Minimax goodness-of-fit testing in multivariate nonparametric regression

Abstract

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 ∈ {pipe}f-f 0{pipe} ≥ r n 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. 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. © 2009 Allerton Press, Inc.