= Solution
A <curved exponential family> restricts a higher-dimensional full canonical family to a lower-dimensional smooth parameter surface:
$$
f(x;\theta)=h(x)\exp\{\eta(\theta)^\top t(x)-A(\eta(\theta))\},\qquad\theta\in\mathbb R^q,\quad\eta(\theta)\in\mathbb R^p,\quad q<p.
$$
The derivative matrix $J(\theta)=\partial\eta/\partial\theta$ has rank $q$. A genuinely curved surface is not merely an affine subfamily under a choice of natural coordinates. With $n$ iid observations the ambient sufficient statistic is $T_n=\sum_i t(X_i)$, and the parameter score and information are
$$
U_\theta=J(\theta)^\top[T_n-n\nabla A(\eta(\theta))],\qquad I_\theta=nJ(\theta)^\top\nabla^2A(\eta(\theta))J(\theta).
$$
The second-derivative terms from $\eta(\theta)$ have zero expectation because the centered sufficient statistic has mean zero. Under regularity, the constrained MLE is still first-order asymptotically normal with this information. Curvature affects higher-order bias, likelihood geometry and efficiency; concavity of the ambient likelihood need not survive along the nonlinear parameter surface.
For example, $N(\mu,\mu^2)$ with $\mu>0$ has $t(x)=(x,x^2)$ and natural parameters $\eta_1=1/\mu$, $\eta_2=-1/(2\mu^2)$, constrained by $\eta_2=-\eta_1^2/2$. It has one free parameter but generally still needs both $\sum X_i$ and $\sum X_i^2$ as sufficient statistics. To see why, a likelihood ratio between two samples is constant in $\mu$ only if their statistic differences satisfy $\Delta T_1/\mu-\Delta T_2/(2\mu^2)=\text{constant}$ for all positive $\mu$, forcing both differences to zero. Lower parameter dimension therefore does not generally reduce the minimal sufficient statistic to that dimension.
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