Curved exponential families

Curved exponential families generalize exponential families while preserving some of the magic, I guess?

Credited to (Efron 2004) and used in Efron (1978) to show some nice behaviour for degrees of freedom-based penalties. Will read soon.

Anirban DasGupta:

There are some important examples in which the density (pmf) has the basic Exponential family form $f(x\mid \theta )=e^{\sum _{i=1}^k{\eta }_i(\theta )T_i(X)-\psi (\theta )}h(x)$, but the assumption that the dimensions of $\mathrm{\Theta }$, and that of the range space of $({\eta }_1(\theta ),\cdots ,{\eta }_k(\theta ))$ are the same is violated. More precisely, the dimension of $\mathrm{\Theta }$ is some positive integer $q$ strictly less than $k$. Let us start with an example.[…]

Suppose $X\sim N(\mu ,{\mu }^2),\mu \ne 0$. Writing $\mu =\theta$, the density of $X$ is $$\begin{array}{c}f(x\mid \theta )=\frac{1}{\sqrt{2\pi }|\theta |}{e}^{-\frac{1}{2{\theta }^2}(x-\theta {)}^2}{I}_{x\in \mathcal{R}}\\ =\frac{1}{\sqrt{2\pi }}{e}^{-\frac{x^2}{2{\theta }^2}+\frac{x}{\theta }-\frac{1}{2}-\log |\theta |}{I}_{x\in \mathcal{R}}\end{array}$$

Writing ${\eta }_1(\theta )=-\frac{1}{2{\theta }^2},{\eta }_2(\theta )=\frac{1}{\theta },T_1(x)=x^2,T_2(x)=x,\psi (\theta )=\frac{1}{2}+\log |\theta |$, and $h(x)=\frac{1}{\sqrt{2\pi }}{I}_{x\in \mathcal{R}}$, this is in the form $f(x\mid \theta )=e^{\sum _{i=1}^k{\eta }_i(\theta )T_i(x)-\psi (\theta )}h(x)$, with $k=2$, although $\theta \in \mathcal{R}$, which is only one dimensional. The two functions ${\eta }_1(\theta )=-\frac{1}{2{\theta }^2}$ and ${\eta }_2(\theta )=\frac{1}{\theta }$ are related to each other by the identity ${\eta }_1=-\frac{{\eta }_2^2}{2}$, so that a plot of $({\eta }_1,{\eta }_2)$ in the plane would be a curve, not a straight line. Distributions of this kind go by the name of curved Exponential family. The dimension of the natural sufficient statistic is more than the dimension of $\mathrm{\Theta }$ for such distributions.

Examples?