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I'm trying to fit a model on a distribution that likely has unbounded moments. Say it's some stable distribution generated from a linear combination of random shocks with unbounded at least variance, but of course, I want to fit a model to figure out what generates this data generating process. Lots of theoretical tools fail with unbounded variance, I can't (I think) use anything that depends on $L^2(P)$, I don't think I can use optimal transport, something like a GAN I don't think would work either. What are my options to do this? I'm thinking of doing something like calculating some functional distance between $\phi_{x-y}(t)$ and $\phi_0(t) = 1$ where x is my data distribution and y is the model I want to match the data distribution and $\phi$ is the characteristic function, but this seems somewhat hacky and I'm not sure what the theoretical implications of using such a method as it does seem like there will be many possible distributions x-y that have the same functional distance. It would be nice to also have a test, perhaps Maximum Mean Discrepancy-like, which is able to handle unbounded moments.

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    $\begingroup$ There is a body of literature on this and what you use depends on why you are looking and what you are looking at. If we were sitting at a table, I would end up with a long string of questions for you. If you are concerned with the data generating process, then you are likely constrained to a Bayesian method. The only material downside of Bayesian model selection methods is that they require you to set a prior distribution over the parameter and the model space. If you have a lot of dimensionality, you pretty much have to use proper priors instead of informative ones. $\endgroup$ – Dave Harris Feb 17 at 5:15
  • $\begingroup$ For more clarification, I'm mainly attempting to model the distribution of income which has power-law tails and looks like some levy distribution. I don't anticipate a high dimensional problem and Bayesian approach is fine. Would you have any resources/links? $\endgroup$ – www3 Feb 17 at 16:04
  • $\begingroup$ I will look early next week when I really have some free time. If the distribution is not analytic and cannot be approximated well by an analytic function, then the only Bayesian choice would be the method of histograms in some form. There are some Frequentist methods that use the characteristic function. $\endgroup$ – Dave Harris Feb 19 at 4:26
  • $\begingroup$ Ok thanks! Totally no rush. $\endgroup$ – www3 Feb 19 at 19:45

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