Obtaining similar null distributions in the normal linear model using computational methods

Hypothesis testing when the null hypothesis belongs to the univariate or multivariate normal linear model is discussed. More specifically it is shown how data can be replicated from the null distribution conditional on the sufficient statistics for the parameters of the null hypothesis at hand. This distribution will be called the similar null distribution of the data. It is shown how the similar null distribution of the data can be used to obtain level alpha tests for any test statistic that is of interest. The p-value that is obtained using the distribution of the test statistic conditional on the sufficient statistics is called the similar p-value.