Problem: Prove that if the distribution of $X=(X_1,...,X_n)$ is in an exponential family, then $T(X)=(T_1(X),...,T_J(X))$ is sufficient for $\theta$.
Progress: Knowing that an exponential family can be written as $f(x;\theta)=1(x\in A)\exp\{\sum_{j=1}^J\gamma_j(\theta)T_j(x)+d(\theta)+S(x)\}$, is this just the result of using the Neyman-Fisher Factorization Theorem? This seems too trivial so I must be missing something.
Any help or verification would be appreciated!
Yes $f(x;\theta)=\exp\left\{\sum_{j=1}^J\gamma_j(\theta)T_j(x)+d(\theta)\right\}\exp\{S(x)\}$ has been factored into a component that is only a function of $x$ and functions of $\theta$ that depend on $x$ only through functions of $x$.