Testing for additivity and joint effects in multivariate nonparametric regression using Fourier and wavelet methods

We consider the problem of testing for additivity and joint effects in multivariate nonparametric regression when the data are modelled as observations of an unknown response function observed on a d-dimensional lattice and contaminated with additive Gaussian noise. We propose tests for additivity and joint effects, appropriate for both homogeneous and inhomogeneous response functions, using the particular structure of the data expanded in tensor product Fourier or wavelet bases studied recently by Amato & Antoniadis (2001) and Amato, Antoniadis & De Feis (2002). The corresponding tests are constructed by applying the adaptive Neyman truncation and wavelet thresholding procedures of Fan (1996), for testing a high-dimensional Gaussian mean, to the resulting empirical Fourier and wavelet coefficients. As a consequence, asymptotic normality of the proposed test statistics under the null hypothesis and lower bounds of the corresponding powers under a specific alternative are derived.