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Old 06-11-2011, 12:55 PM
Fiip27 Fiip27 is offline
Join Date: May 2010
Posts: 4
Default TIA Review Problem 235

How do they calculate the Nelson-Aalen estime of H(X), used with the survival function, for Jack to be 1/3 and Jill to be 1/3 + 1/4?

I'm interpreting this problem as 5 regular observations, plus 2 censored observations at 25 and 35.
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Old 06-11-2011, 01:04 PM
daaaave daaaave is offline
David Revelle
Join Date: Feb 2006
Posts: 3,012

I'm not including Jack and Jill in the risk set when trying to make an estimate about them. I'll reword the problem to make it clearer.

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Old 06-11-2018, 01:31 PM
ericp ericp is offline
Join Date: Aug 2007
Posts: 283

This is the most recent post on this problem that I could find. I still don't understand how the survival for Jack is 1/3 (assuming then I could figure out Jill).
If I back out to find H from S= 1/3 I get H = -1.098612289.
that is close to 1/7 + 1/6 + 1/5 + 1/4 + 1/3 which makes sense to me only if both Jack and Jill ran beyond 70 laps.
I really have no clue how this problem was solved. The solution just says "the probability that jack runs 50 or more laps is e^-1/3." But How?
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Old 06-11-2018, 02:57 PM
daaaave daaaave is offline
David Revelle
Join Date: Feb 2006
Posts: 3,012

We are given that Jack ran at least 35 laps. If we start a Nelson-Aalen estimate, looking only at what happens after time 35, there are 3 data points (45, 55, and 70), only one of which is below 50, giving a value of H-hat of 1/3, for S-hat = e^{-1/3}.

If you don't feel comfortable starting at 35, you could say that the estimated P[X>50 | X>35] is S-hat(50)/ S-hat(35), and H-hat(50) = 1/5 + 1/4 + 1/3, while H-hat(35) = 1/5 + 1/4, so S-hat(50)/S-hat(35) = e^{-(1/5+1/4+1/3)}/e^{-(1/5+1/4)} = e^{-1/3}. See near the end of C.2.4 for a comparison of these two approaches for conditional problems.

There's also a lengthy discussion of this at

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