May i know what is the difference between the neyman pearson lemma and the likelihood ratio test? they seem like the same thing to me.
For starters, a likelihood ratio test is a test while the NP Lemma is a theorem regarding the optimality of some tests. In some sense, the NP Lemma says that, in the simple-simple case, likelihood ratio tests are most powerful for their size. However, the NP Lemma goes further as well: not all most powerful tests are necessarily likelihood ratio tests since likelihood ratio tests don't include randomized tests.
The NP Lemma does not concern cases other than simple null simple alternative (at least not directly), whereas we can typically construct a likelihood ratio test regardless of the form of the hypothesis. However, the likelihood ratio test may not be optimal (in a most-powerful sense), and indeed it typically won't be once we depart from the scenarios in which the NP Lemma applies.