Analytic Permutation Testing via Kahane–Khintchine Inequalities
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The permutation test is a versatile type of exact nonparametric significance test that requires drastically fewer assumptions than similar parametric tests, but achieves the same statistical power. The main downfall of the permutation test is the high computational cost of running such a test making such an approach laborious for complex data and experimental designs and completely infeasible in any application requiring speedy results. We rectify this problem through application of Kahane–Khintchine-type inequalities under a weak dependence condition and thus propose a computation free permutation test—i.e. a permutation-less permutation test. Our approach is computation-free and valid for finite samples. This general framework is applied to multivariate and functional data as well as the corresponding covariance matrices and operators resulting from theorems in commutative and non-commutative Banach spaces. Extending two-sample to k-sample testing extends the proof techniques from Rademacher sums to Rademacher chaoses. We test this methodology on classic functional data sets including the Berkeley growth curves and the phoneme dataset. We also consider hypothesis testing on speech samples, functional and operator data, under two experimental designs: the Latin square and the complete randomized block design.
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