I have a data set and I have to fit this data set with a stable distribution. The problem is that the stable distributions are known analytically only in the form of the characteristic function (Fourier transform). How can I do this?
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As suggested in the comments,
you can use
One way to fit the $\alpha$ parameter is via the Nagaev transform described by Okoneshnikov. An alternative is the 'Probability of Return' method of Mantegna and Stanley, which is considerably easier.
edit: the other 'classical' method is of Kogon & Williams (S.M. Kogon, Douglas B. Williams, "On Characteristic Function Based Stable Distribution Parameter Estimation Techniques"), see also matlab implementation of K&W
@Vincent's answer sounds good, but here is another approach: Since you know the Fourier transform of the distribution, take the appropriate Fourier transformation of the data, and find parameters that give the best fit in Fourier space.
I think this method should work just as well in theory, and in practice would avoid lots of numerical integration to get the form of the stable distributions. I am not coding up the test now, sorry. Anyone have any insight on this?