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The permutation inference distribution (PID) introduced in Wu and Vos ( 2019 ), The Journal of Nonparametric Statistics 31(3), 722-742) is extended to the joint PID for multiple regression. The joint PID is used for both hypothesis testing and the construction of confidence regions and its computational burden is similar to that of conducting a single hypothesis test. Asymptotic normality results show that PID confidence regions are asymptotically ellipsoidal and exact. In finite samples, the PID confidence regions can be used to check the fidelity of normal approximations. In some cases, normal theory-based confidence regions may be adjusted to approach nominal confidence errors. Simulation studies and real data applications are used to evaluate inferences obtained from the PID.

Permutation confidence region for multiple regression and fidelity to asymptotic approximation