Wikipedia claims that the statistic $S(X)$ is minimal sufficient if and only if
$f_{\theta}(x)/f_{\theta}(y) $ is independent of $\theta$ $\iff$ $S(x) = S(y)$.
It is also claimed that this is a direct consequence of Fisher's factorization theorem. However, I do not see how this is a direct consequence.
As a definition of sufficient statistic, I know that the following two are equivalent:
- $S(x) = S(y)$ $\implies$ $f_{\theta}(x)/f_{\theta}(y) $ is independent of $\theta$
- The conditional distribution of $X$ given $S(X)$ does not depend on $\theta$.
As a definition of minimal sufficient, I use that a sufficient statistic $S(X)$ is a minimal sufficient statistic if and only if for all sufficient statistics $S'$, $S'(x) = S'(y)$ implies $S(x)=S(y)$.
I can prove that if $f_{\theta}(x)/f_{\theta}(y) $ is independent of $\theta$ $\iff$ $S(x) = S(y)$, then $S(X)$ is minimal sufficient
The direction $S(x) = S(y)$ $\implies$ $f_{\theta}(x)/f_{\theta}(y) $ is independent of $\theta$ is implies the fact $S$ is sufficient. Now if $S'$ is sufficient, then $S'(x) = S'(y)$ implies $f_{\theta}(x)/f_{\theta}(y) $ is independent of $\theta$, which implies $S(x) = S(y)$. So $S(X)$ is minimal sufficient.
But I am struggeling with showing that if $S(X)$ is minimal sufficient, then $f_{\theta}(x)/f_{\theta}(y) $ is independent of $\theta$ $\iff$ $S(x) = S(y)$. I don't see how the factorization theorem helps here.