I have a homework question as follows, so would appreciate hints and suggestions.
Given two uniform i.i.d. random variables, $X_1$ and $X_2$, with support $[\theta ;\theta +1]$. The hypotheses are as follows - $H_0: \theta = 0$ ; $H_A: \theta \neq 0$.
Part of the assignment is to determine the size and power of a test which always rejects $H_0$ regardless of observed data, and a test which always accepts $H_0$ regardless of observed data.
How do we think about the power of a test in cases like this, where we're not thinking about Z scores or any clearly defined probabilities?
Here is an analogy from a completely different every day scenario that you might want to think about. Consider two decision rules:
A) Tag someone as having a fever regardless of their temperature.
B) Tag someone as not having a fever regardless of their temperature.
In the scenario of a person truly having a fever, decision rule A will be absolutely perfect.
In the scenario of a person truly not having a fever, decision rule B will be absolutely perfect.