Minimax goodness-of-fit testing in multivariate nonparametric regression

We consider an unknown response function f defined on Δ = 0, 1] ^d , 1 ≤ d ≤ ∞, taken at n random uniform design points and observed with Gaussian noise of known variance. Given a positive sequence r _n → 0 as n → ∞ and a known function f ₀ ∈ L ₂(Δ), we propose, under general conditions, a unified framework for goodness-of-fit testing the null hypothesis H ₀: f = f ₀ against the alternative H ₁: f ∈ $ \mathcal{F} $ \mathcal{F} , ∥f − f ₀∥ ≥ r _n , where $ \mathcal{F} $ \mathcal{F} is an ellipsoid in the Hilbert space L ₂(Δ) with respect to the tensor product Fourier basis and ∥ · ∥ is the norm in L ₂(Δ). 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-Woźniakowski norm and a norm constructed from multivariable analytic functions on the complex strip.