Minimax optimal procedures for testing the structure of multidimensional functions

A novel method for detecting some structural characteristics of multidimensional functions is presented. We consider the multidimensional Gaussian white noise model with an anisotropic estimand. Using the relation between the Sobol decomposition and the geometry of multidimensional wavelet basis we can build test statistics for any of the Sobol functional components. We assess the minimax optimality of these test statistics and show that they are optimal in presence of anisotropy with respect to the newly determined minimax separation rates. An appropriate combination of these test statistics allows to test some general structural characteristics such as the atomic dimension or the presence of some variables. Numerical experiments show the potential of our method.
This is joint work with J.-M. Freyermuth, F. Autin, C. Pouet and J. Aston.

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