Discussion of “Random Rates in Anisotropic Regression” by Hoffmann and Lepski

We congratulate the authors for a stimulating paper (referred to as HL in the following). As the authors correctly stated, the number of variables does not affect the optimal rate of convergence in a regular parametric model, but it does affect the optimal rate of convergence in nonparametric models. To be more precise, the optimal rate of convergence in a nonparametric function estimation problem depends on the “effective” nonparametric dimension of the model. For example, the “effective” dimension of a nonparametric additive model is one, no matter how many variables are in the model. Therefore, variable selection in the additive model does not affect the optimal rate of convergence of the model. In light of this, we would like to consider the problem of adaptive estimation in the nonparametric functional ANOVA setup. It is readily seen that the variable selection problem considered in HL is a special case of the problem we consider. Following the example of HL, we will concentrate on the white noise model setting. This can be motivated by the results in Brown and Low (1996), and Nussbaum (1996).