Testing for Symmetry and Conditional symmetry Using Asymmetric Kernels

We derive nonparametric tests of symmetry using asymmetric kernels with either shrinkingor fixed bandwidths. We show how to extend the approach to examine conditional symmetry byderiving conditions under which our tests are applicable to residuals from semiparametric modelswith a (sufficiently smooth) nonparametric link function. As a by-product, we prove the consistencyof the asymmetric kernel estimator of the derivative of the density function. Simulationsshow that the asymptotic tests perform well even in very small samples, entailing better size andpower properties than some of the existing symmetry tests.[...]