Two ways of writing the critical region for Hypothesis Testing I have one confusion in writing critical region. Let us suppose that we have a test condition that rejects the the null hypothesis if  $T(X) > \chi_{2n,\alpha}$. Can we also write this same test as reject $H_0$ if
$T(X) < \chi_{2n,1-\alpha}$.
If yes, how can I understand it intuitively? I have trouble understanding how the second test can be derived from the first one.
 A: (Most likely) no. Your question is a bit hard to answer as you do not say what your hypothesis and your test statistic are, nor does it define $\chi_{2n,\alpha}$, so we will make assumptions.
First, typically, statistics $T(X)$ that have a chi-square null distribution (that is how I read the quantile $\chi_{2n,\alpha}$) are (as the square indicates) squared expressions, i.e., any negative signs get removed by squaring so that, generally, large values of the test statistic provide evidence against the null. A leading example is tesing if all slope coefficients of a regression model are zero, see e.g. here. Another is that no study in a meta-analysis provides evidence against the null, see e.g. here.
Hence, you reject if the statistic is larger than an extreme upper quantile of the null distribution, which, in your notation, should then be $\chi_{2n,\alpha}$. At the 5% level, this notation could indicate the 95% quantile. Then, $\chi_{2n,1-\alpha}$ would be the 5% quantile and rejecting if $T(X) < \chi_{2n,1-\alpha}$ would mean to reject if the test statistic is very small.
Now, it is not impossible to think of applications of test statistics with a chi-square null distribution that reject either if $T$ is small or if $T$ is large (see e.g. here), but, by my argument above, that is not common.
A: They are different tests of size $\alpha$, that have power for different alternatives. They define different rejection regions. They are not the same test, and one cannot be derived from the other.
One has power against alternatives that predict big test-statistics, $T$. The other has power against ones that predict small test-statistics. For sure the first one is more common.
