Discussion
In the context of testing of statistical hypotheses, which one of the following statements is true ?
Answer: A
The true statement is that for simple against simple, the likelihood ratio principle leads to the most powerful test.
The Neyman-Pearson lemma says that the most powerful test of a given size for \(H_{0} : \theta = \theta_{0}\) against \(H_{1} : \theta = \theta_{1}\) rejects when the likelihood ratio \(L(\theta_{1})/L(\theta_{0})\) exceeds a constant.
The others fail. The two error probabilities \(\alpha\) and \(\beta\) need not add up to 1. Randomized tests are used to attain an exact size (level) for discrete distributions, not a desired power. And UMP tests against composite alternatives often do not exist, for example against two-sided alternatives in the normal mean problem.
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