The Annals of Mathematical Statistics

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

D. L. Burdick

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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 $k = 2,3, \ldots, N - 1$.

Article information

Source
Ann. Math. Statist., Volume 43, Number 2 (1972), 428-438.

Dates
First available in Project Euclid: 27 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoms/1177692623

Digital Object Identifier
doi:10.1214/aoms/1177692623

Mathematical Reviews number (MathSciNet)
MR309244

Zentralblatt MATH identifier
0238.62051

JSTOR
links.jstor.org

Citation

Burdick, D. L. Distribution Free Tests for Symmetry Based on the Number of Positive Sums. Ann. Math. Statist. 43 (1972), no. 2, 428--438. doi:10.1214/aoms/1177692623. https://projecteuclid.org/euclid.aoms/1177692623


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