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Question 12, Spring 2007 Exam 4/C *May 27, 2007*

*Posted by Peter in Exam 4/C.*

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**12.** For 200 auto accident claims you are given:

- Claims are submitted
*t*months after the accident occurs,*t*= 0, 1, 2, …. - There are no censored observations.
- is calculated using the Kaplan-Meier product limit estimator.
- , where is calculated using Greenwood’s approximation.
- .

Determine the number of claims that were submitted to the company 10 months after an accident occurred.

**Solution.** There are two key observations we need to make. The first is that we are given the risk set at time *t* = 0, namely . The second observation is that because no observations are censored, the Kaplan-Meier estimator of the survival time takes on a particularly simple form. This is because in the absence of censoring, ; that is, the risk set at time is simply the risk set at time minus those who died in the meantime. Therefore

So with this in mind, we have . Recalling Greenwood’s approximation,

so

Substituting and solving, we obtain .

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