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I want to model the left tail of an unknown distribution with a Generalized Pareto distribution. Somehow I have to select how much of the tail to model.

I am wondering if it is possible to create an IC for selecting between models of the same sample but different sizes and if so, is there an IC for doing so already?

None of the IC that I've read about seem to have considered this although for example AICc takes sample size into consideration for different reasons.

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  • $\begingroup$ you are talking about the left tail. Do you know if it is a heavy tail or not? $\endgroup$ – chuse Apr 29 '15 at 13:20
  • $\begingroup$ also, GEV is a distribution for block maxima/minima, if you are considering a threshold, isn't it better to consider Generalized Pareto Distribution? $\endgroup$ – chuse Apr 29 '15 at 13:23
  • $\begingroup$ My bad, I meant GPD though a similar problem could be formulated for block maxima. It is supposed to be heavy tailed at least $\endgroup$ – Richard Löwenström Apr 29 '15 at 13:27
  • $\begingroup$ have you considered resampling from a canonical form of the distribution? $\endgroup$ – EngrStudent Apr 29 '15 at 13:34
  • $\begingroup$ @EngrStudent The objective is to use peaks over threshold from extreme value theory and model the tail. Regardless, wouldn't resampling be hard given that we are looking at the tails where there may be very little data available? $\endgroup$ – Richard Löwenström Apr 29 '15 at 13:42
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This question is still mildly debated; there are many who study how to fit power laws, beginning by Bauke and mainly by Clauset, Shalizi and Newman, but also critically by Deluca.

In principle there's no a priori method. You can see that the problem is that you would like to use some maximum likelihood, but when you move the threshold the dataset is changing, so comparisons are meaningless.

What Clauset proposes instead is to pick the lower (higher in your case) threshold such that the p-value of your ML fit is under a certain bound that you define. Clearly if you use a high threshold, you will get a better power-law, but you are dropping data in the process, so this minimization in the threshold acts as a "force" agaist that.

I'm talking about power laws here, but you see that the same is valid for Generalized Pareto Distributions. If I may do a bit of advertisement, I used the method I described here.

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  • $\begingroup$ That's how I was thinking too. I'll read your links! By the way, do you think an IC could be formulated if you made certain assumptions about the underlying distribution? $\endgroup$ – Richard Löwenström Apr 29 '15 at 14:24

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