# Factorization Theorem

Let ${\displaystyle f_{\theta }(x)}$ be a statistical model and suppose that

${\displaystyle f_{\theta }(s)=h(s)f_{\theta }(T(s))}$

then ${\displaystyle T(s)}$ is a sufficient statistic for the model.

PROOF

• ${\displaystyle s_{1}}$ and ${\displaystyle s_{2}}$:

${\displaystyle T(s_{1})=T(s_{2})}$

${\displaystyle f_{\theta }(T(s_{1}))=f_{\theta }(T(s_{2}))}$

• ${\displaystyle L(\theta |s_{1})={{h(s_{1})f_{\theta }(T(s_{1}))} \over {h(s_{2})f_{\theta }(T(s_{2}))}}h(s_{2})f_{\theta }(T(s_{2}))=c(s_{1},s_{2})L(\theta |s_{2})}$