Testing symmetry of a nonparametric bivariate regression function

Abstract

We propose a test for symmetry of a regression function with a bivariate predictor based on the L ₂ distance between the original function and its reflection. This distance is estimated by kernel methods and it is shown that under the null hypothesis as well as under the alternative the test statistic is asymptotically normally distributed. The finite sample properties of a bootstrap version of this test are investigated by means of a simulation study and a possible application in detecting asymmetries in grey-scale images is discussed.

Keywords:

• bootstrap
• local polynomial estimation
• nonparametric regression
• testing for symmetry

Acknowledgements

The authors would like to thank Stephanie Söhnel for her assistance with the simulations and Martina Stein, who typed parts of this manuscript with considerable technical expertise. This work has been supported in part by the Collaborative Research Center “Statistical modelling of nonlinear dynamic processes” (SFB 823) of the German Research Foundation (DFG).