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What is the common choice of distribution to fit skewed data? I am in the process of studying the effect of skew on qqplot (and varies other). So I want to know the standard version of skewed distribution.

  1. Ideally it should have mean, variance parameter just like a normal distribution. But an extra skewness control parameter.

  2. Ideally it should have 1st, 2nd, 3rd moment non-zero, but every other moment above are 0.

Does such distribution exist? I find skewed normal, but it seems overly complex and does not have 4th moment =0.

Same applies to a $\text{standard kurtosis} \not= 0$ distribution, where we can set kurtosis freely. Does such distribution exist?

kjetil b halvorsen
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yuhao
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    I’m wondering if you could construct such a distribution via moment-generating functions. – Dave May 15 '20 at 01:50
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    Why those moment requirements? Besides, they are **impossible**. You probably meant cumulants. – kjetil b halvorsen May 15 '20 at 16:21
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    The only distributions with zero fourth (central) moments are constant; for them, all central moments beyond the first are zero. – whuber May 15 '20 at 16:27
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    @kjetil even with setting higher order cumulants to 0, it's not going to work: the normal distribution is the only (nontrivial) distribution having a finite number of non-zero cumulants, so having a distribution with non-zero 3rd cumulant but all higher cumulants zero is also impossible. – Glen_b May 16 '20 at 11:33
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    http://www.scholarpedia.org/article/Cumulants#Uniqueness – Glen_b May 16 '20 at 11:44
  • The OP concerns effects of skew and kurtosis on q-q plots. Here is a link showing a direct mathematical connection in the case of kurtosis. Similar algebra can be developed for skewness. https://stats.stackexchange.com/questions/262453/does-this-q-q-plot-indicates-leptokurtic-or-platykurtic-distribution/354076#354076 – BigBendRegion Jul 13 '20 at 20:10

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