Open Access
February, 1985 On the Maximum of a Measure of Deviation from Independence Between Discrete Random Variables
Zvi Gilula, Gideon Schwarz
Ann. Probab. 13(1): 314-317 (February, 1985). DOI: 10.1214/aop/1176993085

Abstract

The squared $n^k$-dimensional Euclidean distance $f_k$ between a given joint distribution of $k$ random variables with values in $1, \cdots, n$ and the joint distribution of independent variables with the same respective marginals has been suggested as a measure of dependence. The following facts are established for $M_k$, the maximum of $f_k$ over all joint distributions for fixed $k$: (1) $M_k$ is attained among the distributions with all $k$ variables equal to a variable $X$ that takes on just two values. (2) For $k \leq 6, M_k = 1/2 - (1/2)^k$ is attained when the distribution of $X$ is $\{1/2, 1/2\}$. (3) For $k \geq 7, M_k$ is not attained at $\{1/2, 1/2\}$ and strictly exceeds $1/2 - (1/2)^k$. (4) For $k \rightarrow \infty$, the distributions of $X$ where $M_k$ is attained approach $\{0, 1\}$, and $M_k \nearrow 1$.

Citation

Download Citation

Zvi Gilula. Gideon Schwarz. "On the Maximum of a Measure of Deviation from Independence Between Discrete Random Variables." Ann. Probab. 13 (1) 314 - 317, February, 1985. https://doi.org/10.1214/aop/1176993085

Information

Published: February, 1985
First available in Project Euclid: 19 April 2007

MathSciNet: MR770647
zbMATH: 0557.62058
Digital Object Identifier: 10.1214/aop/1176993085

Subjects:
Primary: 62H20
Secondary: 62H05

Keywords: Deviations from independence , Upper bounds

Rights: Copyright © 1985 Institute of Mathematical Statistics

Vol.13 • No. 1 • February, 1985
Back to Top