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Following Critical Phenomena in Natural Sciences of Didier Sornette, I am plotting the maximum value among $N$ variables. Doing this I can see the tail behaviour of a distribution (page 19 of C.P.N.S. of Sornette). Calling $\lambda$ the maximum value among the first $N$ variables, in the case the tail is of the form $f(x)\sim e^{-x/a}$ then $\lambda\sim a \ln N$. In the case the tail of the distribution is of the form of power law $f(x)\sim x^{-(a+1)}$ then $\lambda\sim N^{1/a}$. Now I have many samples of data coming from the same physical phenomenon, but they show two different behaviour. Ones with tail of power law type and ones with exponential tail type. Does this means that are presents two different mechanisms generating the same physical phenomenon?

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If they have different distributions then they are different in some way. A physical explanation for the difference is not a statistical explanation. The answwer lies in the science and not in statistics. This may not be an appropriate question for this site and you may want to migrate it elsewhere. A moderator can help you find the most suitable site. – Michael Chernick Jul 28 '12 at 9:59
up vote 3 down vote accepted

The difference between power-law and exponential decrease is a common argument to consider that two phenomena are underpinned by different physical mechanisms. This comes from the fact that power-law phenomena are scale-invariant, which usually involves a mechanism of self-similarity which is hard to envision for exponential phenomena.

That said, a very strong assumption in all the above is that sampling is independent. I don't know which is the physical phenomenon you study, but if there is a chance that the records have some sort of dependence (people collecting the data, observation sites, measurement instrument etc.), I would start by exploring the sampling biases before physical realities.

Some pages of the chapter are missing on Google books, so I don't know how you drew your conclusions, but a lot depends on this. By analyzing the tails of the distribution, you inherently focus on rare events, and it is easy to get fooled by the noise. In particular, simple log-log plots are very "dangerous" because the curve becomes more and more noisy. So, before concluding that there are two different physical mechanisms, you should convince yourself that the esimation method is robust. Perhaps using your methods on different subsamples of the same sample will give you an idea of the variability that you can expect.

After all this, if your conclusions still hold, then yes, I would believe that your phenomenon comprises two alternative, mutually exclusive physical mechanisms.

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@gu11aume I think you did a nice job of digging out some real statistical issues that makes this question suitable for discussion here. But the ultimate answer to the question depends on understanding the subject matter. The difference is there but is it correct to attribute cause to two different physical mechanisms. That is what is outside the scope of statistics. In consulting situations the statistician and the client need ot recognize this. – Michael Chernick Jul 28 '12 at 11:01
@gui11aume The data rapresents the magnitude (the whole rain) of rainfalss in differents points of the planets. I performed an auto correlation study in order to avoid time dependent sample of data. So, in general, i can say :yes are not correlated, in the sense of space and time. – emanuele Jul 28 '12 at 11:26
@MichaelChernick i think this is a question that belongs to the shadow side of statistics and physics. I have no problem to migrate the question, but i think that someone can tell me that this argument do not well fit a pure physical argument. – emanuele Jul 28 '12 at 11:29
Rainfall? Sister! (I'm analyzing rainfall patterns from ~100 stations in the desert southwest of the U.S.). On a more appropriate note, there are at least two different physical mechanisms. One is rainfall generated by frontal systems moving across the area, e.g., cold fronts from Alaska or warm fronts from Hawaii, and the other is thunderstorm activity. Depending on where the station is located, the larger rainfall events may be primarily generated by one or the other, e.g., just east of San Diego there's little thunderstorm activity, near Phoenix big rains are almost all thunderstorms. – jbowman Jul 28 '12 at 15:43
I am glad we figured this out!! @emanuele it seems you know who to talk to now :-) – gui11aume Jul 28 '12 at 15:50

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