Factorization Theorem

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

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

then $T(s)$$T(s)$ is a sufficient statistic for the model.

PROOF

• $s_{1}$$s_{1}$ and $s_{2}$$s_{2}$:

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

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

• $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})$$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})$