Optimal testing in functional analysis of variance models

Abstract

We consider the testing problem in a general functional analysis of variance model. We test the null hypotheses that the main effects and/or the interactions are zeros against the composite nonparametric alternative hypotheses that they are separated away from zero in L2-norm and also possess some smoothness properties. We adapt the minimax functional hypothesis testing procedures for testing a zero signal in a Gaussian \signal plus noise model to derive asymptotically (as the noise level goes to zero) minimax nonadaptive and adaptive functional hypothesis testing procedures for the main effects and/or the interactions based on the empirical wavelet coefficients of the data. Wavelet decompositions allow one to characterise different types of smoothness conditions assumed on the response function by means of its wavelet coefficients for a wide range of various function classes. In order to shade some light on the theoretical results obtained, we carried out a small simulation study to examine the finite sample performance of the proposed functional hypothesis testing procedures. We also apply these tests to two real-life data examples arising from endocrinology and from neuropsychology. Concluding remarks and hints for possible extensions of the proposed methodology are also given.