To that point, skew and excess Kurtosis are just two of an infinite number of moments, so obviously they do not characterize the distribution. As someone else here suggested, one can look at the Fourier (or other) Transform, but then you are again left with evaluating the difference between two functions or distributions: knowing that the FT of a Gaussian is a Gaussian in its dual space doesn't help with "how close" a t-domain distribution F(t) is to a t-domain Gaussian G(t), you've just moved the problem into dual space.
We have a tendency to want to reduc... (read more)
To that point, skew and excess Kurtosis are just two of an infinite number of moments, so obviously they do not characterize the distribution. As someone else here suggested, one can look at the Fourier (or other) Transform, but then you are again left with evaluating the difference between two functions or distributions: knowing that the FT of a Gaussian is a Gaussian in its dual space doesn't help with "how close" a t-domain distribution F(t) is to a t-domain Gaussian G(t), you've just moved the problem into dual space.
We have a tendency to want to reduc... (read more)