Functional Response Designs via the Analytic Permutation Test

Abstract: Vast literature on experimental design extends from Fisher and Snedecor to the modern day. When data lies beyond the assumption of univariate normality, nonparametric methods including rank based statistics and permutation tests are enlisted. The permutation test is a versatile exact nonparametric significance test that requires drastically fewer assumptions than similar parametric tests. The main downfall of the permutation test is high computational cost making this approach laborious for complex data and sophisticated experimental designs and completely infeasible in any application requiring speedy results such as high throughput streaming data. We rectify this problem through application of concentration inequalities and thus propose a computation free permutation test -- i.e. a permutation-less permutation test. This general framework is applied to multivariate, matrix-valued, and functional data. We improve these concentration bounds via a novel incomplete beta transform. We extend our theory from 2-sample to $k$-sample testing through the use of weakly dependent Rademacher chaoses and modified decoupling inequalities. We test this methodology on classic functional data sets including the Berkeley growth curves and the phoneme dataset. We further consider analysis of spoken vowel sound under two experimental designs: the Latin square and the randomized block design.