Cyclic Permutation Test: A Nonstandard Exact Test For Linear Models
4Lihua Lei (Stanford University)
https://simons.berkeley.edu/talks/cyclic-permutation-test-nonstandard-exact-test-linear-models
Statistics in the Big Data Era
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 applies 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 nonstandard cyclic permutation group under the null hypothesis. The power can be further enhanced by solving a secondary non-linear traveling 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 from 1908 to 2018, highlighting the novelty of our test. This is a joint work with Peter Bickel.