Suppose we have a test statistic designed for testing a null about two groups of samples.
If we apply permutation test to the groups of samples and the test statistic, will that change the null hypothesis?
What I heard of before is that permutation test is to test if the two groups of samples are coming from the same distribution, where "same" can be understood in various ways. So does it mean that after applying the permutation test, the original null will be changed?
An example, consider the case when the test statistic is to test if the locations of two groups of samples are the same.
the t-test tests if the mean of two normally distributed groups of samples have the same mean. Will applying permutation change its null, for example, from equal mean to equal distribution?
the Wilcoxon rank-sum test tests if the mean ranks of the two groups are the same. Will applying permutation change its null, for example, from equal mean rank to equal distribution?
What does permutation test bring to a location test?