Minimax Goodness-of-Fit Testing in Multivariate Nonparametric Regression

We consider an unknown response function $f$ defined on $\mathrm{\Delta }=[0,1{]}^d$, $1\le d\le \mathrm{\infty }$, taken at $n$ random uniform design points and observed with Gaussian noise of known variance. Given a positive sequence $r_n\to 0$ as $n\to \mathrm{\infty }$ and a known function $f_0\in L_2(\mathrm{\Delta })$, we propose, under general conditions, a unified framework for the goodness-of-fit testing problem for testing the null hypothesis $H_0:f=f_0$ against the alternative $H_1: f\in\CF, |f-f_0|\ge r_n$,where$\CF$isanellipsoidintheHilbertspace$ L_2(\Delta)$withrespecttothetensorproductFourierbasisand$|\cdot|$isthenormin$ L_2(\Delta)$. We obtain both rate and sharp asymptotics for the error probabilities in the minimax setup. The derived tests are inherently non-adaptive. Several illustrative examples are presented. In particular, we consider functions belonging to ellipsoids arising from the well-known multidimensional Sobolev and tensor product Sobolev norms as well as from the less-known Sloan-Wo$\acute{\mathrm{z}}$niakowski norm and a norm constructed from multivariable analytic functions on the complex strip. Some extensions of the suggested minimax goodness-of-fit testing methodology, covering the cases of general design schemes with a known product probability density function, unknown variance, other basis functions and adaptivity of the suggested tests, are also briefly discussed.