[R] Fitting power-law: how to choose xmin ?

[saray]

Hello.

I am trying to find probability density distribution that best fits my
data. Therefore, I am trying to fit several models (like gamma,
pareto, log-normal, ...) and then choose the best one using Akaike
Information Criterion (AIC).

In order to find the parameters for the power-law distribution I am
using power.law.fit {igraph}. As power-law distribution is not defined
in that range (0,infinity), one has to choose a lower bound. As
default R taking xmin to be the smallest value in the data. Now, my
problem is that such a selection of xmin will give the power-law
advantage on all the other models, since all the others defined on (0,
infinity) and the biggest fitting error is obtained in the tails. So,
here are my questions:
1. How can I choose the best xmin such that the model selection will
not be biased in the favour of power-law?
2. How can I make sure that the lower bound I have chose is good
enough.
3. I think that there is no risk in choosing xmin too small because
all other models support the range (0,infinity). Is that correct? Can
I choose xmin = 0.000000001 for example ?

Thank you very much in advance,
Saray Shai