Parameter Estimation for the Crow-AMSAA Model

Maximum Likelihood Estimators

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 T* is the termination time and is given by:

Taking the natural log on both sides:

(4)

And differentiating with respect to yields:

Set equal to zero and solve for :

(5)

Now differentiate Eqn. 4 with respect to :

Set equal to zero and solve for :

(6)

Biasing and Unbiasing of Beta

Eqn. 6 returns the biased estimate of . The unbiased estimate of can be calculated by using the following relationships. For time terminated data:

For failure terminated data:

Crow-AMSAA (N.H.P.P.) Example 1

Two prototypes of a system were tested simultaneously with design changes incorporated during the test. The data collected over the entire test is presented in Table 5.1. Find the Crow-AMSAA parameters and the intensity function using Maximum Likelihood Estimators.

Table 5.1 - Developmental test data for two identical systems

Failure Number

Failed Unit

Test Time Unit 1 (hr)

Test Time Unit 2 (hr)

Total Test Time (hr)

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 (N.H.P.P.) Example 1

For the failure terminated test, using Eqn. 6:

where:

Then:

From Eqn. 5:

Therefore, becomes:

Figure 5.4 shows the plot of the failure rate. If no further changes are made, the estimated MTBF is or 46 hr.

Figure 5.1: Failure rate plot for Crow-AMSAA (N.H.P.P.) Example 1 using Maximum Likelihood Estimation

 

See Also:
Crow-AMSAA (N.H.P.P.)

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