This paper proposes new smooth tests for symmetry about an unknown center in the spirit of Neyman (1937). We transform the data using the probability integral transformation with respect to the unknown center, so that the transformed data follow the standard uniform distribution under the null hypothesis of symmetry. Extending Neyman (1937)’s smooth test for uniformity, we show that only the odd-ordered orthogonal moments of the transformed data are required. We also systematically characterize the parameter estimation uncertainty associated with existing estimators of the center. Our smooth tests converge to the convenient chi-square distribution under the null hypothesis, and they possess nontrivial asymptotic power against local alternatives that converge to the null at the parametric rate. We then discuss the special case in which the center is known. We perform extensive numerical simulations to compare the finite sample performance of our test statistics when employing different estimators of the center. The empirical size and power of our tests are satisfactory, even for small sample sizes. Finally, to illustrate the practicality of the proposed tests, an empirical application is conducted to test the symmetry of infant birth weight distributions.
