Sufficient Statistics of the Gamma Distribution
Theorem
Given a random sample $\mathbf{X} := \left( X_{1} , \cdots , X_{n} \right) \sim \Gamma \left( k, \theta \right)$ that follows the Gamma distribution.
The sufficient statistic $T$ for $\left( k, \theta \right)$ is as follows. $$ T = \left( \prod_{i} X_{i}, \sum_{i} X_{i} \right) $$
Proof
$$ \begin{align*} f \left( \mathbf{x} ; k, \theta \right) =& \prod_{k=1}^{n} f \left( x_{k} ; k, \theta \right) \\ =& \prod_{i=1}^{n} {{ 1 } \over { \Gamma ( k ) \theta^{k} }} x_{i}^{k - 1} e^{ - x_{i} / \theta} \\ =& \left( {{ 1 } \over { \Gamma ( k ) \theta^{k} }} \right)^{n} \left( \prod_{i=1}^{n} x_{i} \right) ^{k - 1} e^{ - \sum_{i} x_{i} / \theta} \\ \overset{k}{=}& \left( {{ 1 } \over { \Gamma ( k ) \theta^{k} }} \right)^{n} \left( \prod_{i=1}^{n} x_{i} \right) ^{k - 1} \cdot e^{ - \sum_{i} x_{i} / \theta} \\ \overset{\theta}{=}& {{ 1 } \over { \theta^{nk} }} \exp \left( - \sum_{i} x_{i} / \theta \right) \cdot \left( {{ 1 } \over { \Gamma ( k ) }} \right)^{n} \left( \prod_{i=1}^{n} x_{i} \right) ^{k - 1} \end{align*} $$
According to the Neyman factorization theorem, $T := \left( \prod_{i} X_{i}, \sum_{i} X_{i} \right)$ is the sufficient statistic for $\left( k, \theta \right)$.
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