1) In Neyman-Pearson lemma, for a fixed alpha, it gives the test with thehighest power(most powerful test).

How about the "Likelihood Ratio Test"? (which is seemingly a generalization of the N-P lemma) What does it give? Does it also give the test with thehighest power? Given any alpha, there are an infinite number of decision rules. What's sospecialabout the decision rulegiven by the Likelihood Ratio Test?

2) There is a theorem related to the likelihood ratio test:

Now I don't understand...What is the meaning of "free parameters" as stated in the theorem?

Thank you for explaining!

Note: This topic is also being discussed in the SOS mathematics cyberboard.