A Test for Symmetry Using the Sample Distribution Function

Let X₁, , Xₙ be n independent random variables with a common continuous distribution function (df) F. Let Fₙ denote the sample df of the X's. Let F be the class of all continuous df's and F₁ denote the df's in F which are symmetric about zero. To test the hypothesis F F₁ a common test is a weighted sign test of the form aₖ sgn Xₖ which has been studied by van Eeden and Bernard (1957), Hajek (1962), and Hajek and Sidak (1967). Usually the test included in nonparametric texts is for aₖ equal to the rank of |Xₖ| and is known as the Wilcoxon one-sample or signed rank test. This test is consistent against certain alternatives including the case when F is symmetric about some 0. That the test is not consistent against all alternatives in F - F₁ is evident from a discussion of its properties in Noether (1967). In this paper a test statistic for the hypothesis F F₁ is constructed in the spirit of the Kolmogorov-Smirnov statistics and shown to be consistent against all alternatives in F - F₁. Its df for both the finite and asymptotic cases are included along with the df's of two closely associated "one-sided" test statistics.