An Assumption-Free Exact Test For Fixed-Design Linear Models With Exchangeable Errors
share
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 α for arbitrary fixed design matrix and arbitrary exchangeable errors, whenever 1 / α is an integer and n / p > 1 / α - 1. The test applies the marginal rank test on 1 / α linear statistics of the outcome vectors where the coefficient vectors are determined by solving a linear system such that the joint distribution of the linear statistics is invariant 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. We show that CPT has comparable power with existing tests through extensive simulation studies. 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 non-triviality of our test.
READ FULL TEXT