Sufficient statistic equivalence

Question

Let $\theta'$, $\theta \in \Theta$ such that $\theta' \neq \theta$. I want to prove that $T$ is a sufficient statistic if and only if $$\frac{f(x,\theta')}{f(x,\theta)}$$ is a function dependent only on $T(x)$.

I tried to use factorization theorem but for different parameters functions can be different and it leads nowhere.

Answer 1

From the factorisation theorem we know that $T$ is sufficient if and only if:

$$f_\theta(x) = h(x) g_\theta(T(x)).$$

So we then have:

$$R_{\theta', \theta}(x) \equiv \frac{f_\theta'(x)}{f_\theta(x)} = \frac{h(x) g_\theta'(T(x))}{h(x) g_\theta(T(x))} = \frac{g_\theta'(T(x))}{g_\theta(T(x))},$$

which depends on $x$ only through $T(x)$.