In a testing task, suppose we have already chosen a test statistic $T(X)$, and know its distribution under null hypothesis.
Let $\mu$ be the mean of the null distribution of $T(X)$. Why is it reasonable to use $|T(X) - \mu| \geq c$ for some $c$ as the testing procedure to reject the null hypothesis?
I think that that rejecting null if and only if $|T(X) - \mu| \geq c$ requires the null distribution of $T(X)$
to have its mass centered around its mean $\mu$. If the null distribution of $T(X)$ has none of its mass in an interval around its mean $\mu$, and most of its mass at a certain distance away from its mean $\mu$, it seems not reasonable to reject null if and only if $|T(X) - \mu| \geq c$, doesn't it?
to be symmetric around its mean $\mu$. If the null distribution of $T(X)$ isn't symmetric around its mean $\mu$, will $T(X) \in (a, b)$ perhaps be better than $|T(X) - \mu| \geq c$ as rejection region?
Thanks and regards!