TITLE: Definitions for an exponential family to be curved or flat?
QUESTION [14 upvotes]: I was wondering how a curved exponential family is defined? Also how is a flat exponential family  defined?

Is "curved" or "flat" defined for a family of probability distributions, or for a
parametrization of a family of probability distributions? By the
latter, I mean if it is possible that, for two parametrizations of
the same family of probability distributions, a parametrization is "curved" while the other parametrization isn't?
I searched in some books, but their definitions aren't the same,
and I am wondering if they are equivalent and why?

From Casella and Berger's Statistical Inference, p115:  
From Casella and Berger's Statistical Inference, again, p137~138: 

is this a definition of "curved"?
From Bickle and Doksum's Mathematical Statistics Vol I, p56~57 

From a note by Charles J. Geyer 

An exponential family is convex (also called flat) if its natural
  parameter space is a convex subset of the full natural parameter space
  (dom c, where c is the cumulant function). 
An exponential family is curved if it is a smooth submodel of a full
  exponential family that is not itself a flat exponential family,
  where smooth means the natural parameter space is specified as the
  image of a twice continuously differentiable function from Rp for
  some p into the full natural parameter space.



Thanks and regards!

REPLY [5 votes]: I actualy understand in this way, that  full family means its parameter space contains all the possible parameters while curved family may have a restricted parameter space... and examples given by hejseb seem to explain this quite nicely.