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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.

  1. 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.

  2. 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!

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In the manifold context, injective only (unless the manifold in question is an open set of $\mathbb R^n$).

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Thanks, (1) why "unless the manifold in question is an open set of Rn"? (2) Do you mean it may not be surjective? Why is it? –  Tim Apr 28 '13 at 1:15
    
When you have an open set of $\mathbb R^n$, it parametrizes itself by the identity map. Otherwise you must cover upba manifold by various pieces, each of which is the bijective image of such an open subset. –  Ted Shifrin Apr 28 '13 at 1:19
    
Thanks! (1) So you are talking about the case when a parameterization needs more than one mappings to cover the manifold, right? If a parameterization only needs one mapping, then it must be surjective, is it? (2) why is every mapping in a parameterization injective? (3) in other mathematical branches other than manifold, is a parameterization (if there is only one mapping in it) of some object defined to be surjective? How about injective? –  Tim Apr 28 '13 at 1:29
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