 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.