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Are a standard Gaussian and a skew Gaussian nested? I'd say yes, because when we set the skewness parameter $\alpha=0$ in the skew normal we get the standard Gaussian.

Also, are the normal/skew normal and sinh-arcsinh distributions nested? In this case I also think they are, as setting the skewness parameter to zero and the kurtosis parameter to one reduces the sinh-arcsinh distribution to a standard Gaussian.

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  • $\begingroup$ Could you explain what you mean by "nested"? After all, any two distributions are automatically members of infinitely many one-parameter distribution families. $\endgroup$
    – whuber
    Dec 7, 2015 at 17:27

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Yes, the normal is a particular case of the skew normal distribution:

$$s(x;\alpha) = 2\phi(x)\Phi(\alpha x),$$

then, $s(x;0) = 2\phi(x)\Phi(0\cdot x) = 2\phi(x)\Phi(0) = 2\phi(x)/2 = \phi(x).$

No, the skew normal is not a particular case of the sinh-arcsinh normal with unit kurtosis parameter. There are several types of "skew normal" distributions, it is just that Azzalini called his distribution "the skew normal" for marketing purposes. A good summary of skew distributions can be found in this pdf.

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