LME: is the Gaussian distribution sensitive to outliers?

Hi all,

In Y=X*beta + Z*b + e

b is assumed to be following Gaussian distribution.

Is there a way to test whether that's a good fit?

Is the Gaussian distribution estimation sensitive to outliers?

Is there a way to make it a Gaussian-mixture?

How do we have b to follow Gaussian distribution with different variance
for different groups?

Your pointers are greatly appreciated! Thanks a lot!

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