- In statistical inference, it is important to know the distribution of some statistics under null hypothesis \((H_0)\), so that quantities like p-values can be derived.
- The null distribution is available theoretically in some cases.
- For example, assume \(X_i\sim \mathcal{N}(\mu,\sigma^2), i=1,\dots,n\). Under \(H_0: \mu=0\), we have \(\bar{X} \sim \mathcal{N}(0, \sigma^2/n)\). Then \(H_0\) can be tested by comparing \(\bar{X}\) with \(\mathcal{N}(0,\sigma^2/n)\).
- When null distribution cannot be obtained, it is useful to user permutation test to "create" a null distribution from data.