Distribution Free Tests for Symmetry Based on the Number of Positive Sums

Abstract

Let $X_1, X_2, \ldots, X_N$ be independent identically distributed random variables with common continuous distribution function $F$. Designate by $\mathscr{J}$ a nonempty set of subsets of the integers $\{ 1, 2,\ldots, N\}$ and by $\mathscr{Y} = \mathscr{Y}(\mathscr{J})$ the mapping which assigns to each set $I \in \mathscr{J}, I = \{t_1, t_2, \ldots, t_k\}$ the partial sum $\sum{t_j \in I}X_{t_j}$. Define the random variable $N = N(\mathscr{J})$ as the number of positive sums in the range of $\mathscr{Y}. N(\mathscr{J})$ has been shown to be distribution free when $F$ is the distribution function of a symmetric random variable if $\mathscr{J} = \{1,2, \ldots, N\}$ or $\mathscr{J} = \text{power set of} \{1,2, \ldots, N\}$. Several other nontrivial examples of this phenomenon have been discovered--all by different methods. This paper presents a unified method that derives all previously known results, provides a constructive method for obtaining infinitely many essentially different sets $\mathscr{J}$ with this property, and finally provides a powerful necessary condition on any such set $\mathscr{J}$ that yields a complete characterization of those sets $\mathscr{J}$ for which $N(\mathscr{J})$ is distribution free and $\mathscr{J}$ contains all $k$ element subsets of $\{1,2, \ldots, N\}$ where $k = 2,3, \ldots, N - 1$.