Grouped Data for the Crow-AMSAA Model

For analyzing grouped data, we follow the same logic described in the Duane model. If Eqn. (1) is linearized:

According to Crow [9], the likelihood function for the grouped data case, (where n1, n2, n3, ..., nk failures are observed within each group and k is the number of groups), is: MATH And the MLE of λ based on this relationship is: MATH (32)where n is the total number of failures from all the groups.
And the estimate of β is the value $\widehat{\beta }$ that satisfies: MATH (33)

Crow-AMSAA Example 4

Consider the grouped failure times data given in Table 5.2. Solve for the Crow-AMSAA parameters using MLE.

Table 5.2 - Grouped failure times data

Run Number	Cumulative Failures	End Time (hr)	ln(Ti)	ln(Ti)2	ln(θi)	ln(Ti) • ln(θi)
1	2	200	5.298	28.072	0.693	3.673
2	3	400	5.991	35.898	1.099	6.582
3	4	600	6.397	40.921	1.386	8.868
4	11	3000	8.006	64.102	2.398	19.198
		Sum =	25.693	168.992	5.576	38.321

Solution to Crow-AMSAA Example 4

To obtain the estimator of β, Eqn. (33) must be solved numerically for β. Using RGA, the value of $\widehat{\beta }$ is 0.6315. Now plugging this value into Eqn. (32), the estimator of λ is: MATH Therefore, the intensity function becomes: