Definitions for an exponential family to be curved or flat?

I was wondering how a curved exponential family is defined? Also how is a flat exponential family defined?

1. 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?

2. 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!

Solution 1:

In my class we used Casella & Berger, and it wasn't very obvious to me what the definition meant since it's quite technical. If you look at example 3.4.4 (p. 113), you see that in the end what they get is

$$ f(x|\boldsymbol{\theta})=h(x)c(\boldsymbol{\theta})\text{exp}[w_1(\boldsymbol{\theta})t_1(x)+w_2(\boldsymbol{\theta})t_2(x)]\\ f(x|\boldsymbol{\theta})=h(x)c(\boldsymbol{\theta})\text{exp}\left[\sum_{i=1}^2w_i(\boldsymbol{\theta})t_i(x)\right]\\ $$

so we sum two terms ($k=2$) and the vector $\boldsymbol{\theta}=(\mu, \sigma^2)$ is of dimension 2. Hence it is a full exponential family distribution. Now, consider the case where we instead have $x\sim N(\mu, \mu^2)$. Then what you end up with is

$$ f(x|\boldsymbol{\theta})=h(x)c(\boldsymbol{\theta})\text{exp}\left[\sum_{i=1}^2w_i(\mu)t_i(x)\right]\\ $$ so clearly in this case $d=1<k=2$. Thus, it is a curved exponential. This can also be seen by the fact that the parameter space is not an open set, it's just a curve. Maybe you should take a look at exercise 3.33, perhaps it could be helpful to get your hands dirty a little.

Edit: Also, example 3.4.8 is basically what I just said too.

This might also be helpful.

Solution 2:

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.