Testing for symmetries in multivariate inverse problems

We propose a test for shape constraints which can be expressed by transformations of the coordinates of multivariate regression functions. The method is motivated by the constraint of symmetry with respect to some unknown hyperplane but can easily be generalized to other shape constraints of this type or other semi-parametric settings. In a first step, the unknown parameters are estimated and in a second step, this estimator is used in the L2-type test statistic for the shape constraint. We consider the asymptotic behavior of the estimated parameter and show that it converges with parametric rate if the shape constraint is true. Moreover, we derive the asymptotic distribution of the test statistic under the null hypothesis and furthermore propose a bootstrap test based on the residual bootstrap. In a simulation study, we investigate the finite sample performance of the estimator as well as the bootstrap test.