Read: Harris, D.E. (2017) The Distribution of Returns. Journal
of Mathematical Finance , 7, 769-804.
The paper derives all distributions that should be present. In practice, these are mixture distributions. The use of stable distributions came about when it was concluded that it probably was not a Cauchy distribution. Unfortunately, the methodology used to do the testing was seriously flawed in most cases. In a separate paper, it has been shown that no admissible non-Bayesian statistic exists that is also computable with regard to the bulk of all finance problems. There are deep issues in probability theory that makes such questions difficult to solve.
The simplified version of the issue goes like this. The distributions involved have two distinct issues. The first is that they lack sufficient statistics, except order statistics of course. As a result, any point statistic must lose information except Bayesian statistics where a cost function has been applied to a predictive distribution. So if you use any form of Frequentist solution, then you must lose information in order to create the measurement. This is not a problem for Bayesian methods as they always use all the information that exists in the sample and the Bayesian likelihood is always minimally sufficient.
The second issue has to do with truncation. For the distributions involved, if there were no truncation, then the median and the mode would be collocated and either could be thought of as the center of location. There is no mean, of course.
This allows the use of order statistics, it may still be the case they are not admissible for other reasons, but order statistics do not lose information other than the information loss that comes from converting things into ranks in the first place. The median is the mean rank and so you can discuss minimum variance unbiased estimators with regard to rank statistics even though the underlying data lacks a variance.
The difficulty comes from truncation. The difference between the median and the mode for US stocks from 1925-2016 is two percent per annum and the resulting difference in measurement of risk results in a four percent understatement of actual risk by using a non-Bayesian method. This can be confirmed by changing the problem from an analytic problem to a geometry problem. This bias is preserved under the transformation to logs and so log measures overstate return and understate risk.
The mode remains the center of location, but the median is now shifted as part of the left tail is missing. This is not problematic for Bayesian methods but catastrophic for Likelihoodist and Frequentist methods.
To heap insult onto injury, distribution tests are subject to the Jeffreys-Lindley paradox. Distribution tests are sharp null hypotheses. The CRSP universe has around sixty million end-of-day trades. The paradox is a theorem that states that all true sharp null hypotheses will be falsified once the sample size is large enough. Most non-Bayesian distribution tests must falsify true nulls if the sample size is large.