Theorem 5.2.11 Suppose $X_1,\dots, X_n$ is a random sample from a pdf or pmf $f(x\mid \theta)=h(x)c(\theta)\exp(\sum_{i=1}^kw_i(\theta)T_i(x))$ is in exponential family. Define statistics $T_i=\sum_it_i(X_j)$ where $i=1,\dots, k$. If the set $\{(w_1(\theta),\dots,w_k(\theta))\}$ contains some open subset of $\mathbb{R}^k$, then the distribution of $(T_1,\dots, T_k)$ is an exponential family of the form $g(u_1,\dots, u_k\mid\theta)=H(u_1,\dots, u_k)c(\theta)^n \exp(\sum_iw_i(\theta)u_i)$
Q: I am looking for a proof of the theorem or reference of proof. How is containing open set for $(w_1(\theta),\dots,w_k(\theta))$ used to derive transformation? All I could see is that somehow Jacobian between $(T_i)$'s and $(X_i)$'s is accounted by part of $H$ but this may not be 1-1.