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Sufficient Statistics for the Beta Distribution 📂Probability Distribution

Sufficient Statistics for the Beta Distribution

Theorem

Given a random sample X:=(X1,⋯ ,Xn)∼Beta(α,β)\mathbf{X} := \left( X_{1} , \cdots , X_{n} \right) \sim \text{Beta} \left( \alpha, \beta \right) that follows a beta distribution,

the sufficient statistic TT for (α,β)\left( \alpha, \beta \right) is as follows. T=(∏iXi,∏i(1−Xi)) T = \left( \prod_{i} X_{i}, \prod_{i} \left( 1 - X_{i} \right) \right)

Proof

f(x;α,β)=∏k=1nf(xk;α,β)=∏k=1n1B(α,β)xkα−1(1−xk)β−1=1B(α,β)(∏k=1nxk)α−1(∏k=1n(1−xk))β−1 \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:=(∏iXi,∏i(1−Xi))T := \left( \prod_{i} X_{i}, \prod_{i} \left( 1 - X_{i} \right) \right) is the sufficient statistic for (α,β)\left( \alpha, \beta \right).

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