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I've recently been learning about MPTs (most powerful tests), UMPTs (uniformly most powerful tests) and LRTs (likelihood ratio tests), and do not totally understand in which context the different tests are used.

Here, I define an MPT as: $Lx(θ$1)/$Lx(θ$0)

An LRT as: $Lx(\hat{θ})/Lx(\hat{θ}$0)

And a UMPT as just an MPT that can cater for composite one-sided alternative hypotheses.

I have two main questions regarding which hypotheses to use each of these for:

  1. Are LRTs only used when the hypotheses are of the form $H$0$: θ ∈ Θ$0 vs $H$1$: θ ∈ Θ$0c ? As in, can LRTs only be used in the case that the different hypothesis ranges are collectively exhaustive (and mutually exclusive of course)? Would we be able to use an LRT for a non-collectively exhaustive case, such as $H$0$: θ = θ$0 vs $H$1$: θ > θ$0 ?

  2. When we are faced with the test Ho: $H$0$: θ ≤ θ$0 vs $H$1$: θ > θ$0, we are usually asked to just create a regular MPT of the likelihood of some arbitrary $θ$1 (where $θ$1 > $θ$0) divided by the likelihood of $θ$0, and then prove that the final test does not depend on $θ$1, as well as prove that P(reject $θ$0) ≤ a, for all $θ$$θ$0. But isn't Ho: $H$0$: θ ≤ θ$0 vs $H$1$: θ > θ$0 a test that warrants an LRT too? Because we are testing for the true $θ$ being in the range ($θ$$θ$0) versus being in the range ($θ$ > $θ$0).

  3. What test form do we use in the case of Ho: $H$0$: θ ≤ θ$0 vs $H$1$: θ > θ$1 ?

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