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I have data on runners who run marathons; for each runner I have their final times on a number of races. I would like to predict how fast they are running considering outliers i.e.

  • he's running faster than x hours with probability:

x | P(x)

1:30 | 0%

2:00 | 1%

2:30 | 30%

3:00 | 66%

4:00 | 3%

So, I have a curve for each athlete for the probability he's running the marathon in x hours.

How can I evaluate the curve?

Background: Currently I'm doing my evaluation using RMSE i.e. I guess that he will run in 2:45h and evaluate the error.

This doesn't make much sense for me, since in reality he's running usually in 2:30 hours but he has some outliers which influences the mean downwards to 2:45 hours.

I would like to evaluate my curve, which encodes that I have downwards outlier with certain probabilities.

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    $\begingroup$ I did some pretty extensive editing to fix the grammar of your post; I hope I did not mess up the meaning. $\endgroup$ – Peter Flom Dec 2 '12 at 14:22
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    $\begingroup$ It sounds like you're asking for individual distribution functions of the predicted times, discretized to half-hour intervals. Did I understand right? If so you'd need some kind of model that reduced the potential differences in two CDFs down to some finite set of things that relate to the differences in distribution, such as (for example) assuming shifts in scale or location and scale exist (and maybe depend on a few parameters) but the shapes are similar. $\endgroup$ – Glen_b Dec 2 '12 at 15:38
  • $\begingroup$ Actually, it should not be discretized, it should be continuous. $\endgroup$ – user1141785 Dec 3 '12 at 9:37
  • $\begingroup$ The idea with the shifted/scaled CDFs is exactly what I need. But how can I evaluate that my CDFs are scaled/located correctly and that my predictions are correct? $\endgroup$ – user1141785 Dec 3 '12 at 9:54
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    $\begingroup$ Your statements seem contradictory to me. On the one hand, you say "This doesn't make much sense for me, since in reality he's running usually in 2:30 hours but he has some outliers which influences the mean downwards to 2:45 hours." And at the same time you say "I would like to predict how fast they are running considering outliers". Do you want the outliers to be considered or not? Or maybe weighted down? Please clarify. $\endgroup$ – user765195 Jan 1 '13 at 16:32
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I am not sure what you mean by "evaluate the curve" but there are a variety of measures of central location that are less influenced by outliers than the mean is. Best known is the median. Others to consider are the trimmed mean and the Winsorized mean. For the trimmed mean you only consider the middle range of times - you can discard any range of outliers, but fairly typical are 5% or 10%. That is, you would discard (e.g.) the fastest and slowest 5% and then take the mean. The Winsorized mean is similar, except rather than discard the fastest and slowest, you set them equal to the fastest or slowest that is not discarded.

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