An AssumptionFree Exact Test For FixedDesign Linear Models With Exchangeable Errors
Abstract
We propose the Cyclic Permutation Test (CPT) to test general linear hypotheses for linear models. This test is nonrandomized 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 nonstandard cyclic permutation group under the null hypothesis.The power can be further enhanced by solving a secondary nonlinear 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 centurylong efforts on this problem, highlighting the novelty of our test.
 Publication:

arXiv eprints
 Pub Date:
 July 2019
 arXiv:
 arXiv:1907.06133
 Bibcode:
 2019arXiv190706133L
 Keywords:

 Statistics  Methodology
 EPrint:
 Accepted by Biometrika