Dear all,

I am currently reading about permutation/randomization tests and have some difficulties to understand why they are exact. More precisely, I consider two groups of independent random variables with means \(\mu_1\) and \(\mu_2\) and variances \(\sigma^2_1\) and \(\sigma^2_2\), which are assumed to be equal if \( \mu_1 = \mu_2\) holds. To test the one-sided hypothesis \(H_0: \mu_1 \leq \mu_2\) versus \(H_1: \mu_1 > \mu_2\), I apply a permutation test with the same test statistic as for a two sample t-test with unequal variances and unequal sample sizes.

Due to exchangeability, it was no big deal to prove and to understand that the level of significance of the permutation test is equal to \(\alpha\) for \( \mu_1 = \mu_2\). However, I don't understand why the level of significance is less than \(\alpha\) for \( \mu_1 \leq \mu_2\)!?

Can someone give me a hint?

I am currently reading about permutation/randomization tests and have some difficulties to understand why they are exact. More precisely, I consider two groups of independent random variables with means \(\mu_1\) and \(\mu_2\) and variances \(\sigma^2_1\) and \(\sigma^2_2\), which are assumed to be equal if \( \mu_1 = \mu_2\) holds. To test the one-sided hypothesis \(H_0: \mu_1 \leq \mu_2\) versus \(H_1: \mu_1 > \mu_2\), I apply a permutation test with the same test statistic as for a two sample t-test with unequal variances and unequal sample sizes.

Due to exchangeability, it was no big deal to prove and to understand that the level of significance of the permutation test is equal to \(\alpha\) for \( \mu_1 = \mu_2\). However, I don't understand why the level of significance is less than \(\alpha\) for \( \mu_1 \leq \mu_2\)!?

Can someone give me a hint?

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