  We propose the Cyclic Permutation Test (CPT) to test general linear
hypotheses for linear models. This test is non-randomized and valid in finite
samples with exact Type I error $\alpha$ for an arbitrary fixed design matrix
and arbitrary exchangeable errors, whenever $1 / \alpha$ is an integer and $n /
p \ge 1 / \alpha - 1$. The test involves applying the marginal rank test to $1
/ \alpha$ linear statistics of the outcome vector, where the coefficient
vectors are determined by solving a linear system such that the joint
distribution of the linear statistics is invariant with respect to a
non-standard cyclic permutation group under the null hypothesis.The power can
be further enhanced by solving a secondary non-linear travelling salesman
problem, for which the genetic algorithm can find a reasonably good solution.
Extensive simulation studies show that the CPT has comparable power to existing
tests. When testing for a single contrast of coefficients, an exact confidence
interval can be obtained by inverting the test. Furthermore, we provide a
selective yet extensive literature review of the century-long efforts on this
problem, highlighting the novelty of our test.
