Conditional L(p)-quantiles and their application to the testing of symmetry in non-parametric regression

The idea of using regression quantiles to test symmetry in a linear regression model is generalized to the non-parametric regression setting. The properties of the L(p)-quantiles, defined through an asymmetric L(p)-loss function, are derived. The asymptotic normality of the kernel estimates of the conditional L(p)-quantiles in the non-parametric regression setting is obtained and their application to the testing of symmetry is discussed.