Testing symmetry of an unknown density function by kernel method

This investigation proposes consistent nonparametric tests for the hypothesis that a distribution is symmetric about a known median, taken without loss of generality to be zero. These tests are based on the L L2 no2 norm and the celebrated kernel method of density estimation. The test statistics when appropriately centered and scaled are asymptotically normal under the null hypothesis. Generalization of the procedure to the case of testing symmetry of residuals in the linear regression model is given offering a more advantageous procedure to that of Fan and Gencay (1995). The procedure is also shown to apply to the case of testing “bivariate symmetry”, cf. Hollander (1971).