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Sufficient Statistics of the Gamma Distribution 📂Probability Distribution

Sufficient Statistics of the Gamma Distribution

Theorem

Given a random sample X:=(X1,⋯ ,Xn)∼Γ(k,θ)\mathbf{X} := \left( X_{1} , \cdots , X_{n} \right) \sim \Gamma \left( k, \theta \right) that follows the Gamma distribution.

The sufficient statistic TT for (k,θ)\left( k, \theta \right) is as follows. T=(∏iXi,∑iXi) T = \left( \prod_{i} X_{i}, \sum_{i} X_{i} \right)

Proof

f(x;k,θ)=∏k=1nf(xk;k,θ)=∏i=1n1Γ(k)θkxik−1e−xi/θ=(1Γ(k)θk)n(∏i=1nxi)k−1e−∑ixi/θ=k(1Γ(k)θk)n(∏i=1nxi)k−1⋅e−∑ixi/θ=θ1θnkexp⁡(−∑ixi/θ)⋅(1Γ(k))n(∏i=1nxi)k−1 \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:=(∏iXi,∑iXi)T := \left( \prod_{i} X_{i}, \sum_{i} X_{i} \right) is the sufficient statistic for (k,θ)\left( k, \theta \right).

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