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I am currently fitting above mentioned functions to my data and I can observe, that both lognormal and Weibull are better fits than powerlaw. In literature, it is often suggested, that it is hard to distinguish these functions, but I can in my data.

I do not completely understand the differences between them though. What might be an explanation for the underlying process that generates my data that I can infer from these results.

Hope someone can help me.

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  • $\begingroup$ I think it may be pretty difficult to back out much detail about the underlying process. If you have explicit models for your process that lead to different distributions, you can then distinguish between these models. However, a huge number of models can lead to a log-normal distribution. Remember that a log-normal distribution is just the end result of multiplying a lot of (positive) independent identically distributed variables (by the central limit theorem). $\endgroup$ – AJK Oct 7 '13 at 23:07
  • $\begingroup$ The underlying process I look at is the distribution of how many times users request specific sites of a website. So for example, one page got requested 1000 times, the next one 200 times and so on. $\endgroup$ – fsociety Oct 8 '13 at 7:27
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See these lecture notes from the complex systems course taught by Prof. Peter Dodds. It has some great links to literature, etc..

In a nutshell, random multiplicative growth can lead to lognormal distributions and there are certain things one can enforce within the dataset (minimum number of occurrences among other things, see some of this wonderful literature in the references!) that can lead to a power-law distribution.

As mentioned by AJK, this can be hairy business and as you probably know there are entire research groups (generally categorized as complex systems groups) that are dedicated to determining what gives rise to different distributions and how to classify them. There is a rich history here.

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