |
209.8 |
186.9 |
396.7 |
5.98318 |
19 |
2 |
226.9 |
194.2 |
421.1 |
6.04287 |
20 |
1 |
232.2 |
206.0 |
438.2 |
6.08268 |
21 |
2 |
267.5 |
233.7 |
501.2 |
6.21701 |
22 |
2 |
330.1 |
289.9 |
620.0 |
6.42972 |
Solution to Crow-AMSAA (NHPP) Example 1
For the failure terminated test, using Eqn. (6):
where:
Then:
From Eqn. (5):
Therefore, λi(T) becomes:
Figure 5.1 shows the plot of the failure rate. If no further changes are made, the estimated MTBF is 0.0217906 or 46 hr.
Figure 5.1: Failure rate plot for Example 5-1 using Maximum Likelihood Estimation |
