Testing symmetry in nonparametric regression models

In a recent paper Ahmad and Li (1997) proposed a new test for symmetry of the error distribution in linear regression models and proved asymptotic normality for the distribution of the corresponding test statistic under the null hypothesis and consistency under fixed alternatives. The present paper has three purposes. On the one hand we derive the asymptotic distribution of the statistic considered by Ahmad and Li (1997) under fixed alternatives and demonstrate that asymptotic normality is still valid but with a different rate of convergence. On the other hand we generalize Ahmad and Li's (1997) test of a symmetric error distribution to general nonparametric regression models. Moreover, it is also demonstrated that a bootstrap version of the new test for symmetry has good finite sample properties.