Permutation inference distribution for linear regression and related models

For linear regression and related models, a permutation inference distribution (PID) is introduced. Like the confidence distribution in the Bayesian/Fiducial/Frequentist inference framework, the PID allows the construction of both confidence intervals and p-values. For two-sample problems and pairwise comparisons in ANOVA models, a fast Fourier transformation method can be used to find the exact PID. In general, however, random permutations are required except for small samples where all permutations can be generated. Simulation studies and real data applications are used to evaluate inferences obtained from the PID. PID methods are close to standard parametric methods when the errors are iid and normal. For skewed and heavy tailed errors, PID methods are superior to bootstrap and standard parametric methods.