Testing a power-law hypothesis from the averaged distribution

Question

A way to test the existence of power-law in the distribution is given in the following paper: http://arxiv.org/abs/0706.1062

The gist of the whole procedure is as follows:

1. Let $\textbf{X}$ be the set or vector of observations of discrete or continuous variable and we want to check whether these elements have come from the power-law distribution $k^{-\alpha}$.
2. The most basic (and worst) procedure would be to find (with an appropriate binning) the distribution of numbers from $\textbf{X}$ and plot it on log-log scale. If the points in the plot seem to be present approximately on straight line, then the distribution is power-law. Then one can visually fit the straight line to the approximately straight part of the plot and find out index $\alpha$.
3. A better way is to use either a cumulative distribution function or logarithmic binning and again fit a straight line visually.
4. Finally, one can use one of the above and then use say 'least square fitting' to find the exponent $\alpha$.
5. However, all these methods assume that the underlying distribution is indeed a power-law. This may not be always true. A better way that is described in the above mentioned paper is like this: Using a method of maximum likelihood, one first finds a lowest point $k_{min}$ above which power-law holds and the corresponding index $\alpha$. Then we generate a large number of synthetic power-law distributed samples using this value of $\alpha$ and compare their fluctuations with the fluctuation of the actual data. If the fraction of synthetic data sets with more fluctuation than original data is large ($>0.1$) then usually one considers this as an indicative of a power-law.

This whole process, however, uses an actual vector of observations $\textbf{X}$ and not the distribution generated using values of $\textbf{X}$.

Now consider a following situation: I have a simulation of a random process and I would like to check the existence of power-law in that process. However, if this process indeed could be simulated, I would simulate it not once but say $1000$ times and average all the resulting distributions to see the actual shape of the underlying distribution.

So far, so nice. But this invites a devil now. While averaging the distributions, we have simply thrown away individual vectors $\textbf{X}$ and they can't be averaged since every permutation of a vector $\textbf{X}$ contains the same information. Hence the method described above can't be used since it uses $\textbf{X}$ to establish a power-law.

One way to solve the problem would be the following: After we find the average distribution $p(k)$, construct a dummy vector $\textbf{X}_d$ that contains values $k$ and the number of values would be proportional to $p(k)$. Generally we would choose closest integer to $nP(k)$ where $n$ is the sample size.

One obvious problem with this approach is that when values of $p(k)$ are very small, it is not very clear how many values of $k$ should be included in $\textbf{X}_d$ ($0$ or $1$). Both choices seem to give bias over the actual underlying distribution.

How can one solve this problem without great computational efforts?

Answer 1

I don't quite understand the problem. Does your simulation output random samples or a distribution? How do you average your distributions?

If your simulation outputs random samples from some fixed distribution, it seems that you could throw the samples from all of your 1000 runs into one long vector, and test that one vector using the method you describe above.