The following Corollary is used in "Theory of Point Estimation" by Lehmann to prove a theorem. However I'm unsure how to prove this Corollary (it's left as a problem, so proof is omitted). The corollary states:

A necessary and sufficient condition for a statistic $U(X)$ to be sufficient is that for any fixed $\theta_1, \theta_2$ , the ratio $p_{\theta_1} (x)/ p_{\theta_2}(x)$ is a function only of $U(X)$.

Clearly if $U(X)$ is sufficient, then the ratio is a function only of $U(X)$ as a direct result of the factorisation theorem.

However I'm not sure how to prove that $p_{\theta_1} (x)/ p_{\theta_2}(x)$ being a function only of $U(X)$ implies sufficiency. Intuitively we would expect this to be the case, also because of the factorisation theorem, but I'm not sure how I'd prove this formally.


Let $R$ be the support of the distribution of $X$, and suppose that do not depend on the parameter $\theta$. Fix one value of $\theta_2$, and let $\theta_1$ vary over the parameter space. Write $p_{\theta_2}(x)= g(x)$ (for clarity). Then we get $$ p_{\theta_1}(x)=g(x) h(U(x);\theta_1) $$ for some function $h$, and $g$ (not depending on $\theta_1$). But that is the factorization theorem.


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