Curved exponential family
= Curved exponential family
A $(p,q)$ curved <exponential family>, with $q<p$, restricts a $p$-parameter canonical family to a smooth rank-$q$ parameter surface, usually a nonaffine surface. For example, $N(\mu,\mu^2)$ with $\mu>0$ has sufficient statistics $(Y,Y^2)$ and natural parameters $(1/\mu,-1/(2\mu^2))$, constrained by $\eta_2=-\eta_1^2/2$. Canonical concavity in the ambient parameters need not imply concavity along the curved parameter surface.