Test of symmetry in nonparametric regression

The minimax properties of a test verifying a symmetry of an unknown regression function f from n independent observations are studied. The underlying design is assumed to be random and independent of the noise in observations. The function f belongs to a ball in a Holder space of regularity beta. The null hypothesis accepts that f is symmetric. We test this hypothesis versus the alternative that the L-2 distance from f to the set of symmetric functions exceeds rootr(n)/2. As shown, these hypotheses can be tested consistently when r(n)=O(n(-4beta/(4beta+1))).