A Test for Symmetry Using the Sample Distribution Function

Abstract

Let $X_1,\cdots, X_n$ be $n$ independent random variables with a common continuous distribution function (df) $F$. Let $F_n$ denote the sample df of the $X$'s. Let $\mathfrak{F}$ be the class of all continuous df's and $\mathfrak{F}_1$ denote the df's in $\mathfrak{F}$ which are symmetric about zero. To test the hypothesis $F \varepsilon \mathfrak{F}_1$ a common test is a weighted sign test of the form $\sum a_k \operatorname{sgn} X_k$ 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 $a_k$ equal to the rank of $|X_k|$ 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 $\mu \neq 0$. That the test is not consistent against all alternatives in $\mathfrak{F} - \mathfrak{F}_1$ is evident from a discussion of its properties in Noether (1967). In this paper a test statistic for the hypothesis $F \varepsilon \mathfrak{F}_1$ is constructed in the spirit of the Kolmogorov-Smirnov statistics and shown to be consistent against all alternatives in $\mathfrak{F} - \mathfrak{F}_1$. Its $\operatorname{df}$ for both the finite and asymptotic cases are included along with the $\operatorname{df}$'s of two closely associated "one-sided" test statistics.