On the non-identifiability of unified skew-normal distributions

In this note, we investigate the non-identifiability of the multivariate unified skew-normal distribution under permutation of its latent variables. We show that the non-identifiability issue also holds with other parametrizations and extends to the family of unified skew-elliptical distributions and more generally to selection distibutions. We provide several suggestions to make the unified skew-normal model identifiable and describe various sub-models that are identifiable.