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 Hölder 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 $\sqrt{r_n/2}$. As shown, these hypotheses can be tested consistently when $r_n=O(n^{-4\beta/(4\beta+1)})$.