Problem on interpretation of data 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?
 A: 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.
