Chi-Square Distribution's Sufficient Statistics
Theorem
Let’s assume we have a random sample $\mathbf{X} := \left( X_{1} , \cdots , X_{n} \right) \sim \chi^{2} (r)$ that follows a chi-squared distribution. The sufficient statistic $T$ for $r$ is as follows. $$ T = \left( \prod_{i} X_{i} \right) $$
Proof
Relationship between gamma distribution and chi-squared distribution: $$ \Gamma \left( { r \over 2 } , 2 \right) \iff \chi ^2 (r) $$
Sufficient statistic for the gamma distribution: Let’s say we have a random sample $\mathbf{X} := \left( X_{1} , \cdots , X_{n} \right) \sim \Gamma \left( k, \theta \right)$ that follows a gamma distribution.
The sufficient statistic $T$ for $\left( k, \theta \right)$ is as follows.
$$ T = \left( \prod_{i} X_{i}, \sum_{i} X_{i} \right) $$
Chi-squared distribution is essentially a gamma distribution, and since the sufficient statistic for $k = r/2$ in the gamma distribution is $\prod_{i} X_{i}$, the sufficient statistic for the chi-squared distribution is also $\prod_{i} X_{i}$.
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