I know version of Karlin-Rubin theorem* which makes assumption that if $f_\theta(x)$ is joint density of variables $X_1, ... X_n$ (our sample), then $l(x) = \frac{f_{\theta_1}(x)}{f_{\theta_0}(x)}$ (called likelihood ratio) is non-decreasing function of some sufficient statistic $T(x)$ for $\theta_1 > \theta_0$.
I think I've seen examples where people don't use such $l(x)$ but for example:
find some sufficient statistic $T(x)$ first
then calculate likelihood ratio of density of $T(x)$ instead of joint density of $X$s
Why can we do it? Are these two approaches really equivalent?
Are there some other versions of K-R theorem (or it's assumptions) that can be used to solve problems in possibly easier way?
- I know that the wikipedia page uses $x$ for denoting sufficient statistic which I call $T(x)$, but I'm almost sure that my professor uses different version of theorem: "first write $l(x)$ which is ratio of densities of $X_1, ..., X_n$ and then show that it is a function of some $T(x)$."
