Please note: I posted this first on Mathoverflow. As it might fit better on stats.stackexchange I reposted it there. Here's the link: Post on stats.stackexchange.com/

For my thesis, I currently have to fit some heavy-tailed data. As the fitted (positive, continuous) distributions will be used in a numerical integration procedure that involves Fourier transformation, I am restricted to distributions with analytic characteristic function. Some first analysis (Hill Plot etc) showed that the tails can be fitted by a stable distribution quite well. However, closer to zero this is not the case. So I played a little with a mixture (or mixed – a term that seems to be heavily overloaded in statistics) of stable and an exponential distribution, i.e.: $f(x)=c_1 f_{\text{exp}}(x) + c_2 f_{\text{stable}}(x)$, where $c_1 + c_2=1$. This seems to improve the fit significantly. The question remains, how to fit the mixed distribution. From what I've read, it seems reasonable to consider minimum-distance estimators, like Anderson-Darling to achieve a maximum-goodness of fit. I did not find any implemented algorithms for the minimum-distance procedure. So I wanted to use some numerical optimization algorithm that allows constraints (which I need) and implement it myself.

Does this approach make sense? Recommendations for the optimization method? I do not have an analytic Jacobian, of course. Is there any tested, implemented method? Should I use a different approach? MLE is “involved” as there is no distribution function of the stable distribution

Remark: Computational effort is not relevant. However, as this is only a minor issue of the thesis, I'd rather spend not too much time on it. Parameter estimation of heavy-tailed distributions is a vast field, combined with a lack of experience this can end in a disaster. So I'd be happy if someone points me into the right direction.