On the maximum of a measure of deviation from independence between discrete random variables. (English) Zbl 0557.62058

Gilula, Zvi; Schwarz, Gideon

Ann. Probab. 13, 314-317 (1985).

Summary: The squared \(n^ k\)-dimensional Euclidean distance \(f_ k\) between a given joint distribution of k random variables with values in 1,...,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 \(k\to \infty\), the distributions of X where \(M_ k\) is attained approach \(\{\) 0,1\(\}\), and \(M_ k\nearrow 1\).

MSC:

62H20	Measures of association (correlation, canonical correlation, etc.)
62H05	Characterization and structure theory for multivariate probability distributions; copulas

Keywords:

deviations from independence; upper bounds; measure of dependence