Minimax Goodness-of-Fit Testing in Multivariate Nonparametric Regression
Abstract
We consider an unknown response function f defined on =[0,1]d, 1 d∞, taken at n random uniform design points and observed with Gaussian noise of known variance. Given a positive sequence rn 0 as n∞ and a known function f0 ∈ L2(), we propose, under general conditions, a unified framework for the goodness-of-fit testing problem for testing the null hypothesis H0: f=f0 against the alternative H1: f∈, \|f-f0\| rn, where is an ellipsoid in the Hilbert space L2() with respect to the tensor product Fourier basis and \|·\| is the norm in L2(). 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-Wozniakowski 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.