# Are these models nested?

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.

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

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