consider you've got X that follows heavy-tailed distribution(for instance its values are following Pareto distribution), so how could one tell where does actually the heavy tail start from?
btw, when i plot P(X) i can see intuitively from where the big jump starts, but is there a more rigorous way of finding tail's starting point? thanks in advance.
This question is recurrent in extreme value analysis. It is the so-called "choice of the optimal sample fraction in extremes". You can google these words, you will find a lot of references on this problem. The most simple solution relies on a graphical tool, a kind of quantile-quantile plot, but more sophisticated solutions relying on bootstrap also exist.
I have never seen this question before, and I am not clear on what use an answer would offer. However, if the distribution is symmetric I would find it natural to view the value(s) where the pdf of X intersects the corresponing pdf of a normal variate.
I am not sure that it would be helpful to give a definition. But there could be too many points of intersection of two densities to use Kenneth's idea. There is a rule of thumb for unimodal distributions, not always true.
The densities for two distributions that differ in location only intersect at one point; if they differ only in spread, they intersect in two points; if they differ only in skewness they intersect in three points; and if they differ only in kurtosis they intersect in four points. By "only" I mean the other three characteristics (of location, spread, skewness, kurtosis) are the same. This might depend on how these measures are defined.
For Kenneth's idea to work you would need to specify which two of the four intersections to use.
This question is recurrent in extreme value analysis. It is the so-called "choice of the optimal sample fraction in extremes". You can google these words, you will find a lot of references on this problem. The most simple solution relies on a graphical tool, a kind of quantile-quantile plot, but more sophisticated solutions relying on bootstrap also exist.
Regarding Terry's comments. X and the normal variate would have the same location and both lack skewness (due to symmetry). I should have stated that I presumed the distribution of X to be unimodal. They also have the same variance. Hence, the pdf:s intersects at two points if the kurtosis of X exceeds 3. And that is a requirement I think for claiming it to be fat-tailed. In a q-q plot the underlying straight line would be intersected at the three points. The first intersection defines the end of the left fat-tail, the second the common center of X and the normal variate, and the third the starting point of the right fat tail. This is a naive definition of the tails, I certainly would not claim the definition to be good, complete or useful!
The applications indicated by Stéphane was novel and interesting to me.
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