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Testing symmetry of a nonparametric bivariate regression function

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.