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Let $\{P_\theta: \Theta \subset \mathbb{R^k}\}$ be a regular statistical family. Let $T=(T_1,\ldots,T_k)$ has density of form: $\displaystyle f=c(\theta)\cdot h(t) \cdot \exp \left\{\sum_{j=1}^{k}\theta_jt_j\right\} $. Let $\ell_1\ge0, \ldots , \ell_k \ge 0$ and $\ell_1+\cdots+\ell_k=\ell$. Prove that:

$$E_\theta \left[ \prod_{i=1}^k T_i^{l_i} \right] = c(\theta)\frac{\partial^\ell}{\partial \theta_1^{\ell_1} \cdots \partial \theta_k^{\ell_k}}\frac{1}{c(\theta)}.$$

My attempt:

$f=h(t)\cdot \exp \left\{\sum_{j=1}^ k\theta_jt_j - \ln\frac 1 {c(\theta)} \right\}$ is in canonical form and we have $(T_1,\ldots,T_n)$ is sufficient statistic hence $$(ET_1,\ldots,ET_n)= \left( \frac{\partial}{\partial \theta_1}\ln\frac 1 {c(\theta)}, \ldots, \frac{\partial}{\partial \theta_k}\ln\frac 1 {c(\theta)}\right) = \left(c(\theta) \cdot\frac{\partial}{\partial \theta_1}\frac{1}{c(\theta)}, \ldots , c(\theta) \cdot\frac{\partial}{\partial \theta_n}\frac{1}{c(\theta)} \right)$$ and here I stuck.

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    Simply differentiate $$\frac1{c(\theta)}=\int h(t) \cdot \exp \left\{\sum_{j=1}^{k}\theta_jt_j\right\}dt$$ to get $$\frac{\partial^\ell}{\partial \theta_1^{\ell_1} \cdots \partial \theta_k^{\ell_k}}\frac{1}{c(\theta)}=\int \prod_{i=1}^k t_i^{\ell_i} \cdot h(t) \cdot \exp \left\{\sum_{j=1}^{k}\theta_jt_j\right\}dt=\frac1{c(\theta)}\int \prod_{i=1}^k t_i^{\ell_i} \cdot f(t) dt$$ – 2017-01-03

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