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September, 1987 On Measuring Internal Dependence in a Set of Random Variables
Robert A. Koyak
Ann. Statist. 15(3): 1215-1228 (September, 1987). DOI: 10.1214/aos/1176350501


To measure dependence in a set of random variables, a multivariate analog of maximal correlation is considered. This consists of transforming each of the variables so that the largest partial sums of the eigenvalues of the resulting correlation matrix is maximized. A "maximalized" measure of association obtained in this manner permits statements to be made about the strength of internal dependence exhibited by the random variables. It is shown, under a weak regularity condition, that optimizing transformations exist and that they satisfy a geometrically interpretable fixed point property. If the variables are jointly Gaussian, then the identity transformation is shown to be optimal, which extends Kolmogorov's result for canonical correlation to the principal components setting.


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Robert A. Koyak. "On Measuring Internal Dependence in a Set of Random Variables." Ann. Statist. 15 (3) 1215 - 1228, September, 1987.


Published: September, 1987
First available in Project Euclid: 12 April 2007

zbMATH: 0631.62069
MathSciNet: MR902254
Digital Object Identifier: 10.1214/aos/1176350501

Primary: 62H20

Rights: Copyright © 1987 Institute of Mathematical Statistics


Vol.15 • No. 3 • September, 1987
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