As a "bonus" question (i.e. volontary), I got a question regarding invariance in my inference class I'm taking. So I'm trying to understand invariance, and uniformly most powerful invariant tests (UMPI tests). In terms of hypothesis testing, I have understood that invariance means that one-to-one transformations does not affect the decision (reject/not reject null hypothesis) we make. The question is:
Let X1 ,...., Xn be i.i.d N($\mu$, $\sigma$). Consider the problem of testing H0 : $\mu$ = 0 vs H1 : $\mu$ $\ne$ 0, and the group of transformations defined by gc(Xi) = cXi, c $\ne$ 0. Show that the testing problem is invariant with respect to the mentioned group. Derive a UMPI test.
Hope that there is someone here with more understanding of these types of problems, I'm really interested in understanding this but the only book I found that deals with this, I'm having a hard time understanding (Mathematical statistics, by Shao).