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Recently, I read papers that perform power-law fitting on their empirical data (estimate the alpha), some of them report corresponding p-value for the Kolmogorov-Smirnov test, but many of them do not.

I am completely new to this kind of work and I am able to perform power-law fitting thanks to the program from Clauset et al. However, according to the K-S test and its corresponding p-value, my data seems doesn't follow a pure power-law distribution.

Then I use the python program powerlaw to do a truncated power-law fit, which seems a better fit visually, but this package doesn't provide a K-S test function.

And my question is:

  1. How to perform a Kolmogorov-Smirnov test on the truncated power-law distributed data? or maybe other tests?
  2. How to estimate the xmax of a truncated power-law?
  3. Is it normal that a power-law fitted line does not go through the data points, but lies above them?

I think this is a very hard problem for me, is there some concrete tutorial on data fitting and testing?

Here is my data and the power-law fitted line (alpha=1.92) enter image description here

⬇ My data and the truncated power-law fitted line (alpha=1.66, beta=0.0009) enter image description here

⬇ The truncated power-law fit of the data from a paper enter image description here

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