A parameterization is a mapping used in differential geometry for describing a manifold, and in statistics for describing a family of distributions, and may be used for other applications I don't know or list here.
I was wondering if a parameterization is defined to be surjective? I guess yes, because, in statistics, it seems that if a parameterization is injective, we can then take its inverse. I am not sure if my understanding is correct, and if it is the same case in other areas than statistics.
Is it not defined to be injective? I think yes, because for example, in statistics, there is another concept identifiability for an injective parameterization and unidentifiability for an noninjective parameterization. I am not sure if my understanding is correct, and if it is the same case in other areas than statistics.
Thanks and regards!
In the manifold context, injective only (unless the manifold in question is an open set of $\mathbb R^n$).