Sufficient Statistics for the Beta Distribution
Theorem
Given a random sample $\mathbf{X} := \left( X_{1} , \cdots , X_{n} \right) \sim \text{Beta} \left( \alpha, \beta \right)$ that follows a beta distribution,
the sufficient statistic $T$ for $\left( \alpha, \beta \right)$ is as follows. $$ T = \left( \prod_{i} X_{i}, \prod_{i} \left( 1 - X_{i} \right) \right) $$
Proof
$$ \begin{align*} f \left( \mathbf{x} ; \alpha, \beta \right) =& \prod_{k=1}^{n} f \left( x_{k} ; \alpha, \beta \right) \\ =& \prod_{k=1}^{n} {{ 1 } \over { B(\alpha, \beta) }} x_{k}^{\alpha - 1} \left( 1 - x_{k} \right)^{\beta - 1} \\ =& {{ 1 } \over { B(\alpha, \beta) }} \left( \prod_{k=1}^{n} x_{k} \right)^{\alpha - 1} \left( \prod_{k=1}^{n} \left( 1 - x_{k} \right) \right)^{\beta - 1} \end{align*} $$
According to the Neyman factorization theorem, $T := \left( \prod_{i} X_{i}, \prod_{i} \left( 1 - X_{i} \right) \right)$ is the sufficient statistic for $\left( \alpha, \beta \right)$.
■