I'm trying to work through a theorem in the Lehmann statistical inference book and I'm confused about a proof. They are proving that a set of tests are UMP unbiased level-alpha tests for a series of hypotheses in a multiparameter exponential family.

Write the following exponential family as:

$f_{X}(x\mid {\boldsymbol {\theta }})=h(x)g({\boldsymbol {\theta }})\exp {\Big (\boldsymbol {\theta }}\cdot \mathbf {T} (x){\Big )}$

Suppose that the first element of ${\boldsymbol {\theta }}$, $\theta_1$ is known. Then we can re-write the family as: $f_{X}(x\mid {\boldsymbol {\theta }})=h(x)g({\boldsymbol {\theta }})\exp {\Big (\theta_1T_1(x) + \displaystyle\sum_{i=2}^n\theta_iT_i(x)}{\Big )}$

My question is, how do you know now that the $T_i$ are a sufficient statistic for the $\theta_i$, $i=2,3,\ldots,n$? I"m guessing the result must be obvious from the factorization theorem, but I'm not sure how to employ it with the $\theta_1T_1(x)$ term at the beginning of the exponential.

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    $\begingroup$ Hint: $ae^{b+c} = ae^be^c$. Pull the $\theta_1T_1(x)$ out to make some $\hat{h}(x) = h(x)\ldots$ $\endgroup$ – Matthew Gunn Jan 20 '17 at 1:52
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    $\begingroup$ Matthew Gunn nailed it, but if you're still stuck I can provide a detailed answer tomorrow or the following day. $\endgroup$ – David Kozak Jan 21 '17 at 1:07
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    $\begingroup$ I think I see, $\hat{h}(x) = h(x) \exp(\theta_1T_1(x)$ and then the sufficiency of the remaining $T_i$ for the remaining $\theta_i$ follows from the factorization theorem? $\endgroup$ – user21359 Jan 21 '17 at 19:45

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