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!