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I performed a fit of some SP500 returns with two heavy tailed distributions, using MATLAB. These are like two guess about what distribution has generated the data. This is the output

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In both cases standard error for estimated parameters are reported. This sound me as suggestion to use z-test for hypothesis testing about parameters. It seems me that in t-Student case, at least for tail index greater than $2$ (finite variance), this can work. However stable distribution deal with infinite variance (exception if tail index equal to $2$). This is not a problem? ML estimator maintains asymptotic normality? If not, exist one way for parameters inference (for example $\beta$) with above output only?

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    $\begingroup$ downvoters can explain what is wrong in the question and how to improve it? $\endgroup$
    – markowitz
    Commented Jun 3, 2022 at 16:14
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    $\begingroup$ I see at least two significant problems here. First, there appear to be four questions rather than one. Second, I can't really tell because the English is too distorted. $\endgroup$
    – whuber
    Commented Jun 4, 2022 at 13:25
  • $\begingroup$ Questions appear to be several. However it seems me easy to see that them are so strongly related that, in essence, deal with the same doubt. However I delete last part in order to accept your suggestion. About my english I'm sorry, corrections would be appreciated. $\endgroup$
    – markowitz
    Commented Jun 4, 2022 at 14:34
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    $\begingroup$ I tried to improve the english. $\endgroup$
    – markowitz
    Commented Jun 5, 2022 at 14:58

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I got the answer. It is yes, ML estimator maintain asymptotic normality even in stable distribution case. So with output above it is easy to compute the CI and make inference about parameters in usual way.

For detail we can read Nolan (2001): https://link.springer.com/chapter/10.1007/978-1-4612-0197-7_17 especially par 3.1

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