Papers › An Assumption-Free Exact Test For Fixed-Design Linear Models With Exchangeable Errors
An Assumption-Free Exact Test For Fixed-Design Linear Models With Exchangeable Errors
Lihua Lei, Peter J. Bickel
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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 an arbitrary fixed design matrix and arbitrary exchangeable errors, whenever 1 / α is an integer and n / p ≥1 / α- 1. The test involves applying the marginal rank test to 1 / α 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.
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