Difference powerlaw, lognormal and streteched exponential (Weibull) function

#1
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.

The underlying process I look at is the distribution of how many times users request a resource of a system. So for example, one resource got requested 1000 times, the next one 200 times and so on.

Hope someone can help me.
 

BGM

TS Contributor
#2
how many times users request a resource of a system
It seems that you are modelling a discrete distribution rather than continuous; of course you may use a round-offed continuous distribution to model it.

The shape of the distribution in general is very similar. Maybe you can look at the tail of these distribution to see whether it is heavy tail or light tail.
 
#3
It seems that you are modelling a discrete distribution rather than continuous; of course you may use a round-offed continuous distribution to model it.

The shape of the distribution in general is very similar. Maybe you can look at the tail of these distribution to see whether it is heavy tail or light tail.
Yep, I am modeling a discrete distribution and also use the appropriate methods for fitting powerlaw to discrete distributions.

I know that these three types of distributions are very similar. But, there have to be differences that I want to exploit in order to better understand the underlying process causing these observations.

As mentioned, it is quite surprising that both lognormal and Weibull are better fits to the data than powerlaw. This has to have some kind of explanation as they are normally quite similar fits. Could one idea be that powerlaw is scale-free with no finite boundary, while the others have some kind of natural limit?