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I am looking for a generalized normal distribution with mean $\mu$ and variance $\sigma^2$ but with an additional parameter for the kurtosis to add fat-tails. I can't find it - can anybody help?

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Depends how you want to generalize, but for instance the Student t distribution becomes the standard normal distribution for high degrees of freedom. – Raskolnikov Dec 21 '10 at 10:33
@Raskolnikov: Ideally I want to let the parameters for mean and variance stay the same but having an additional parameter for kurtosis. – vonjd Dec 21 '10 at 11:00
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That's can be obtained with any distribution with at minimum two free parameters. Just relabel the parameters so that the mean and variance are $\mu$ and $\sigma^2$. Your constraints are too weak to give anything interesting. – Raskolnikov Dec 21 '10 at 11:22
I cannot follow: The other constraint is that when the third parameter is e.g. 3 (or 0) we are back to the normal distr. – vonjd Dec 21 '10 at 11:45

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I think Pearson type VII distribution might be what your are looking for. You will need to reparametrize it, but wikipedia page has the necessary formulas.

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this was really very helpful - Thank you. Do you know if there is also a log-pearson type vii distribution? I can't find any references in google... – vonjd Dec 21 '10 at 16:36
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@vonjd, you're welcome. Thank helpful people who mentioned this distribution in kurtosis wikipedia page, that is how I found it. If by log-pearson you mean distribution of random variable whose log is pearson VII, you can derive its distribution function similarly to log-normal. Hence no readily available references. – mpiktas Dec 21 '10 at 20:47

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