I have a set of data so I am willing to fit a distribution to this set of data. Now the ones in consideration are the Gaussian and the Von Mises distributions respectively. I have done most of the stuff needed and in my case i have seen (visually) that the Von Mises fits my data better. It is a long procedure so I prefer not to go through everything. My question is:

Is there any way to quantitavely prove that the Von Mises is better than the Gaussian? I am aware of the individual Goodness of Fit tests for each of the distributions but as I understand this is not a common measure of both. As a result I cannot say that e.g. the Von Mises is 2.3% better overall or something similar.

I hope you understand my problem

Any thoughts will be very much appreciated

Thanks

Alex